GED Math Study Guide 2021!

why do we classify numbers why do we give them names like integers irrational numbers or negative numbers for the same reason we classify anything we want to make sure that everyone has an understanding of what specific numbers are called and what they mean after all there's a difference between 25 and negative 32 and 4 to the sixth power in this mometrix video we'll provide an overview of numbers and their classifications numbers are our way of keeping order we count the amount of money we have we measure distance we use percentages to indicate a sale numbers are an integral part of our everyday existence whether they are whole numbers rational numbers or the first type of numbers we're going to look at real numbers a real number is any value of a continuous quantity that can represent distance on a number line essentially it's any number you can think of 50 is a real number one billion is a very large real number real numbers encompass three classifications of numbers which we'll talk about in a little bit whole numbers rational numbers and irrational numbers are all real numbers imaginary numbers are not real numbers they are complex numbers that are written as a real number multiplied by an imaginary unit i for instance the square root of negative 1 calculates as the imaginary number i and the square root of negative 25 is 5 i even though imaginary numbers aren't real numbers they do have value electricians use imaginary numbers when working with currents and voltage imaginary numbers are also used in complex calculus computations so just because these numbers are called imaginary doesn't mean they aren't useful whole numbers are numbers that we count with 1 2 3 4 and 5 are all whole numbers so are negative 17 in zero whole numbers do not have fractions or decimals all whole numbers are called integers integers can be positive or negative whole numbers all integers and whole numbers are part of a bigger group called rational numbers this group also includes fractions and decimals that means that three-fifths and 7.25 are rational numbers rational numbers can also be positive or negative rational numbers have opposites which are called irrational numbers these numbers can't be written as a simple fraction pi is the most famous irrational number we have a close approximation of how to calculate pi but it's just a close approximation pi is renowned for going on and on forever that's why it's an irrational number you can't easily write it as a fraction natural numbers are those that are positive integers although there is some debate as to whether natural numbers start at zero or one negative numbers are well exactly that they are the numbers below zero there are several other number classifications as well numbers are divided into even and odd numbers if you can divide a number by 2 that number is even so 24 36 and 74 are all even numbers because if you divide them by 2 you get 12 18 and 37. even numbers always end with 0 2 4 6 or 8. odd numbers can't be divided by 2 and leave a whole number any odd number divided by 2 will leave a fraction so 17 divided by 2 is 8.5 23 divided by 2 is 11.5 all odd numbers will end in 1 3 5 7 or 9. numerators and denominators form fractions which are compromised of two integers the number on top is the numerator the number on the bottom is the denominator the numerator the top number shows how many parts we have the denominator the bottom number shows how many parts make a whole let's say you have six apples and three of the apples get eaten the number of apples you have left over would be displayed as three-sixths you would then divide three the top number into six the bottom number to determine the percentage of remaining apples in this case the number is 50 so that's our look at numbers in their classifications from whole numbers to irrational numbers we need to know what to call numbers so we can know what they mean i hope that this overview was helpful to you hey guys today we're going to take a look at the mathematical operations addition subtraction multiplication and division these four operations serve as the fundamental building blocks for all math so it is crucial to have a solid understanding to build upon let's dive in we use addition and subtraction to solve many real world situations addition and subtraction are simply the mathematical terms used to describe combining and taking away when we add we are combining or increasing when we subtract we are taking away or decreasing as a reminder the symbol we use for addition is a plus sign the answer to an addition problem is called the sum the symbol we use for subtraction is a minus sign and the answer to a subtraction problem is called the difference essentially addition and subtraction are opposite operations one adds value and the other deducts value one strategy for visualizing these two operations is to use a number line we will use a number line to illustrate the following examples let's imagine a situation that involves the sale of popcorn for this scenario let's assume that you are trying to raise money by selling bags of popcorn and you start with 20 bags when your first customer arrives they wish to purchase four bags of popcorn this means that your remaining number of bags will decrease we can represent this situation with a simple equation that involves subtraction we started with 20 bags and we decreased by 4 or subtracted 4. our subtraction equation is written as 20 minus 4 equals 16. on a number line we can represent this deduction by starting at 20 and then moving backwards 4 units in the negative direction each jump backwards represents subtraction by 1. let's say you started with 20 bags of popcorn and ended up with six bags left at the end of the day you need to replenish your stock in order to keep up your sales so you make four more bags of popcorn how many bags of popcorn do you now have available to sell for this scenario since we are looking at an increase of bags we will use addition this situation can be described using the equation six plus four equals ten you had six bags initially and then combined that amount with four more bags now you have 10 bags in all on a number line addition is represented by jumps to the right in the positive direction each jump to the right represents the addition of one unit so in this example we would be starting at 6 and jumping 4 units to the right we can see that we land on 10. it is important to notice that when using addition the order of the values does not matter for example 10 plus 30 is the same as 30 plus 10. the placement or arrangement of the values has no effect on the outcome both arrangements would equal 40. however the same is not true for subtraction does 30 take away 10 mean the same thing as 10 take away 30 clearly not we can see that the order matters when dealing with a situation involving subtraction the technical term for this quality is known as the commutative property essentially the property is true for operations where the values can move around commute and the outcome of the expression or equation will not change the commutative property applies to addition but not to subtraction another operation that also shares the commutative property is multiplication let's discuss multiplication together with division as we did for addition and subtraction multiplication and division are similar to addition and subtraction in that they perform opposite functions the function of multiplication is to represent multiple groups of a certain value whereas division is designed to show the separating or subdividing of a value into smaller groups as a reminder the symbol we use for multiplication is a times sign the answer to a multiplication problem is called the product the symbol we use for division is a division sign and the answer to a division problem is called the quotient multiplication is essentially a convenient and time efficient way to show what's called repeated addition for example if you need to fill 30 bags of popcorn and each bag requires 60 kernels it could take hours to count up how many kernels you need in total by just using addition a faster and more efficient way to make this calculation would be to use repeated addition instead of counting each seed independently we would group them up and add the groups together the calculation would then become 30 groups of 60. this grouping for the purpose of repeated addition is the multiplication process at its core thirty groups of sixty is written as thirty times sixty which is one thousand eight hundred so one thousand eight hundred kernels are required to fill up thirty bags of popcorn both addition and multiplication are commutative because the order does not affect the answer 30 groups of 60 gets us the same result as 60 groups of 30. our last operation division can be considered multiplication's opposite when we use division we are essentially splitting up a large group into smaller subgroups for our popcorn example we can use division to answer the following question how many bags of popcorn can i make using 1 800 kernels if each bag requires 60 seeds this situation requires us to divide the large value 1 800 into groups of 60. each smaller subgroup will now represent a bag of popcorn one thousand eight hundred divided into groups of sixty is represented as one thousand eight hundred divided by sixty in this case the answer is thirty so thirty bags of popcorn can be made with our 1 800 kernels as you can see division is not commutative because the order of the values plays a crucial role in determining the answer 1 800 divided by 60 is not the same thing as 60 divided by one thousand eight hundred okay that's all for this review of mathematical operations thanks for watching and happy studying [Music] before we dive in let's review the basic parts of a fraction remember a fraction simply represents a part of a whole it has a numerator and a denominator which tells us what the part is and what the whole is let's look at the fraction 3 4 as an example we can see that the 3 is our numerator and the 4 is our denominator so the fraction 3 4 is really saying 3 parts out of 4 parts total it can also be helpful to visualize 3 as simply one-fourth plus one-fourth plus one-fourth it is very important to remember that a denominator of four does not represent the value of four a denominator of four represents the value of one that is divided up into four equal parts or fourths the type of fraction we're working with here represents a value less than one whole three-fourths is not quite one if we had four-fourths that would be equivalent to 1 but we only have 3 out of 4 parts we see and use fractions that are less than 1 all the time in our daily lives whether it's for things like recipes or keeping track of time recipes often call for amounts such as one half teaspoon of salt and we often keep track of time in terms of quarter hours like a quarter past three for three fifteen though we observe this type of fraction very frequently in our daily lives it is not the only type of fraction consider the following scenario you're ordering pizza for a big celebration there will be lots of hungry guests at this celebration so you order three pizzas each pizza is cut into six slices this means that each pizza has six equal parts and as a fraction six would be considered our whole or our denominator if your first guest eats two slices we would represent this as the fraction two-sixths two parts out of six parts total but what if that first guest was really hungry and grabbed seven slices again each pizza was cut into six equal slices so six remains as our whole or denominator but this time our part is seven in this scenario our numerator is larger than our denominator seven over six fractions with a numerator larger than their denominator are referred to as improper fractions essentially improper fractions equal a value that is more than one one whole pizza would be represented by six over six or six sixths seven over six represents seven sixths which is more than one pizza this could be visualized as one over six plus one over six plus one over six plus one over six plus one over six plus one over six plus one over six equals seven over six it can also be written in another form called a mixed number an improper fraction and a mixed number will represent the same amount but simply be written in a different form for example the improper fraction 7 over 6 could also be written as the mixed number 1 and 1 6. mixed numbers and improper fractions share the same amount but as a mixed number the parts are collected and consolidated into as many groups of one whole as possible for example four over four would be grouped together as one seven over seven would also be grouped together as one any value where the numerator is equivalent to the denominator would be expressed simply as one in our pizza example the guest took seven slices from a group of pizzas that were sliced into sixths we said this could be represented as the improper fraction seven over six or visualized as one over six plus one over six plus one over six plus one over six plus one over six plus one over six plus one over six as a mixed number we would group six of these six in order to form six over six or one whole by grouping six over six together we can see that one over six is left over we would write our mixed number as one and one-sixth let's try a few more examples let's write the following improper fractions as mixed numbers four thirds can be visualized as one over three plus one over three plus one over three plus one over three we know that three over three is equal to one so let's group three of these thirds together we are now left with one and one third as our mixed number three halves can be visualized as one-half plus one-half plus one-half we then know that two halves two over two makes one whole and we're left with one half left over so three halves as a mixed number is one and one half let's try one more example seven fourths seven fourths is the same as one fourth plus one-fourth plus one-fourth plus one-fourth plus one fourth plus one fourth plus one-fourth now we know that four-fourths are grouped as one whole so these four-fourths pulled over to equal one and we're left with one two three fourths seven fourths written as a mixed number is one and three-fourths this process will take place in reverse in order to convert between a mixed number to an improper fraction for example if we started with a mixed number one and three-fourths and we wanted to convert it to an equivalent improper fraction we would take a look at the whole number in this case it is one this whole number is really representing the denominator that's in the fraction in this case it's four so the one is equal to four over four when we combine these four fourths with the three-fourths we end up with seven fourths total or seven over four that's all there is to it now i know what you're thinking what does this phrase actually mean quite a bit actually because that saying provides the key to remembering an important math concept the order of operations the order of operations is one of the more critical mathematical concepts you'll learn because it dictates how we calculate problems it gives us a template so that everyone solves math problems the same way let's start off with a simple question what is an operation an operation is a mathematical action addition subtraction multiplication division and calculating the root are all examples of a mathematical operation let's take a look at this problem looks easy right well it wouldn't be so easy if we didn't understand the order in which the math operation occurs if we didn't have rules to determine what calculations we should make first we'd come up with different answers should you start by subtracting 4 minus 6 and then multiplying by 7 no the order of operations tells us how to solve a math problem and this brings us back to aunt sally operations have a specific order and this is what please excuse my dear aunt sally helps us to understand it's an acronym p e m d a s or pymdas that tells us in which order we should solve a mathematical problem so first is please which stands for parentheses so we solve everything inside of the parentheses first then e excuse which is for exponents we solve that after we solve everything in a parentheses multiplication which is the my and this happens from left to right and then division which is the dear which also happens left to right and then we have addition and subtraction which also happens from left to right and this is ant and sally okay so now that we know the order of operations let's apply it to our problem that we have here and solve so if let's kind of go down our list we don't have parentheses and we don't have exponents but we do have multiplication so we do that before we do any addition and subtraction so let's go ahead and multiply seven times four that gives us 28 and now we're subtracting six which gives us 22. now let's look at another problem without the operations you could calculate this as 7 plus 7 which is equal to 14 times 3 which is equal to 42. and this would be wrong remember you multiply before you add therefore the equation should look like this [Music] so when we do problems like this we can use parentheses to group together our numbers that are going to take place first so in this case it's 7 times 3 and when we do that we get 21 and we have plus 7 left over when we add those together we get 28 and that's our answer let's look at some more complex problems the order of operations dictates how to solve this problem remember you multiply exponents first here's the wrong way to solve the [Music] problem why is that wrong because you violated the order of operations you do not multiply first you perform an operation on the exponent first this is how it should be done [Music] see solving the equation in the right order provides the correct answer let's try out one more problem this one is a little bit more challenging but it perfectly illustrates the order of operations remember the order what do we do first the number inside the parentheses so 8 times 6 is equal to 48 and then we subtract 15 and that gives us 33. here's how the problem looks now [Music] so our next step is multiplication and division so let's perform all of our multiplication and division problems and then see what we have left [Music] now we finish with addition and subtraction so here's what we have and our answer is 37. there is an exception if an equation only has one expression you don't have to follow the order of operations here are some examples of single expressions 10 plus 10. well there are no other operations so you just know to go ahead and add them together and you get 20. same thing with subtraction multiplication or division all of those are single expressions before we dive in we will need to have a solid understanding of place value decimals and place value go hand in hand so it can be tough to make sense of one without having some background experience with the other take a look at this place value chart with this example of 3528.74 we can see that each movement to the right of the decimal point drops us down by a factor of 10. we move from tenths to hundredths to thousandths on the other hand as we move to the left of the decimal point we increase by a factor of 10 each time we move from tens to hundreds to thousands this base 10 number system is not the only number system in use today but it is very common and widely used around the world let's look at another example this time let's use the number 136.289 this number as a value is read as 136 and 289 thousandths let's break this value up into its different parts in the terms of place value in this example we look at each digit and then take note of where it is in relation to the decimal point to determine the total value of the number the digit 2 is located in the tenths position so it does not represent the value two but two tenths the digit one is located in the hundreds position so it does not represent a value of one but a value of one hundred when you break apart a decimal value into its different parts you are essentially thinking about the number in its expanded form you are looking at each digit and identifying how much this digit represents based on its location the location of each digit reveals its value or place value this is similar to the process of writing a number in expanded form this example would be one hundred plus thirty plus six plus two tenths plus eight hundredths plus nine thousandths in expanded form this can sometimes be helpful in order to see each digit as an independent value the place value as a system allows us to write and express numbers with extreme accuracy for example instead of simply rounding the amount of fuel needed for an important mission to the moon as approximately 950 gallons we could instead say with complete accuracy and confidence that the amount needed for a safe trip would be exactly 950.458 gallons another helpful way to visualize place value is by using money we know that one dollar can be represented by 10 dimes we also know that one dollar can be represented by 100 pennies this is similar to saying that one whole can be represented by ten tenths or by one hundred hundredths one dollar equals ten dimes one dime is one tenth of a dollar one dollar equals 100 pennies one penny is 1 100 of a dollar with this understanding of place value we are ready to dive into the topic of decimals operations with decimals is a topic that most of us use in our daily lives especially when dealing with money for example five dollars two dimes and five pennies is represented by the decimal number 5.25 if we remember our place values the five in the ones place represents five dollars the two in the tenths position represents twenty cents or two tenths and the five in the hundredths position represents five cents or five hundredths in the real world it is often convenient to work with fractions and mixed numbers other times it makes more sense to work with decimals if we want to talk about measuring amounts in a recipe it is generally helpful to use fractions on the other hand when we are dealing with something like temperatures it is often more efficient to use decimals using fractions mixed numbers or decimals does not change the amount it simply changes the form for example if a recipe calls for four-fifths cups of flour this is the same amount as the decimal point eight these values are equivalent all decimal values can be written as fractions mixed numbers included for example when writing the decimal value 5.25 as a mixed number we simply need to look at each digit and then note its location with the value 5.25 we would represent the first 5 in the ones position as simply 5. then we look at the values to the right of the decimal we see 0.25 this reaches out to the hundredths place so it is representing 25 hundredths or as a fraction 25 over 100 so 5.25 as a mixed number would be 5 and 25 hundredths we can simplify the fraction making it five and one-fourth as you can see using decimals is something we do on a regular basis and being able to convert between decimals and fractions can ensure that our number is as accurate as possible you may have heard a math teacher or two say math is a language if that is true then algebraic rules and notation should be considered the grammar and punctuation of the language of math some students who struggle with math are confused by how to apply the rules and interpret the notation in this video we will focus on the notation and interpretation of exponents [Music] this video will focus on the meaning of exponents that are natural numbers also referred to as counting numbers which are 1 2 3 things like that other types of exponents are interpreted differently and will be covered in other videos let's start by quickly reviewing some terminology an exponent is written as a superscript on a number or algebraic expression which is referred to as the base there are a few ways to verbalize a power this simple example can be read as five squared five to the second five raised to the second power or five raised to the power of 2. in any case the exponent should be interpreted as repeated multiplication whatever is defined as the base should be multiplied by itself however many times the exponent implies for example 5 squared equals five times five this can then easily be evaluated as twenty-five let's look at some more examples seven cubed this time the base is seven and the exponent is three this can be read as seven to the third or seven cubed raising a base of seven to the power of three means to multiply seven by itself three times seven times itself three times equals three hundred forty-three let's try another one but this one will look a little different negative 2 squared it is important to point out that parentheses are being used with this example to define the base negative 2 is being raised to the second power the interpretation is the same simply multiply negative two by itself twice multiplying two negative numbers using parentheses results in a positive value but if we were to take the parentheses away and instead say negative 2 squared then our answer would be negative 4. why because the 2 is squared before the effects of the negative can take place so we have 2 times 2 equals 4 and then the negative affects it let's try another one with negatives negative 5 cubed this is equal to negative 5 times negative 5 times negative 5 which gives you negative 125. powers of 10 are used frequently in math and science applications scientific notation uses powers of 10 to express very large or small values in an efficient and organized manner but we'll dive into that topic in another video exponents are also used to raise algebraic expressions to powers but the meaning is the same multiply whatever the base is by itself however many times that is indicated by the exponent here are a few examples x cubed is equal to x times x times x as you can see the notation x cubed is a more efficient way to write the expanded version of x times x times x 2x cubed is equal to 2 times x times x times x notice that the base here is x adding parentheses to an expression changes the meaning if you put in parentheses 2x cubed means 2x times 2x times 2x which is equal to 2 times 2 times 2 times x times x times x which when you simplify that gives you 8 x cubed let's try one more example before we go x plus 2 squared once again the parentheses here are used to define the base to be x plus two the exponent of two instructs you to multiply this base by itself twice so it equals x plus two times x plus two multiplying these two binomial expressions results in the quadratic expression x squared plus four x 4. so as long as you can identify the base the multiplication of that base by itself becomes pretty straightforward as mentioned the interpretation of notation and math is half the battle practicing the basics is key in order to gain confidence for more complicated math content hi everyone welcome to this mometrix video on factors factors in math refer to a number or numbers that produce a given number when multiplied for example when you multiply 6 times 7 together you get 42 well 6 and 7 are the factors that contributed to the outcome of 42 but those are not the only potential factors to getting 42. like i said factors refer to numbers that produce a given number when multiplied so any two numbers that i multiply together to get 42 are considered factors of 42. let's see what else we have since all of these numbers 1 2 3 6 7 14 21 and 42 can be multiplied together to get 42 they are all considered factors of 42. another way that we could look at this is through division let's say you are asked the question is 9 a factor of 45 well to find out you can divide 45 by 9 and determine whether or not the remainder is zero like this in other words is your answer a whole number if you find a number with a zero remainder then both nine and that number are factors of forty-five since we can see that after the division the remainder is zero this tells us that the divisor nine is a factor of the dividend forty-five so the answer to our question is 9 a factor of 45 is yes now let's take a look at another one is 8 a factor of 45 [Music] when we divide 45 by 8 the divisor we are left with is a remainder of 5. so 8 is not a factor of 45 this means that there are not eight equal parts to 45 so these are examples of what factors are but there are also types of factors common factors greatest common factors and prime factors these are all exactly what they sound like common factors are factors that two numbers share or have in common for example let's look at the factors of 35 and 40. for 35 we have 1 and 35 because 1 times 35 is equal to 35 we also have 5 and 7 because 5 times 7 is equal to 35 that's about it let's move over to 40. for 40 we have 1 and 40 because 40 times 1 is equal to 40. we also have 2 times 20 because when you multiply those you get 40. we also have 4 times 10 and 5 times 8 all of which end up equaling 40. our factors for 40 are 1 2 4 5 8 10 20 and 40. to determine the common factors between 35 and 40 we just need to look at each list of factors for each number and determine which ones are the same so in the case of 35 and 40 the only factors that they share between these two lists are one and five so one and five are the common factors the next type of factor is a greatest common factor this is the highest value common factor you look at the list of common factors like we have here and determine which number is largest in the case of 35 and 40 one and five are the only factors that they have in common and five has a higher value than one therefore five is the greatest common factor now the last type of factor is a prime factor a prime factor is any number that can only be divided by 1 and itself to produce a whole number for example 11 is a prime factor because it can only be divided by one and itself when you multiply itself by one you end up with 11. there's no other combination of factors that will result in 11. hi and welcome to this video about the algebra of polynomial and rational expressions in this video we will explore multiplying polynomials factoring polynomials dividing polynomials rational expressions and operations with rational expressions before we get heavily into polynomial algebra let's recap a few important terms a polynomial is a mathematical expression of one or more algebraic terms each of which consists of a constant multiplied by one or more variables raised to a non-negative integral power such as a plus b x plus c x squared the key pieces to remember are number one polynomials consist of one or more terms of the general form a x raised to the nth power that are added or subtracted the sample in the definition has three terms number two the coefficients of the terms can be any type of number and number three the powers of the terms can only be non-negative integers including zero let's look at some quick examples two x squared minus four x plus five is a polynomial expression but 2x raised to the 2.5 power minus 4x plus 5 is not a polynomial expression why because of the decimal exponent remember the exponents in a polynomial expression have to be integers point four x plus one is a polynomial expression but x raised to the negative third power plus one is not because of the negative exponent now that we've looked at some polynomial expressions let's look at the different types a monomial is a polynomial consisting of one term for example x and two are monomials a binomial is a polynomial consisting of two terms x plus one and x cubed plus four are both binomials a trinomial is a polynomial consisting of three terms 12 x raised to the fourth power plus three x cubed plus one is a trinomial a degree is the highest exponent in a polynomial 2 is a degree 0 polynomial x plus 2 is a degree 1 polynomial and x squared plus 3 is a degree 2 polynomial now let's move on to multiplying one of the traditional methods taught for multiplying polynomials is foil this method highlights the need for organization multiplying polynomials is an application of the distributive property the key idea is that every term of each polynomial needs to be multiplied by every term of every other polynomial one way to help visualize and organize polynomial multiplication is with an area model which looks like this let's multiply three x plus four by negative two x minus five so three x times negative two x gives you negative six x squared and then 4 times negative 2x gives you negative 8x then here you get 3x times negative 5 equals negative 15 x and then here you have 4 times negative 5 which gives you negative 20. so the product is negative six x squared minus eight x minus fifteen x minus twenty which then simplifies to negative 6 x squared minus 23 x minus 20. the area model demonstrates how to multiply every term of one polynomial by every term of the other and it is expandable for polynomials of any size let's try another example our terms on the left are seven b to the fifth and negative two b squared and our terms on the top are two a cubed negative four a squared a and negative twelve we need to simply multiply all the terms then add the resulting terms together [Music] none of these terms are like terms so they can't be combined so our answer is [Music] so this is our final answer as we'll see throughout this video working with polynomials is almost identical to working with simple numbers when we think of factors for example we may think of numbers that are multiplied to create another number for example 2 and 3 are factors of 6 20 and 5 are factors of 100 often we look for numbers prime factorization the prime factorization of 30 is 2 times 3 times 5 because all the factors are prime numbers think of factoring as undoing the area model we just explored often the terms of polynomials have common factors consider this polynomial each term has a common factor of 5 a squared [Music] so this can be rewritten as five a squared times four b plus two a plus three b cubed plus one this is the prime factorization of the polynomial it cannot be factored any further sometimes factoring happens in groups it appears as if this polynomial cannot be factored rearranging to group the a terms and the b terms we have this [Music] pulling out the common factors we have this [Music] now we can see the common factor of two x plus one so we have this two x plus one times a plus b squared we can always check our factoring work by multiplying the factors to be sure we end up with the original polynomial polynomial division is closely related to multiplication and factoring consider the number six its prime factors are two and three two times three equals six so six divided by two is three and six divided by three is two this polynomial [Music] has prime factors of three x plus four and negative two x minus five negative six x squared minus twenty three x minus twenty divided by three 3x plus 4 gives you negative 2x minus 5. and then if you do negative 6x squared minus 23x minus 20 divided by negative 2x minus 5 that gives you 3x plus 4. one polynomial can be divided by another as long as the degree of the dividend is greater than or equal to the degree of the divisor here's how it works first make sure the exponents of both polynomials are in descending order and put a 0 in for any missing in the sequence for example two x squared plus one would be two x squared plus zero x plus one [Music] divide the first terms here the question is how many times does 3x go into negative 6x squared then we're going to multiply 3x plus 4 by negative 2x then we'll subtract the product [Music] then we bring down the next term in the divisor and we repeat how many times does 3x go into negative 15x multiply and then subtract there's a remainder of zero so negative two x minus five and three x plus four are factors of negative six x squared minus 23x minus 20 as we already knew rational expressions are closely related to rational numbers the word rational comes from the word ratio just as rational numbers are defined as having the form integer over integer rational expressions are defined by the form polynomial over polynomial here are some examples of rational [Music] expressions [Music] when common factors occur in the numerator and denominator of a rational function they can be divided to make one this is often taught as cancelling out but it actually stems from the fact that for example two divided by two equals one we can remove the common factors from these rational expressions [Music] rational expressions behave like fractions to add or subtract them we need to get a common denominator for example [Music] since one expression has 3x as the denominator and the other has x squared a common denominator of 3x times x squared can be created [Music] remember to multiply each numerator by the same factor so that we are essentially multiplying each expression by one now multiply through and simplify [Music] to multiply them we multiply straight [Music] across [Music] and to divide them we multiply by the reciprocal [Music] and there you have it hi and welcome to this review of the pythagorean theorem we're going to go over how to use it properly and also go over some special triangles called pythagorean triples that could save you some time when taking tests so let's get started first things first what is the pythagorean theorem the pythagorean theorem is a squared plus b squared is equal to c squared now this is used to find the length of the side of a right triangle when we know the length of the other two sides the triangle has to be a right triangle which means that it has an angle that measures exactly 90 degrees like this one [Music] the theorem is very easy to remember and just as easy to use in the theorem a b and c are the lengths of the three sides of the triangle but which is which let's start by figuring out where to find a b and c in a triangle to start you can tell that you're dealing with a right triangle because you see this little square in one of the angles that's the symbol for a right angle or 90 degree angle and any triangle that has a right angle must be a right triangle so now we have to decide where to put the three side lengths the key is to start with c which is always on the side across from the right angle this is called the hypotenuse and it's always the longest side you may be asking yourself if c is always across from the right angle how do i tell which of the other two is a and which is b it's a good question and the answer is it doesn't matter either of these two sides which are called legs can be used as a and then just use the other one for b let's pick three centimeters for a and our b will be down here and we know that's four centimeters so this is what our theorem looks like when we have it filled in 3 squared plus 4 squared is equal to c squared now we can just evaluate 3 squared and 4 squared which means multiplying 3 times 3 and 4 times 4 to get the following [Music] so now we have 9 plus 16 is equal to c squared so what's that little addition tells us 25 is equal to c squared now here's where it gets a little tricky we know that c squared is equal to 25 but we want to know what c is not what c squared is so how can we get rid of that little two well we use the inverse or opposite operation of squaring something and that inverse operation is the square root since it's an equation whatever we do to one side of the equation we must do to the other so i'm going to take the square root of both sides the square root of c squared is c and the square root of 25 just happens to be 5. look at that it's always good to check our answer to see if it makes sense since we're finding c it should be longer than any of the other two sides and 5 is greater than both 3 and 4. also because of the triangle inequality theorem which is something we'll get to in a later video the hypotenuse must be less than the sum of the other two sides which means 5 has to be smaller than 3 plus 4 which it is so you might have noticed that the answer to this problem was a nice neat integer 5. this is actually kind of rare if we look at random triangles but it's not rare in a math problem you might see on a test it happens whenever a problem uses a pythagorean triple the triangle we just looked at is the most common kind of a 3 4 5 right triangle the legs measure 3 and 4 and the hypotenuse is 5. sometimes it will be disguised by multiplying all the numbers by 2 which means we would get a 6 8 and 10 or multiplied by 10 which means we would have a 30 40 and 50 lengths or maybe any other number you don't need to know this to solve a pythagorean theorem problem but it's a nice shortcut to save you some time or allow you to check your answer another way [Music] other pythagorean triples include 5 12 13 8 15 and 17 and 7 24 25 there are many more but these ones are the ones you will see most often of course there are right triangles that aren't pythagorean triples let's look at one just so you can see what the answer will look like [Music] once again we're solving for the hypotenuse the longest side which is opposite the 90 degree or right angle this is length c when we plug in 5 and 7 for a and b we get this 25 plus 49 is equal to c squared so let's add 25 and 49 to get 74 which is equal to c squared then we take the square root of each side and find out that the square root of 74 is equal to c now even though 74 isn't a perfect square this is the answer if you take the square root on a calculator you only get an approximation of the answer which is about [Music] 8.60232526 forever we can do this approximation to do our checks it's greater than either of the other two sides and it's less than the two sides added together but when writing the answer we should use the square root form thanks for watching and happy studying

Channel: Mometrix Test Preparation

Views: 4,571

Rating: 4.9389315 out of 5

Keywords:

Id: CzsaK9I4Bfg

Length: 59min 31sec (3571 seconds)

Published: Fri Mar 12 2021