Grade 8 Math Q1 Ep1: Factoring Polynomials

[Music] [Music] good day everyone and welcome to deaf ed tv i am teacher joshua and i will be your guide in sharpening your scales and enhancing your minds in order to face the challenges here in grade 8 mathematics ready your self-learning mojos your paper and your pen with you let us have a wonderful day of learning the first lesson that we will tackle is factoring polynomials specifically the greatest common monomial factor but before that i would like you to observe the following images can you give a description that is common to all of them they are all orange in color what else they're all good for your health how about this what is common among these four even if they are of different colors we know that all of these are crayons what if i ask you to think what is common with 2 4 and 6 they are all even numbers or divisible by 2. if we look at their factors two has one and two four has one two and four six has one two three and six observe that aside from one all of them has a factor of 2. how about on this expressions x 2x 3x and 4x they all have the same variable x like in english in science we can also find commonalities in algebra specifically the greatest common factor it is just what you did in the constants 2 4 and 6 and expressions x 2x 3x and 4x easy right in this lesson we will factor polynomials with a common monomial factor let us start with a review on how to get the greatest common factor and one way to find the greatest common factor of two or more numbers is prime factorization let us determine the greatest common factor of 8 and 12. first thing to do is write each number as a product of prime factors eight is equal to the product of two times two times two well twelve is equal to two times two times 3 do you see the same number in each set of factors next is we identify common prime factors of the given numbers we can show them by encircling the common factors by pairs last is to multiply the common factors so what is the greatest common factor of 8 and 12 the greatest common factor is 4 which is the product of the common factors 2 times 2. how about the greatest common factor of algebraic expressions where there are variables and exponents since a variable represents an unknown value we cannot use prime factorization to find their greatest common factor so how can we find the gcf of n cubed n raised to 5 n raised to 6 and n raised to 9. what do you remember about a variable n raised to an exponent it means that n raised to an exponent k means n is multiplied to itself k times based on the given exponent so n cubed is n times n times n same goes to the expression n raised to five it is equivalent to the product of n written five times also with the other expressions what did you observe how many n is common in the factors the greatest common factor is n cubed or n raised to 3 from n times n times n let us review the following terms according to the compact oxford english dictionary the word polynomial comes from the latin greek words poly meaning many and nomen which means name or term therefore polynomial means many terms a polynomial is an algebraic expression that shows a sum or difference of two or more terms containing whole number exponents on the variable a variable is a symbol which represents an unknown value a constant is a symbol or a number with fixed value in an algebraic term numbers form the numerical coefficient when symbols form the literal coefficient in this example the expression is 4 x squared minus 2. this is a polynomial since the variables have non-negative whole number exponents this polynomial has two terms four x squared and two the first term has a numerical coefficient four and a literal coefficient x squared an exponent refers to the number of times a number is multiplied by itself the last term is a constant term since there are no variables multiplied to the number now we will determine the greatest common factor of algebraic expressions with numerical and literal coefficients find the greatest common factor of four x cubed and eight x squared four x cubed can be expressed as a product of 2 times 2 times x times x times x and 8 s squared is equal to 2 times 2 times 2 times x times x next step is we identify common factors we can separate and align the numerical and literal coefficients to make this easy can you identify which is common among the prime factors how many two are common how many x are common we can encircle these pairs to help us determine the common factors the common factors are 2 2 x and x last is to get the product of these common factors the product is the greatest common factor of the given four x cubed and eight x squared we will multiply two times two times x times x the greatest common factor of four x cubed and eight x squared is four x squared [Music] now let us try this example find the greatest common factor of these expressions 15 y raised to the power of 6 and 9z the first step is to enumerate the prime factors what is the prime factorization of 15y raised to 6. the numerical coefficient can be factored as 3 times 5 while y raised to 6 means that y is multiplied to itself 6 times how about 9z 9z is equal to 3 times 3 times z can you identify all common factors note that 3 is the only common factor hence the greatest common factor of 15y raised to the power of 6 and 9z is 3. that is one method we can use to determine the greatest common factor of polynomials but there are some cases where it seems that the given pair of monomials do not have a common factor observe these monomials 4 a squared and 9 b squared let us start with the numerical coefficients four and nine four can be expressed as the product two times two and nine can be expressed as three times three do they have common factors in a glance they seem to have no common factors but remember that one is a factor of any number hence the greatest common factors of four and 9 is 1. next are there any common factors for the literal coefficients a squared and b squared since the expressions do not have common variables we can conclude that 1 is the greatest common factor of the given monomials therefore the greatest common monomial factor of 4a squared and 9b squared is one and if the gcf of the expressions is one we can say that they are relatively prime how about if we get the greatest common factor of three monomials what is the greatest common factor of six a squared b squared three a b squared and fifteen a cubed b squared first get their prime factors six a squared b squared is equal to two times three times a times a times b times b how about three a b squared three a b squared is just three times a times b times b note that we use two variables here so it can be helpful to separate them in our solution then we have 15 a cubed b squared this monomial can be written as 3 times 5 multiplied with a times a times a multiplied with b times b because of the indicated exponents now identify the common factors we will have these numbers and variables and lastly we write the product of these factors what would be our final answer the greatest common factor of the expressions 6 a squared b squared 3 a b squared and 15 a cubed b squared is the product 3 times a times b times b which is three a b squared [Music] we've exercised our eyes on finding common characteristics of things and we exercise our minds by reviewing prime factorization and the greatest common factor we may now proceed on how we can use these skills let us have an exercise in getting the greatest common factor of these expressions let us start with the greatest common factor of 12a and 18 abc the greatest common factor of 12a and 18 abc is 6a next what is the greatest common factor of 4x 12x squared and 8 the answer is 4. how about the greatest common factor of 6x 14y and 15z since all three expressions do not have a common prime factor we can say that they are relatively prime hence the greatest common factor of 6x 14y and 15z is 1. notice that in the previous examples prime factorization is used to find the greatest common factor of the given pair of monomials the next examples illustrate how to factor polynomials by getting the greatest common monomial factor let us write the expression 6x plus 3x squared in factored form we can factor this polynomial by getting the greatest common factor of each term and i will introduce a new method to acquire the common factors we can divide the solution into two parts one for identifying the greatest common factor of the polynomial and then finding the other factor with the help of the identified expression 6x plus 3x squared is a binomial 6x is the first term and 3 x squared is the second term the numerical coefficient of 6x in the first term is 6 and 3 in the second term observe the literal coefficients the simplest way to identify their greatest common factor is to get the common variables with the least exponent by prime factorization we will get a greatest common factor 3x to factor 6x plus 3x squared simply divide each term of the given polynomial by the common factor 6x divided by 3x and 3x squared divided by 3x by applying division and the quotient rule in the loss of exponent 6x divided by 3x is equal to 2 since x divided by x is just 1. then what is 3x squared divided by 3x 3 divided by 3 is 1 and x squared divided by x is just x do not forget the signs to complete our polynomial factor last step is to write the polynomial in factored form the polynomial 6x plus 3x squared can be written in factored form as 3x multiplied by the quantity 2 plus x remember that getting the greatest common monomial factor is different from writing a polynomial in its factored form factoring is often called the reverse process of multiplying polynomials where we write a polynomial as a product of two or more simpler polynomials now let us try this example right 12x cubed y raised to 5 minus 20 x raised to 5 y squared z in complete factored form again we can break it down into parts so we can focus on each step while factoring first find the greatest common monomial factor 12 is equal to 2 times 2 times 3 and 20 is equal to 2 times 2 times 5. then we get the common variables in each term which has the least exponent so what is the greatest common monomial factor it is 4 x cubed y squared then find the other factor by dividing each term of the polynomial by the greatest common monomial factor the operation can be shown as such we divide 12 by 4 and subtract exponents of expressions with similar bases the first term of the factor will be 3y cubed minus the second term what do you think will it be 20 divided by 4 is 5 then applying the loss of exponents we will have x squared and z since y squared divided by y squared is 1. step 5 write the polynomial in factored form in this example we have shown that 12 x cubed y raised to 5 minus 20 x raised to 5 y squared z is equivalent to 4 x cubed y squared times the quantity three y cubed minus five x squared z [Music] let us try the next example get the factors of two x plus three y squared what is the greatest common factor of two and three since these numerical coefficients and constant are relatively prime the greatest common factor is one how about the literal coefficients they do not have any common variables hence the greatest common factor is also one similar to our previous example in finding the greatest common monomial factor there will be instances that it is one we call these polynomials prime polynomials and there you have it we have discussed how to factor polynomials by getting the greatest common monomial factor let us have a recap one factoring is the reverse process of multiplication where we write a polynomial as a product of two or more simpler polynomials two the greatest common monomial factor of two or more expressions is the product of the gcf of the numerical and literal coefficients three prime polynomials are polynomials whose greatest common monomial factor is one and can only be written as a product of one and the polynomial itself number four to factor a polynomial using its greatest common monomial factor we can consider these steps a find the greatest common monomial factor of the numerical coefficients b find the variables with the least exponent that appears in each term of the polynomial it serves as the gcf of the literal coefficients c get the product of the greatest common factor of the numerical coefficient and the variables the product serves as the greatest common monomial factor of the given polynomial b find the other factor by dividing the given polynomial by its greatest common monomial factor and e write the final factored form of the polynomial since we are near the end of your lesson prepare your pens and your paper because it is important to evaluate what you have learned i will give you 10 seconds to answer each item [Music] write a polynomial factor in the blank to complete each statement number one seven p squared minus seven p is equal to seven p times the quantity of what polynomial the answer is p minus one number two 18 x y plus three y is equal to blank times the quantity of six x plus one the polynomial 18xy plus 3y can be written as the product 3y times the binomial 6x plus 1. number three 15 m cubed minus 15 m squared plus 20 m is equal to 5 m times what polynomial the polynomial factor in this item is three m squared minus three m plus four number four seventeen x raised to five minus fifty one x raised to four minus thirty four x in factored form is blank times the quantity of x raised to four minus three x cubed minus 2 [Music] the greatest common monomial factor is 17x number five what is the factor of the polynomial 35 x raised to 5 y squared plus 21 x raised to 4 y plus 14 x cubed y squared if the greatest common factor is equal to 7 x cubed y the answer is five x squared y plus three x plus two y how are you in the short seat work did you get a high score nevertheless i know you have done a great job remember that it is okay to commit mistakes at the start the important thing is you learn from these mistakes and apply that knowledge to become better in math and in life as an additional exercise in practice at home try answering the activity on the self learning module 1a at page 10 activity 1 break the grade i hope that you have learned a lot in our episode today note that there are many ways to factor polynomials and you must focus on the key concepts and process in factoring these polynomials with a little more practice i believe that you can ace any lesson in mathematics next topic will be factoring polynomials that we call special products remember math is not only about numbers and operations it is an exercise for our minds for us to be critical logical and responsible thinkers again this is teacher joshua reminding you to keep safe have a nice day and see you next time [Music] you

Channel: DepEd TV - Official

Views: 447,912

Keywords: DepEd TV, DepEdTV, Grade 8, Math

Id: UV6IfPAQM30

Length: 24min 30sec (1470 seconds)

Published: Sun Oct 04 2020