Re: Finding prime factors of binomial and trinomial?

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Posted by T.Gracken on September 25, 2002 at 13:33:41:

In Reply to: Finding prime factors of binomial and trinomial? posted by Mitch on September 24, 2002 at 22:09:56:

: I'm struggling with learning how to do these types of problems, can any one show me how to work these two problems step by step?

: 1) find prime factors of the trinomial
: 16x^2-40xy+25y^2

there are many ways to approach this. one way is to look at "forms"
that is: if a trinomial has the "form" A² - 2AB + B², then it factors to (A - B)²

breakdown of previous form:

first term is A². can you write the first term as "something times itself"? in this case, yes (4x) times itself, so A can be replaced with (4x)

third term is B². can you write the third term as "something times itself"? in this case, yes (5y) times itself, so B can be replaced with (5y)

middle term is 2AB. if you use the A and B from above, is the middle term equal to 2AB? in this case, yes. 2(4x)(5y) = 40xy.

so you have (what is often called a special product or) "A² - 2AB + B²" form. so replace A and B in (A - B)² to get (4x - 5y)²

and you have the prime polynomial factors (4x - 5y) and 4x - 5y). How do you know that they are prime polynomial factors? because they are first degree polynomials with no common factors.

: 2) find prime factors of the binomial
: 12x^2y^2-27

another "form" can be used here, but first the common factors must be extracted.

that is, 3 is a common factor of both terms, so using the distributive property, you can rewrite the expression as

3(4x²y² - 9).

now, notice inside the parenthesis, you have the "form" A² - B², which can be factored as (A - B)(A + B)

right??? A would be 2xy and B would be 3, so rewrite to get

3(2xy-3)(2xy+3)

and you have the three prime polynomial factors. since monomials are considered prime polynomials and first degree polynomials with no common factors are also prime polynomials we have found all the prime factors.

ask your instructor/teacher how to determine if a polynomial is prime when it is of degree two or greater.

: Thanks, very fustrated,
: Mitch

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: : I'm struggling with learning how to do these types of problems, can any one show me how to work these two problems step by step? : : 1) find prime factors of the trinomial : : 16x^2-40xy+25y^2 : : there are many ways to approach this. one way is to look at "forms" : that is: if a trinomial has the "form" A2 - 2AB + B2, then it factors to (A - B)2 : breakdown of previous form: : first term is A2. can you write the first term as "something times itself"? in this case, yes (4x) times itself, so A can be replaced with (4x) : third term is B2. can you write the third term as "something times itself"? in this case, yes (5y) times itself, so B can be replaced with (5y) : middle term is 2AB. if you use the A and B from above, is the middle term equal to 2AB? in this case, yes. 2(4x)(5y) = 40xy. : so you have (what is often called a special product or) "A2 - 2AB + B2" form. so replace A and B in (A - B)2 to get (4x - 5y)2 : and you have the prime polynomial factors (4x - 5y) and 4x - 5y). How do you know that they are prime polynomial factors? because they are first degree polynomials with no common factors. : : : 2) find prime factors of the binomial : : 12x^2y^2-27 : : another "form" can be used here, but first the common factors must be extracted. : that is, 3 is a common factor of both terms, so using the distributive property, you can rewrite the expression as : 3(4x2y2 - 9). : now, notice inside the parenthesis, you have the "form" A2 - B2, which can be factored as (A - B)(A + B) : right??? A would be 2xy and B would be 3, so rewrite to get : 3(2xy-3)(2xy+3) : and you have the three prime polynomial factors. since monomials are considered prime polynomials and first degree polynomials with no common factors are also prime polynomials we have found all the prime factors. : ask your instructor/teacher how to determine if a polynomial is prime when it is of degree two or greater. : : : : Thanks, very fustrated, : : Mitch

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