Download R.5 Factoring Polynomials

R.5 Factoring Polynomials
In this section we learn techniques used to factor polynomial expressions. This is an essential algebraic skill used to solve equations. Factoring a polynomial is basically the reverse of multiplying polynomials. There are several techniques that we will learn. The first is the reverse of the distributive property.
Factoring out the greatest common factor (GCF)
Ex: AB + AC
4x2 ­ 6x3 + 2x
5w2y3­ 15w4y + 20w3y2
To check, make sure that the terms remaining in the parentheses have nothing in common, then distribute the GCF and check that you get the original polynomial.
In each of the examples above, the GCF was a monomial (one term), but it may not always be, as in the following examples:
x(x2 + 6) ­ 5(x2 + 6)
11(y + 7) + 3m(y + 7)
Aug 22­2:02 PM
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Factoring by grouping
This technique involves factoring the GCF out of the first pair of terms, factoring the GCF out of the second pair of terms, and then factoring out the common binomial.
Ex: ax + am + 5x + 5m
6x2 ­ 2x + 9x ­ 3
To check your factorization, multiply the two binomials together using FOIL. The terms may not be in the same order, but they should all be there with their original signs. Note that the order of the factors is not important.
Factor each polynomial by grouping.
2x2 + 6xw ­ xy ­ 3wy
x3 ­ x2 ­ 7x + 7
Aug 22­2:30 PM
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Factoring trinomials with a leading coefficient of 1
In this technique, we're factoring polynomials of the form x2 + bx + c. Our factorization will be two binomials, the first terms of which will be x. In order to find the second term of each binomial, we need to find two numbers whose product is the constant term above (c) and whose sum is the coefficient of x (b).
Ex: x2 + 7x + 10
We need to find two factors of 10 which have a sum of 7.
To check, we multiply our factors together using FOIL.
Factor each trinomial.
A polynomial that is not factorable with integers is called a prime polynomial.
Aug 22­2:39 PM
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Factoring trinomials with a leading coefficient which is not 1
In this technique, we're factoring polynomials of the form ax2 + bx + c. We will do this using the ac method, outlined below.
Factor each trinomial using the ac method.
2x2 + 11x + 5
12x2 ­ 25x + 12
3x2 ­ 7x ­ 6
2x2 + 5x ­ 25
Aug 22­2:46 PM
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Factoring special forms
There are four special forms of polynomials which are factored according to the following formulas.
1) Difference of two squares: a2 ­ b2 = (a ­ b)(a + b)
Ex: x2 ­ 25
9x2 ­ 49y2
Note: The sum of two
squares (a2 + b2) is prime.
2) Perfect square trinomial: a2 + 2ab + b2 = (a + b)2
a2 ­ 2ab + b2 = (a ­ b)2
Ex: x2 + 6x + 9
4x2 ­ 20xy + 25y2
3) Sum of Two Cubes: a3 + b3 = (a + b)(a2 ­ ab + b2)
Ex: x3 + 8
27x3 + y3
4) Difference of Two Cubes: a3 ­ b3 = (a ­ b)(a2 + ab + b2)
Ex: x3 ­ 27
8y3 ­ 125z3
Aug 22­2:56 PM
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Many polynomials will need to be factored using a combination of the techniques discussed in this section. A general factoring strategy is as follows:
1) Factor out the GCF, if there is one.
2) Determine whether the polynomial is a special form, and if it is, factor it accordingly.
3) If it is a trinomial with leading coefficient not 1, use the ac method to factor it.
4) If the polynomial has four terms, factor by grouping.
5) Make sure that each factor is factored completely.
Factor each polynomial completely.
5x2 + 10x ­ 75
2m3 ­ 200m
4mx2 + 5x2 ­ 36m ­ 45
x14 ­ x11
2m2 ­ 48m + 288
x3y4 + 27y4
Aug 25­11:54 AM
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