# Factoring Polynomials

- Is the polynomial in descending order?
- Does the polynomial have a GCF that needs to be factored out using the Distributive Law?
- Do I recognize one of the three patterns of polynomials?
- Difference of Squares factors to conjugate binomials.

x2 - 25 = (x + 5)(x - 5) - Perfect Square Trinomial, all terms positive factors to Square of a Sum.

x2 + 6x + 9 = (x + 3)2 - Perfect Square Trinomial, middle term negative factors to Square of a Difference.

x2 - 8x + 16 = (x - 4)2

- Difference of Squares factors to conjugate binomials.
- When the lead coefficient = 1
- When the sign of the constant term is positive (+):

x2 + 5x + 6- find the factors of the constant term (+ 6) that add to the middle coefficient (+ 5).
- write the factors as the second term of two binomials with the variable as the first
term:

(x + 2)(x + 3) then - Check by multiplying.

- When the sign of the constant term is negative (-): x2 + 5x - 6

- find the factors of the constant term (- 6) that subtract to the middle coefficient (+ 5). The larger of the two factors of 6 will have the same sign as the 5!!
- write the factors as the second term of two binomials with the variable as the first
term:

(x - 1)(x + 6) then - Check by multiplying.

- When the sign of the constant term is positive (+):
- When the lead coefficient does NOT equal 1.
- Our goal is to write the quadratic as a four term polynomial so that we can use the
Distributive Law to factor by Grouping. The quadratic expression in our example is
in the form of

ax2 + bx + c

4x2 + 17x - 15

In our example, a = 4, b = 17, and c = - 15. - Multiply a times c: 4 * - 15 = - 60
- Find the factors of ac that combine to equal b:

+20 and - 3 multiply to - 60 and add to + 17 - Replace bx with using the factors as coefficients:

4x2 - 3x + 20x - 15 - Group the terms two by two with a plus sign in between:

(4x2 - 3x) + (20x - 15) - Use the Distributive Law to factor out the GCF from each binomial:

x(4x - 3) + 5(4x - 3) - Use the Distributive Law to factor out the GCF from binomial:

x(4x - 3) + 5(4x - 3) becomes

(4x - 3)(x + 5) - Check by multiplying.

- Our goal is to write the quadratic as a four term polynomial so that we can use the
Distributive Law to factor by Grouping. The quadratic expression in our example is
in the form of
- Another example:

Factor: 6x2 - 7x - 20

a = 6, c = - 20; ac = -120

8 * -15 = -120; 8 - 15 = -7

6x2 - 15x + 8x - 20

Group and

(6x2 - 15x) + (8x - 20)

factor with DL

3x(2x - 5) + 4(2x - 5)

factor with DL

3x + 4)(2x - 5)

Check by multiplying.