# Factoring Polynomials

1. Is the polynomial in descending order?
2. Does the polynomial have a GCF that needs to be factored out using the Distributive Law?
3. Do I recognize one of the three patterns of polynomials?
1. Difference of Squares factors to conjugate binomials.
x2 - 25 = (x + 5)(x - 5)
2. Perfect Square Trinomial, all terms positive factors to Square of a Sum.
x2 + 6x + 9 = (x + 3)2
3. Perfect Square Trinomial, middle term negative factors to Square of a Difference.
x2 - 8x + 16 = (x - 4)2
4. When the lead coefficient = 1
1. When the sign of the constant term is positive (+):
x2 + 5x + 6
1. find the factors of the constant term (+ 6) that add to the middle coefficient (+ 5).
2. write the factors as the second term of two binomials with the variable as the first term:
(x + 2)(x + 3) then
3. Check by multiplying.
2.  When the sign of the constant term is negative (-): x2 + 5x - 6
1. 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!!
2. write the factors as the second term of two binomials with the variable as the first term:
(x - 1)(x + 6) then
3. Check by multiplying.
5. When the lead coefficient does NOT equal 1.
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.
2. Multiply a times c: 4 * - 15 = - 60
3. Find the factors of ac that combine to equal b:
+20 and - 3 multiply to - 60 and add to + 17
4. Replace bx with using the factors as coefficients:
4x2 - 3x + 20x - 15
5. Group the terms two by two with a plus sign in between:
(4x2 - 3x) + (20x - 15)
6. Use the Distributive Law to factor out the GCF from each binomial:
x(4x - 3) + 5(4x - 3)
7. Use the Distributive Law to factor out the GCF from binomial:
x(4x - 3) + 5(4x - 3) becomes
(4x - 3)(x + 5)
8. Check by multiplying.
6. 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.