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?
a. Difference of Squares factors to conjugate binomials.
x2 - 25 = (x + 5)(x - 5)
b. Perfect Square Trinomial, all terms positive factors to Square of a Sum.
x2 + 6x + 9 = (x + 3)2
c. Perfect Square Trinomial, middle term negative factors to Square of a Difference.
x2 - 8x + 16 = (x - 4)2
4. When the lead coefficient = 1
a. 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.
b. 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.
a. 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.
b. Multiply a times c: 4 * - 15 = - 60
c. Find the factors of ac that combine to equal b:
+20 and - 3 multiply to - 60 and add to + 17
d. Replace bx with using the factors as coefficients:
4x2 - 3x + 20x - 15
e. Group the terms two by two with a plus sign in between:
(4x2 - 3x) + (20x - 15)
f. Use the Distributive Law to factor out the GCF from each binomial:
x(4x - 3) + 5(4x - 3)
g. Use the Distributive Law to factor out the GCF from binomial:
x(4x - 3) + 5(4x - 3) becomes
(4x - 3)(x + 5)
h. Check by multiplying.
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.
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