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8−2
Factoring: Factoring Using the Distributive Property
Main Idea:
Factor polynomials by using the Distributive Property.
Solve quadratic equations in the form ax2 + bx = 0.
Notes:
1. Factoring a polynomial using the Distributive Property.
a. Find the greatest common factor of all the monomial terms.
b. Rewrite each term as the product of the greatest common factor and the
term’s remaining factors.
c. Use the Distributive Property to factor out the greatest common factor.
d. Example: 8bc2 + 24bc
i.
GCF: 8bc2 = 2 • 2 • 2 • b • c • c, and 24bc = 2 • 2 • 2 • 3 • b • c, so the GCF
is 2 • 2 • 2 • b • c, or 8bc.
ii. GCF times remaining factors: 8bc(c) + 8bc(3).
iii. Use Distributive Property: 8bc(c + 3).
2. Factoring by grouping: using the Distributive Property to factor polynomials of four
or more terms.
a. A polynomial can be factored by grouping if all of the following are present:
i. Four or more terms;
ii. Terms with common factors can be grouped together; and
iii. The common factors are identical or are additive inverses of each other.
b. ax + bx + ay + by = x(a + b) + y(a + b) = (a + b)(x + y).
c. Example: 12y3 + 9y2 + 8y + 6
i.
Group terms with common factors: (12y3 + 8y) + (9y2 + 6).
ii. Factor GCF using Distributive Property: 4y(3y2 + 2) + 3(3y2 + 2).
iii. Use Distributive Property a second time to factor a common binomial
factor: (4y + 3)(3y2 + 2).
d. A polynomial will sometimes factor to include binomials that are additive
inverses (e.g., (x − 2) and (2 − x); rewriting the additive inverse as the original
term times −1 may allow factoring by grouping.
3. Zero Product Property.
a. If the product of two factors is 0, then at least one of the factors must be 0.
b. For any real numbers a and b, if a • b = 0, then either a = 0, b = 0, or
both a and b = 0.
c. The Zero Product Property can be used to solve equations of the form ab = 0
or equations that can be written in this form by factoring.
i.
Set each factor equal to 0 and solve the resulting equations.
ii. Example: (x + 3)(x − 5) = 0.
(1) Set each factor equal to 0: x + 3 = 0 and x − 5 = 0.
(2) Solve: x = −3 and x = 5 are the solutions, or roots, or the equation.
d. Be careful not to divide out factors in equations that should be solved using
the Zero Product Property.
i.
x2 = 7x.
(1) Do not divide both sides by x to simplify, since x could be zero and
division by zero is not defined; dividing both sides by x would also
eliminate a possible root that would be found using the Zero Product
Property.)
(2) Restate the equation as x2 − 7x = 0, then factor as x(x − 7) = 0.
(3) The equation can now be solved by setting each factor equal to zero
and solving the resulting equations (x = 0; x = 7).
ii. (x − 2)(x + 4) = 0.
(1) Do not divide both sides by either factored binomial since that would
eliminate a possible root.
(2) Set both factors equal to zero, then solve the two resulting equations
(x = 2; x = −4)