5.12: Solving Equations by Factoring
Zero Product Property: For all real numbers a and b:
ab = 0 if and only if a = 0 or b = 0. (Multiplicative Property of Zero)
(A product of factors is zero if and only if one or more of the factors is zero). (Converse)
-The zero-product property is true for any number of factors. You can use this property to solve certain equations.
Polynomial Equations: both sides of an equation are polynomials. Polynomial equations are usually named by the term
of highest degree.
If a ≠0:
𝑎𝑥 + 𝑏 = 0 is a linear equation.
𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 is a quadratic equation.
𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑 = 0 is a cubic equation.
-Many polynomial equations can be solved by factoring and then using the zero-product property. Often, the
first step is to transform the equation into standard form in which one side is zero. The other side should be a
simplified polynomial arranged in order of decreasing degree of the variable.
-If a factor occurs twice in the factored form of an equation, it is a double root or multiple root. It is only listed one time
in the solution set.
Ex: #3, p.232
𝟏𝟓𝒏(𝒏 + 𝟏𝟓) = 𝟎
1. Set each factor equal to zero and solve.
15𝑛(𝑛 + 15) = 0
(𝑛 + 15) = 0
15𝑛 = 0
𝑛=0
𝑛 = −15
The solution set is {0, −15}.
Ex: #15, p.232
𝒔𝟐 = 𝟒𝒔 + 𝟑𝟐
1. Transform the equation into standard form.
2. Factor the polynomial.
𝑠 2 − 4𝑠 − 32 = 0
(𝑠 2 − 8𝑠) + (4𝑠 − 32) = 0
𝑠(𝑠 − 8) + 4(𝑠 − 8) = 0
(𝑠 − 8) (𝑠 + 4) = 0
𝑠−8 = 0
𝑠+4 =0
𝑠=8
𝑠 = −4
The solution set is {8, −4}.
3. Set each factor equal to zero and solve.
Ex: #39, p.232
𝟗𝒙𝟑 + 𝟗𝒙 = 𝟑𝟎𝒙𝟐
1. Transform the equation into standard form.
2. Factor completely.
3. Set each factor equal to zero and solve.
9𝑥 3 − 30𝑥 2 + 9𝑥 = 0
3𝑥 ( 3𝑥 2 − 10𝑥 + 3) = 0
3𝑥 ( 3𝑥 2 − 9𝑥 − 𝑥 + 3) = 0
3𝑥 {(3𝑥 2 − 9𝑥 + (−𝑥 + 3)} = 0
3𝑥 {3𝑥(𝑥 − 3) − 1(𝑥 − 3)} = 0
3𝑥 (3𝑥 − 1) (𝑥 − 3) = 0
3𝑥 = 0
3𝑥 − 1 = 0
𝑥−3 = 0
𝑥=0
𝑥=
1
3
The solution set is {0,
𝑥=3
1
,
3
3}.
5.12: Solving Equations by Factoring
(𝒙 − 𝟐) (𝒙 + 𝟑) = 𝟔
Ex: #45, p.233
1. Since you have factors that are equal to something besides
zero, you must turn this into a polynomial (by foiling).
𝑥 2 + 3𝑥 − 2𝑥 − 6 = 6
2. Transform the equation into standard form.
𝑥 2 + 𝑥 − 12 = 0
{(𝑥 2 + 4𝑥) + (−3𝑥 − 12)} = 0
3. Factor the polynomial.
{𝑥(𝑥 + 4) − 3(𝑥 + 4)} = 0
(𝑥 + 4) (𝑥 − 3) = 0
4. Set each factor equal to zero and solve.
𝑥+4=0
𝑥−3=0
𝑥 = −4
𝑥=3
The solution set is {−4, 3}.