Solving Equations
1.3 Factoring Trinomials
Why Factoring Matters
Factoring lets us reverse expansion. In this chapter, that is the move that turns one hard equation into two easy equations.
Definition: Trinomial
A trinomial is a polynomial with exactly three terms, such as
$$x^2+5x+6. $$
Example: Factor a monic trinomial
Factor completely:
$$x^2+5x+6.$$
Show Solution
Look for two numbers whose product is $6$ and whose sum is $5$.
Those numbers are $2$ and $3$, so
$$x^2+5x+6=(x+2)(x+3).$$
Final answer:
$$(x+2)(x+3). $$
MyOpenMath: Practice: factoring monic trinomials (placeholder)
Trinomials with Leading Coefficient Not 1
When the leading coefficient is not 1, the same idea works with an extra rewrite step.
Fact: AC Method
To factor $ax^2+bx+c$:
- Compute $ac$.
- Find two integers with product $ac$ and sum $b$.
- Split the middle term and factor by grouping.
Example: Factor by grouping using AC method
Factor completely:
$$2x^2-7x+3.$$
Show Solution
Here $a=2$, $b=-7$, $c=3$, so
$$ac=6.$$
We need two numbers that multiply to $6$ and add to $-7$: they are $-6$ and $-1$.
Split the middle term:
$$2x^2-6x-x+3.$$
Group:
$$2x(x-3)-1(x-3)=(2x-1)(x-3).$$
Final answer:
$$(2x-1)(x-3). $$
MyOpenMath: Practice: AC method factoring (placeholder)
Special Factoring Patterns
Some expressions factor immediately if you recognize the pattern.
Theorem: Difference of Squares
For real numbers $a$ and $b$,
$$a^2-b^2=(a+b)(a-b).$$
Example: Factor a difference of squares
Factor completely:
$$9x^2-16y^2.$$
Show Solution
Recognize squares:
$$9x^2=(3x)^2,\qquad 16y^2=(4y)^2.$$
Apply $a^2-b^2$ rule:
$$9x^2-16y^2=(3x+4y)(3x-4y).$$
Final answer:
$$(3x+4y)(3x-4y). $$
Theorem: Sum and Difference of Cubes
For real numbers $a$ and $b$,
$$a^3+b^3=(a+b)(a^2-ab+b^2),$$ $$a^3-b^3=(a-b)(a^2+ab+b^2).$$
Example: Factor a difference of cubes
Factor completely:
$$x^3-8.$$
Show Solution
Write $8=2^3$:
$$x^3-8=x^3-2^3.$$
Use $a^3-b^3$:
$$x^3-8=(x-2)(x^2+2x+4).$$
Final answer:
$$(x-2)(x^2+2x+4). $$
MyOpenMath: Practice: special factoring patterns (placeholder)
Completing the Square Preview
Completing the square rewrites a quadratic expression into a perfect-square form.
Definition: Completing the Square
To complete the square in $x^2+bx$, add and subtract
$$\left(\frac{b}{2}\right)^2.$$
Example: Complete the square
Rewrite
$$x^2+6x$$
as a perfect square plus/minus a constant.
Show Solution
Half of $6$ is $3$, and $3^2=9$.
Add and subtract $9$:
$$x^2+6x=x^2+6x+9-9=(x+3)^2-9.$$
Final form:
$$(x+3)^2-9. $$
