To factor the quadratic polynomial [tex]\(x^2 - 4x - 12\)[/tex] completely, we can follow these steps:
1. Identify the quadratic polynomial:
The given polynomial is [tex]\(x^2 - 4x - 12\)[/tex].
2. Look for factors of the constant term (-12) that will add up to the coefficient of the middle term (-4):
We need two numbers that multiply to [tex]\(-12\)[/tex] and add up to [tex]\(-4\)[/tex]. Those numbers are [tex]\(6\)[/tex] and [tex]\(-2\)[/tex]. Let's check:
- Factors of [tex]\(-12\)[/tex]: [tex]\((1, -12), (-1, 12), (2, -6), (-2, 6), (3, -4), (-3, 4)\)[/tex]
- The pair [tex]\((-6)\)[/tex] and [tex]\(+2\)[/tex] multiply to [tex]\(-12\)[/tex] and add up to [tex]\(-4\)[/tex].
3. Write the expression using these factors:
Rewrite the middle term [tex]\(-4x\)[/tex] using [tex]\(-6x\)[/tex] and [tex]\(2x\)[/tex]:
[tex]\[
x^2 - 4x - 12 = x^2 - 6x + 2x - 12
\][/tex]
4. Factor by grouping:
Group the terms to make factoring easier:
[tex]\[
x^2 - 6x + 2x - 12 = x(x - 6) + 2(x - 6)
\][/tex]
5. Factor out the common term:
Factor out the common binomial factor [tex]\((x - 6)\)[/tex] from both groups:
[tex]\[
x(x - 6) + 2(x - 6) = (x + 2)(x - 6)
\][/tex]
Thus, the quadratic polynomial [tex]\(x^2 - 4x - 12\)[/tex] factors completely into [tex]\((x - 6)(x + 2)\)[/tex].
So, the final answer is:
[tex]\[
(x - 6)(x + 2)
\][/tex]
