To factor the quadratic polynomial [tex]\( x^2 + 2x - 24 \)[/tex], follow these steps:
1. Identify the quadratic polynomial:
[tex]\[
x^2 + 2x - 24
\][/tex]
2. Determine the factors of the quadratic polynomial:
We need to factor [tex]\( x^2 + 2x - 24 \)[/tex] into two binomials of the form [tex]\( (x + a)(x + b) \)[/tex] such that:
[tex]\[
x^2 + 2x - 24 = (x + a)(x + b)
\][/tex]
3. Find [tex]\( a \)[/tex] and [tex]\( b \)[/tex] such that:
- Their product is equal to the constant term [tex]\(-24\)[/tex] (i.e., [tex]\( a \times b = -24 \)[/tex]).
- Their sum is equal to the coefficient of the [tex]\( x \)[/tex] term, which is [tex]\( 2 \)[/tex] (i.e., [tex]\( a + b = 2 \)[/tex]).
4. Determine suitable values for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
By examining the factors of [tex]\(-24\)[/tex], we look for a pair of numbers whose product is [tex]\(-24\)[/tex] and whose sum is [tex]\( 2 \)[/tex]. We find that [tex]\( 6 \)[/tex] and [tex]\(-4\)[/tex] satisfy these conditions:
[tex]\[
6 \times (-4) = -24 \quad \text{and} \quad 6 + (-4) = 2
\][/tex]
5. Write the polynomial as a product of binomials:
The factors [tex]\( 6 \)[/tex] and [tex]\(-4\)[/tex] allow us to express the quadratic polynomial as:
[tex]\[
x^2 + 2x - 24 = (x - 4)(x + 6)
\][/tex]
Thus, the factored form of the polynomial [tex]\( x^2 + 2x - 24 \)[/tex] is:
[tex]\[
(x - 4)(x + 6)
\][/tex]
