Answer :
To factor the expression [tex]\(x^2 - 5x - 36\)[/tex], we need to identify two numbers that multiply to the constant term, -36, and add to the linear coefficient, -5.
1. Start by listing the pairs of factors that multiply to -36:
- [tex]\(1 \times -36\)[/tex]
- [tex]\(-1 \times 36\)[/tex]
- [tex]\(2 \times -18\)[/tex]
- [tex]\(-2 \times 18\)[/tex]
- [tex]\(3 \times -12\)[/tex]
- [tex]\(-3 \times 12\)[/tex]
- [tex]\(4 \times -9\)[/tex]
- [tex]\(-4 \times 9\)[/tex]
- [tex]\(6 \times -6\)[/tex]
2. We need to find the pair that also adds up to -5. Let's check the pairs:
- [tex]\(1 + (-36) = -35\)[/tex]
- [tex]\(-1 + 36 = 35\)[/tex]
- [tex]\(2 + (-18) = -16\)[/tex]
- [tex]\(-2 + 18 = 16\)[/tex]
- [tex]\(3 + (-12) = -9\)[/tex]
- [tex]\(-3 + 12 = 9\)[/tex]
- [tex]\(4 + (-9) = -5\)[/tex] (This pair works!)
Thus, the pair [tex]\(4\)[/tex] and [tex]\(-9\)[/tex] multiply to -36 and add to -5.
3. Write the expression [tex]\(x^2 - 5x - 36\)[/tex] using these numbers:
[tex]\[
x^2 - 5x - 36 = (x - 9)(x + 4)
\][/tex]
So, the factors of [tex]\(x^2 - 5x - 36\)[/tex] are [tex]\(x - 9\)[/tex] and [tex]\(x + 4\)[/tex]. Therefore, the expressions that are factors are [tex]\(x-9\)[/tex] and [tex]\(x+4\)[/tex].
1. Start by listing the pairs of factors that multiply to -36:
- [tex]\(1 \times -36\)[/tex]
- [tex]\(-1 \times 36\)[/tex]
- [tex]\(2 \times -18\)[/tex]
- [tex]\(-2 \times 18\)[/tex]
- [tex]\(3 \times -12\)[/tex]
- [tex]\(-3 \times 12\)[/tex]
- [tex]\(4 \times -9\)[/tex]
- [tex]\(-4 \times 9\)[/tex]
- [tex]\(6 \times -6\)[/tex]
2. We need to find the pair that also adds up to -5. Let's check the pairs:
- [tex]\(1 + (-36) = -35\)[/tex]
- [tex]\(-1 + 36 = 35\)[/tex]
- [tex]\(2 + (-18) = -16\)[/tex]
- [tex]\(-2 + 18 = 16\)[/tex]
- [tex]\(3 + (-12) = -9\)[/tex]
- [tex]\(-3 + 12 = 9\)[/tex]
- [tex]\(4 + (-9) = -5\)[/tex] (This pair works!)
Thus, the pair [tex]\(4\)[/tex] and [tex]\(-9\)[/tex] multiply to -36 and add to -5.
3. Write the expression [tex]\(x^2 - 5x - 36\)[/tex] using these numbers:
[tex]\[
x^2 - 5x - 36 = (x - 9)(x + 4)
\][/tex]
So, the factors of [tex]\(x^2 - 5x - 36\)[/tex] are [tex]\(x - 9\)[/tex] and [tex]\(x + 4\)[/tex]. Therefore, the expressions that are factors are [tex]\(x-9\)[/tex] and [tex]\(x+4\)[/tex].
