Sure, I'd be happy to help with factoring the polynomial [tex]\( x^2 - x - 20 \)[/tex]. Here's a detailed, step-by-step solution:
1. Identify the polynomial: We are given the polynomial [tex]\( x^2 - x - 20 \)[/tex].
2. Write the polynomial in standard form: The polynomial is already in standard form, [tex]\( x^2 - x - 20 \)[/tex].
3. Look for two numbers that multiply to give the constant term (-20) and add to give the coefficient of the linear term (-1).
- The constant term (c) is -20.
- The coefficient of the linear term (b) is -1.
4. Identify the pairs of factors of -20:
- [tex]\(1 \times -20\)[/tex]
- [tex]\(-1 \times 20\)[/tex]
- [tex]\(2 \times -10\)[/tex]
- [tex]\(-2 \times 10\)[/tex]
- [tex]\(4 \times -5\)[/tex]
- [tex]\(-4 \times 5\)[/tex]
5. Identify which pair of factors adds up to -1 ([tex]\(b\)[/tex]):
- Out of the pairs, [tex]\((4, -5)\)[/tex] and [tex]\((-4, 5)\)[/tex]:
- [tex]\(4 + (-5) = -1\)[/tex]
6. Rewrite the polynomial using the identified factors:
- The polynomial [tex]\( x^2 - x - 20 \)[/tex] can be rewritten as [tex]\( x^2 + 4x - 5x - 20 \)[/tex].
7. Group terms to factor by grouping:
- [tex]\( x^2 + 4x - 5x - 20 = (x^2 + 4x) + (-5x - 20) \)[/tex]
8. Factor out the greatest common factor (GCF) from each group:
- From the first group [tex]\((x^2 + 4x)\)[/tex], factor out [tex]\(x\)[/tex]: [tex]\( x(x + 4) \)[/tex].
- From the second group [tex]\((-5x - 20)\)[/tex], factor out [tex]\(-5\)[/tex]: [tex]\(-5(x + 4) \)[/tex].
9. Factor out the common binomial factor [tex]\((x + 4)\)[/tex]:
- [tex]\( x(x + 4) - 5(x + 4) = (x + 4)(x - 5) \)[/tex].
Therefore, the factored form of the polynomial [tex]\( x^2 - x - 20 \)[/tex] is [tex]\((x - 5)(x + 4) \)[/tex].
This completes the factoring process.
