To factor the trinomial [tex]\( x^2 - x - 20 \)[/tex], we need to express it as the product of two binomials of the form [tex]\((x + a)(x + b)\)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants.
1. Identify the coefficients:
The trinomial is given as [tex]\( x^2 - x - 20 \)[/tex], so:
- The coefficient of [tex]\( x^2 \)[/tex] is [tex]\( 1 \)[/tex].
- The coefficient of [tex]\( x \)[/tex] is [tex]\( -1 \)[/tex].
- The constant term is [tex]\( -20 \)[/tex].
2. Find two numbers that multiply to the constant term and add up to the coefficient of [tex]\( x \)[/tex]:
We need two numbers [tex]\( a \)[/tex] and [tex]\( b \)[/tex] such that:
- [tex]\( a \cdot b = -20 \)[/tex]
- [tex]\( a + b = -1 \)[/tex]
By examining the factor pairs of [tex]\(-20\)[/tex]:
- [tex]\((-1, 20)\)[/tex]
- [tex]\((1, -20)\)[/tex]
- [tex]\((-2, 10)\)[/tex]
- [tex]\((2, -10)\)[/tex]
- [tex]\((-4, 5)\)[/tex]
- [tex]\((4, -5)\)[/tex]
The pair that adds up to [tex]\(-1\)[/tex] is [tex]\((4, -5)\)[/tex].
3. Write the binomials:
Since [tex]\( a = 4 \)[/tex] and [tex]\( b = -5 \)[/tex], we can write the trinomial as the product of two binomials:
[tex]\[
(x + 4)(x - 5)
\][/tex]
Therefore, the trinomial [tex]\( x^2 - x - 20 \)[/tex] factors to:
[tex]\[
(x - 5)(x + 4)
\][/tex]
So, the correct choice is:
A. [tex]\( x^2 - x - 20 = (x - 5)(x + 4) \)[/tex]
