To factor the trinomial [tex]\( -21 + 4x + x^2 \)[/tex] into the form [tex]\((x - a)(x - b)\)[/tex], follow these steps:
1. Rearrange the given trinomial [tex]\( -21 + 4x + x^2 \)[/tex]:
[tex]\[
x^2 + 4x - 21
\][/tex]
2. Identify the factors of the constant term [tex]\(-21\)[/tex] that add up to the coefficient of the linear term [tex]\(4\)[/tex].
3. List factor pairs of [tex]\(-21\)[/tex]:
[tex]\[
(-1, 21), (1, -21), (-3, 7), (3, -7)
\][/tex]
4. Find the pair that adds to [tex]\(4\)[/tex]:
- Check pairs:
[tex]\[
-1 + 21 = 20
\][/tex]
[tex]\[
1 - 21 = -20
\][/tex]
[tex]\[
-3 + 7 = 4
\][/tex]
[tex]\[
3 - 7 = -4
\][/tex]
The pair [tex]\((-3, 7)\)[/tex] adds to [tex]\(4\)[/tex].
5. Use these numbers to factor the trinomial:
[tex]\[
x^2 + 4x - 21 = (x - 3)(x + 7)
\][/tex]
Therefore, the correct factors that complete the expression are [tex]\(-3\)[/tex] and [tex]\(7\)[/tex].
So the factored form of the given trinomial [tex]\( -21 + 4x + x^2 \)[/tex] is:
[tex]\[
(x - (-3))(x - 7)
\][/tex]
Simplifying this, you get:
[tex]\[
(x + 3)(x - 7)
\][/tex]
