To factor the function [tex]\( f(x) = x^2 + 11x + 30 \)[/tex] and convert it to intercept form, follow these steps:
1. Identify the polynomial: We are given [tex]\( f(x) = x^2 + 11x + 30 \)[/tex].
2. Look for factor pairs of the constant term: In this polynomial, the constant term is 30. We need to find two numbers that multiply to 30 and add up to the coefficient of the [tex]\( x \)[/tex] term, which is 11.
3. Find the correct pair:
- The pairs of factors of 30 are:
[tex]\[
(1, 30), (2, 15), (3, 10), (5, 6)
\][/tex]
- We seek pairs that add up to 11. The pair (5, 6) satisfies this condition because [tex]\( 5 + 6 = 11 \)[/tex].
4. Rewrite the middle term using the factor pair:
[tex]\[
x^2 + 11x + 30 = x^2 + 5x + 6x + 30
\][/tex]
5. Group the terms:
[tex]\[
x^2 + 5x + 6x + 30 = (x^2 + 5x) + (6x + 30)
\][/tex]
6. Factor by grouping:
[tex]\[
(x^2 + 5x) + (6x + 30) = x(x + 5) + 6(x + 5)
\][/tex]
7. Factor out the common binomial:
[tex]\[
x(x + 5) + 6(x + 5) = (x + 5)(x + 6)
\][/tex]
So, the function [tex]\( f(x) \)[/tex] factored in intercept form is:
[tex]\[
f(x) = (x + 5)(x + 6)
\][/tex]
