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Factor the function [tex]\( f(x) = x^2 + 11x + 30 \)[/tex] to convert it to intercept form.

[tex]\[ f(x) = \][/tex]
[tex]\[ \square \][/tex]


Sagot :

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]