Question

Explain in paragraph form if x + 12 is a factor of the function f(x) = x2 + 6x − 72. The paragraph has to be 5 or more sentences. Which method did you use to solve? You can't just say yes or no. You have to explain the step by stçep solution.

Answer 1

Substituting x = -12 into f(x) = x^2 + 6x - 72 results in f(-12) = 0, confirming through the factor theorem that x + 12 is a factor. Factoring the quadratic also shows (x + 12) as a factor.

Step-by-step explanation:

To determine if x + 12 is a factor of the function f(x) = x2 + 6x − 72, we can use the factor theorem. The factor theorem states that for some polynomial f(x), if f(c) = 0 for some value c, then x - c is a factor of f(x). In our case, we substitute x = -12 into the function and simplify: f(-12) = (-12)2 + 6(-12) - 72 = 144 - 72 - 72. After performing the calculations, f(-12) = 0. Therefore, x + 12 is indeed a factor of f(x) because substituting -12 into the function results in zero.

Additionally, one can factor the quadratic equation by looking for two numbers that multiply to the constant term -72 and add up to the coefficient of the linear term, 6. These numbers are 12 and -6. Factoring the quadratic, we have f(x) = (x + 12)(x - 6), which clearly demonstrates that x + 12 is a factor of the given function.