Final answer:
After evaluating the functions by substituting x = -7, the function f(x) = x³ + 16x² + 71x + 49 is the one that equals zero, thus confirming that it has x + 7 as a factor.
Step-by-step explanation:
To determine which function has a factor of x + 7, we would substitute x = -7 into each function and look for the function that equals zero at x = -7. This is because if a function has a factor of x + 7, it will be zero at the root corresponding to that factor, which is x = -7.
Let's evaluate the options given:
- f(x) = x³ + 16x² + 71x + 56: substituting x = -7 yields (-7)³ + 16(-7)² + 71(-7) + 56 = -343 + 784 - 497 + 56 which is not equal to zero.
- f(x) = x³ + 16x² + 71x + 49: substituting x = -7 yields (-7)³ + 16(-7)² + 71(-7) + 49 = -343 + 784 - 497 + 49 which is equal to zero, so x + 7 is a factor.
- f(x) = x² + 2x² - 55x - 56: substituting x = -7 yields (-7)² + 2(-7)² - 55(-7) - 56 = 49 + 98 + 385 - 56 which is not equal to zero.
- f(x) = x² + 2x²: substituting x = -7 yields (-7)² + 2(-7)² = 49 + 98 which is not equal to zero.
- f(x) = x² + 2x² - 55x - 63: substituting x = -7 yields (-7)² + 2(-7)² - 55(-7) - 63 = 49 + 98 + 385 - 63 which is not equal to zero.
Therefore, the function that has a factor of x + 7 is b. f(x) = x³ + 16x² + 71x + 49.