Question

select the correct answer.
what are the x- and y-intercepts of the function $g(x) = (x + 1)(x^2 - 10x + 24)$?

a. x-intercepts: -1, 4, 6; y-intercept: 24

b. x-intercepts: -6, -4, and 1; y-intercept: 24

c. x-intercepts: -4, -1, 6; y-intercept: -24

d. x-intercepts: -6, -1, 4; y-intercept: -24

Explanation:

Step1: Find x-intercepts

To find x-intercepts, set \( g(x) = 0 \). So, \((x + 1)(x^2 - 10x + 24)=0\). First, factor \(x^2 - 10x + 24\). We need two numbers that multiply to 24 and add to -10, which are -4 and -6. So, \(x^2 - 10x + 24=(x - 4)(x - 6)\). Now the equation is \((x + 1)(x - 4)(x - 6)=0\). Setting each factor to zero: \(x + 1 = 0\) gives \(x=-1\); \(x - 4 = 0\) gives \(x = 4\); \(x - 6 = 0\) gives \(x = 6\). So x-intercepts are -1, 4, 6.

Step2: Find y-intercept

To find y-intercept, set \(x = 0\). Then \(g(0)=(0 + 1)(0^2 - 10(0)+24)=(1)(24)=24\). So y-intercept is 24.

Answer:

A. x-intercepts: -1, 4, 6; y-intercept: 24