SAT Math Multiple Choice Question 764: Answer and Explanation
14. Which of the following functions, when graphed in the xy-plane, will intersect the x-axis exactly 3 times?
- A. f(x) = (x2 + 1)(x2 + 1)
- B. f(x) = (x2 - 1)(x2 + 1)
- C. f(x) = x2(x2 - 1)
- D. f(x) = x2(x2 + 1)
Correct Answer: C
Advanced Mathematics (graphing polynomials) MEDIUM
First, notice that the answer choices include the factors x2 - 1 and x2 + 1. The first of these is a difference of squares, so it can be factored further: x2 - 1 = (x + 1)(x - 1). However, x2 + 1 is a sum of squares, which cannot be factored over the real numbers. This enables us to express each function in completely factored form:
(A) f(x) = (x2 + 1) (x2 + 1) = (x2 + 1) (x2 + 1)
(B) f(x) = (x2 - 1) (x2 + 1) =(x + 1)(x - 1)(x2 + 1)
(C) f(x) = x2(x2 - 1) = x2(x + 1)(x - 1)
(D) f(x) = x2(x2 + 1) = x2(x2 + 1) = x2(x2 + 1)
Now we can find all of the x-intercepts by setting each factor to 0 and (if possible) solving for x. Notice that if we do this for the factored form of each function, we see that (A) has no x-intercepts, (B) has intercepts at x = -1 and x = 1, (C) has intercepts at x = 0, x = -1, and x = 1, and (D) has an intercept at x = 0. Therefore, the function in choice (C) is the only one that has three x-intercepts.