# SAT Math Multiple Choice Question 764: Answer and Explanation

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**Question: 764**

**14.** Which of the following functions, when graphed in the *xy*-plane, will intersect the *x*-axis exactly 3 times?

- A.
*f*(*x*) = (*x*^{2}+ 1)(*x*^{2}+ 1) - B.
*f*(*x*) = (*x*^{2}- 1)(*x*^{2}+ 1) - C.
*f*(*x*) =*x*^{2}(*x*^{2}- 1) - D.
*f*(*x*) =*x*^{2}(*x*^{2}+ 1)

**Correct Answer:** C

**Explanation:**

**C**

**Advanced Mathematics (graphing polynomials) MEDIUM**

First, notice that the answer choices include the factors *x*^{2} - 1 and *x*^{2} + 1. The first of these is a difference of squares, so it can be factored further: *x*^{2} - 1 = *(x* + 1)(*x* - 1). However, *x*^{2} + 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*) = (*x*^{2} + 1) (*x*^{2} + 1) = (*x*^{2} + 1) (*x*^{2} + 1)

(B) *f*(*x*) = (*x*^{2} - 1) (*x*^{2} + 1) =(*x* + 1)(*x* - 1)(*x*^{2} + 1)

(C) *f*(*x*) = *x*^{2}(*x*^{2} - 1) = *x*^{2}(*x* + 1)(*x* - 1)

(D) *f*(*x*) = *x*^{2}(*x*^{2} + 1) = *x*^{2}(*x*^{2} + 1) = *x*^{2}(*x*^{2} + 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.