# SAT Math Multiple Choice Question 660: Answer and Explanation

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**Question: 660**

**15.** Which of the following can represent the graph in the *xy*-plane of *y* = *a*(*x* - *b*)(*x* + *c*)^{2}, where *a*, *b*, and *c* are all positive constants?

- A.
- B.
- C.
- D.

**Correct Answer:** D

**Explanation:**

**D**

**Advanced Mathematics (analyzing polynomial graphs) HARD**

By the Zero Product Property (Chapter 9, Lesson 5), the graph of *y* = *a(x* - *b)(x* + *c*)^{2} has zeroes at *x* = *b* and a "double root" at *x* = -*c* (because this expression has two factors of *(x* + *c*)). Since *b* and *c* are both positive, this means that the graph must have one single positive root and a "double" negative root. That is, the graph passes through the *x*-axis at a positive value of *x* and "bounces" off of the *x*-axis at a negative value of *x*. Notice that this eliminates choices (B) and (C). We also know that *a*, the "leading coefficient" of the polynomial, is positive. If the leading coefficient of the polynomial is positive, the polynomial must eventually "shoot up" toward positive infinity; that is, it must go up as we move to the right. This rules out choice (A) and leaves only choice (D) as correct.