# SAT Math Multiple Choice Question 644: Answer and Explanation

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**Question: 644**

**14.** If the polynomial *P*(*x*) has factors of 12, (*x* - 5), and (*x* + 4), which of the following must also be a factor of *P*(*x*)?

- A. 2
*x*^{2}+ 8 - B. 4
*x*^{2}- 20 - C. 6
*x*^{2}- 6*x*- 120 - D.
*x*^{2}- 10*x*+ 25

**Correct Answer:** C

**Explanation:**

**C**

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

The simplest polynomial with factors of 12, (*x* - 5), and (*x* + 4) is *P*(*x*) = 12(*x* - 5)(*x* + 4). The completely factored form (including the prime factorization of the coefficient) of this polynomial is *P*(*x*) = (2)^{2} (3)(*x* - 5)(*x* + 4).

Now, using the methods we discussed in Chapter 9, Lesson 4, we can look at the factored form of each choice:

(A) 2*x*^{2} + 8 = 2(*x*^{2} + 8) (*x*^{2} + 8 is not factorable over the

reals, but it does equal

(B)

(C) 6*x*^{2} - *6x* - 120 = 6(*x*^{2} - *x* - 20) = (2)(3)(*x* - 5)(*x* + 4)

(D) *x*^{2} - 10*x* + 25 = (*x* - 5)(*x* - 5)

Notice that every polynomial in (A), (B), and (D) contains at least one factor that is NOT in the factored form of *P*(*x*). (In (D), the factor (*x* - 5) appears twice, but it appears only once in *P*(*x*).) Only choice (C) contains ONLY factors that appear in *P*(*x*), so it is the only choice that must be a factor of *P*(*x*).