PSAT Math Multiple-Choice Question 717: Answer and Explanation

Question: 717

If the following equation is true for every value of x and if a is a constant, what is the value of a?

(x + 4)(x² + ax + 2) = x³ + x² - 10x + 8

• A. -10
• B. -3
• C. -2
• D. 1

Correct Answer: B

Explanation:

(B) First, use FOIL for the left side of the equation:

x³ + ax² + 2x + 4x­² + 4ax + 8

Next, combine like terms:

x³ + (a + 4)x² + (2 + 4a)x + 8

We know that this has to equal the right side of the original equation:

x³ + (a + 4)x² + (2 + 4a)x + 8 = x³ + x² - 10x + 8

The coefficients of the like terms on both sides of the equation must equal one another. The coefficients on the x³-terms are already equal and the constants are equal. So we need to worry about only the x²-terms and the x-terms. Set the coefficients on the x²-terms equal to one another:

a + 4 = 1

Subtracting 4 from both sides reveals that a = -3, which is choice (B).

We also could have set the coefficients of the x-terms equal to each other to solve for a:

2 + 4a = -10

Subtracting 2 from both sides and then dividing by 4 gives a = -3 as well.