PSAT Math Multiple-Choice Question 710: Answer and Explanation

Question: 710

Which of the following could be an equation for the function graphed in the xy-plane below?

• A. x(x - 2)(x + 1)
• B. x(x - 2)²(x + 1)
• C. x(x + 2)²(x - 1)
• D. x(x + 2)(x - 1)

Correct Answer: B

Explanation:

(B) This function turns 3 times, so it's a quartic function, meaning it must have 4 factors. Right away, we can eliminate choices (A) and (D) because these functions each have only 3 factors; they're cubics. To conceptualize, a quadratic function—a parabola—turns once. A cubic function turns twice.

The function has zeros at -1, 0, and 2. We can use this to solve for the factors.

If x = -1, x + 1 = 0. So (x + 1) is a factor. From this, you can eliminate choice (C), thus leaving choice (B) as the correct answer. However, if you want to see where the rest of the factors come from, read on.

If x = 0, x must be a factor. Why? Because if x = 0, the whole function equals zero; so x must be a factor.

If x = 2, x - 2 = 0. So (x - 2) must be a factor. This factor must actually be squared. In general, if (x - a)^m is a factor of a function, then the function crosses the x-axis at a if m is odd and does not cross the x-axis at a if m is even. In our case, if we look at the graph at x = 2, we can see that the graph never crosses the x-axis at 2; it stays above the x-axis right before and after 2. This means that (x - 2) must be raised to an even power. We know that our graph is quartic. Since we already have 2 other distinct factors, the only option is for the exponent to be 2. In other words, (x - 2)² is a factor. Notice that the graph crosses the x-axis at the other two zeros: -1 and 0. At -1, the graph goes from the positive side to the negative side of the x-axis. The graph goes from the negative side to the positive side of the x-axis at 0. So their corresponding factors must occur an odd number of times (in this case, they each occur once). This matches choice (B).