PSAT Math Multiple-Choice Question 663: Answer and Explanation

Question: 663

If -5m⁵ + 3m³ = 2m⁷, what is the sum of all ­possible values of m²?

• A. -2.5
• B. 0
• C. 0.5
• D. 2.5

Correct Answer: C

Explanation:

(C) We can solve for all possible values of m² by subtracting 2m⁷ from both sides, factoring the left side, and setting the left side equal to 0:

-5m⁵ + 3m³ - 2m⁷ = 0

First, factor out -m³:

-m³(5m² - 3 + 2m⁴) = 0

Rearrange the polynomial inside the parentheses so that the terms are decreasing in degree for easier factoring:

-m³(2m⁴ + 5m² - 3) = 0.

Next factor the inside:

-m³(2m⁴ + 5m² - 3) = -m³(m² + 3)(2m² -1) = 0

Now set each factor equal to 0 to solve for possible values of m²:

-m³ = 0

Dividing both sides by -m tells you that m² = 0, so this is one possible value.

m² + 3 = 0

Subtracting 3 from both sides gives m² = -3. However, you can't square a number and get a negative, so this solution is extraneous.

2m² - 1 = 0

Add 1 to both sides and divide by 2:

This is another possible value. Therefore, the two possible values of m² are 0 and 0.5. Thus, their sum is 0.5, which is choice (C).