A question asking what must be an integer in a polynomial written as a product is about divisibility. Match the constant terms: they come from multiplying the constant pieces of the factors, regardless of the middle coefficient. Then use the given integer conditions to find a quotient that must be an integer. Don’t assume the leading coefficient divides the constant term.
Hints
- Hint 1
The constant term is what remains when you set . In the product , which two pieces remain after the terms containing disappear?
- Hint 2
An integer stays an integer when you change its sign. After matching the constant terms, divide by a nonzero constant from one factor. What does the resulting quotient equal?
Step-by-step
Match the constant terms
Step 1Find the relationship between the constants
No Desmos needed. The constant terms give the needed relationship directly. Both forms represent the same expression. Set ; every term containing vanishes, so the constant terms must match:
- Step 2
Find the quotient that must be an integer
Since , cannot be . Divide both sides by :
The problem says is an integer, and its negative is also an integer. So must be an integer. Choice D.