A polynomial identity writes the same expression as a product and as a sum of powers of . Match the , , and constant terms to find the integer products. Rewrite each expression with those products, using Desmos to check valid integer values where helpful. An integer numerator alone does not make a quotient an integer.
Hints
- Hint 1
The constant term is the part without a variable. In the product, it comes from multiplying by . Match it to : what does that tell you about ?
- Hint 2
The discriminant of is . Replace and with the products from the factors. After combining terms, look for a perfect square.
- Hint 3
A quotient of two integers is an integer only when the denominator divides the numerator. For III, how could you check divisibility using integer factors that satisfy the constant-term equation?
Step-by-step
Match coefficients and test the quotient
Step 1Expand the product
The given identity writes the same quadratic in two forms. Multiply across the factors:
Combine the two terms:
- Step 2
Match the coefficients
A coefficient is the number multiplying a power of . Since both forms are the same quadratic, the matching powers must have matching coefficients:
- Step 3
Check statement I
The constants give . Divide by , which is nonzero:
The problem says is an integer, so I must be an integer.
- Step 4
Rewrite statement II as a square
The discriminant here is . Substitute , , and :
Expand the square:
Combine the middle terms:
Write it as one square:
The quantity inside the parentheses is an integer, so II must be an integer.
- Step 5
Disprove statement III with one valid example
Choose , , , and . These are nonzero integers, and , so they satisfy the constant-term condition. Type those four values, then . Desmos shows , which equals . This is III because and . III need not be an integer, so only I and II must be integers. Choice A.