The product of two binomials, u(x) and v(x), is a binomial of degree 8 with a positive leading coefficient. What must be true about the sum of u(x) and v(x)?

Question

Here are the choices:

A. It is a binomial of degree 4.

B. It is a binomial of degree 8.

C. It is a monomial of degree 4.

D. It is a monomial of degree 8.

Answer 1

For a product of binomials to also be a binomial, the product must be a difference of squares, and therefore u(x) and v(x) are each binomials of degree 4, of the form (ax⁴-b), (ax⁴+b). The b terms cancel out, and what's left is a monomial of degree 4. The answer is C.

Answer 2

It can be either a binomial or monomial; my guess is that the author of the problem intended for it to be a monomial.

Pair 1: x^4 + 1; x^4 - 1; the product is a binomial, x^8 - 1, and the sum is a monomial, 2x^4

Pair 2: 2 x^4 + 2; x^4 - 1; the product is a binomial, 2x^8 - 2, and the sum is a binomial, 3x^4 + 1