Question

Let f(2)=2x²+3 and g(x)=−x² +6x+2.Which statements below are true for f(x)×g(x)? Select all that apply.

A) f(x)×g(x) is a polynomial of degree 4.
B) f(x)×g(x) is a quadratic function.
C) f(x)×g(x) is a cubic function.
D) f(x)×g(x) is a polynomial of degree 6.

Answer 1

The product of the functions f(x) and g(x), both quadratic functions, will yield a polynomial of degree 4. Therefore, only statement A is true. Statements B, C, and D are false since f(x)×g(x) results in neither a quadratic, cubic, nor a polynomial of degree 6.

Step-by-step explanation:

The multiplication of the functions f(x) and g(x) results in a polynomial. To determine the degree of the resulting polynomial, one has to multiply the individual terms of function f(x) with those of g(x). Since f(x) is of degree 2 (with the leading term 2x²) and g(x) is also of degree 2 (with the leading term -x²), their product, f(x)×g(x), will be a polynomial of degree 4. This is because the highest-degree terms, when multiplied (2x² * -x²), produce a term of degree 4.

Therefore, statement A, “f(x)×g(x) is a polynomial of degree 4”, is true. Statement B, which says f(x)×g(x) is a quadratic function, is false since a quadratic function is a polynomial of degree 2. Statement C, which states f(x)×g(x) is a cubic function, is false since a cubic function has degree 3. Lastly, statement D, which says f(x)×g(x) is a polynomial of degree 6, is also false because degree 6 would result from multiplying polynomials of, for example, degrees 3 and 2 or degrees 4 and 2, which is not the case here.