Answer :
Final answer:
The product of the functions f(x) and g(x) is 4x^4 - 24x^3 + 60x^2 - 80x - 70, and the value at x = -1 is 28.
Explanation:
To find the product of the functions f(x) = 2x^2 − 10x − 10 and g(x) = 2x^2 − 2x + 7, we need to multiply each term in f(x) by each term in g(x). This process is similar to the FOIL method used for multiplying binomials, but since we have trinomials, it involves more terms.
The product (f⋅g)(x) is found by expanding as follows:
- (2x^2 − 10x − 10)(2x^2 − 2x + 7)
- = 2x^2(2x^2) + 2x^2(− 2x) + 2x^2(7) − 10x(2x^2) − 10x(− 2x) − 10x(7) − 10(2x^2) − 10(− 2x) − 10(7)
- = 4x^4 − 4x^3 + 14x^2 − 20x^3 + 20x^2 + 70x − 20x^2 + 20x + 70
- = 4x^4 − 24x^3 + 60x^2 − 80x − 70
To find (f⋅g)(-1), we substitute -1 into the equation we just found:
- (f⋅g)(-1) = 4(-1)^4 − 24(-1)^3 + 60(-1)^2 − 80(-1) − 70
- = 4(1) + 24(1) + 60(1) + 80 − 70
- = 4 + 24 + 60 + 80 − 70
- = 98 − 70
- = 28
Hence, the correct product function is 4x^4 − 24x^3 + 60x^2 − 80x − 70 and the value of this function at x = -1 is 28.
