let f(x) 2x^2 +3 and g(x)=-x^2 +6 +2. Which statements below are true for f(x) *g(x)? select all that apply
1. The x^2 term has a coefficient of -1
2. The x^3 term has a coefficient of 12
3. The leading term is 2x^4
4. The constant term is 6
5. The resulting polynomial has degree of 4
6. The x has a coefficient of 1
To find the product of two functions, we multiply the corresponding terms of each function.
f(x) = 2x^2 + 3
g(x) = -x^2 + 6 + 2
f(x) * g(x) = (2x^2 + 3) * (-x^2 + 8)
To determine the properties of the resulting polynomial, we need to multiply the terms individually.
(2x^2) * (-x^2) = -2x^4 (this gives us the x^4 term)
(2x^2) * (8) = 16x^2 (this gives us the x^2 term)
(3) * (-x^2) = -3x^2 (this gives us another x^2 term)
(3) * (8) = 24 (this gives us the constant term)
Therefore, the resulting polynomial is:
f(x) * g(x) = -2x^4 - 3x^2 + 16x^2 + 24
Simplifying:
f(x) * g(x) = -2x^4 + 13x^2 + 24
Looking at the resulting polynomial, we can determine the following:
1. The x^2 term has a coefficient of -1: False
2. The x^3 term has a coefficient of 12: False (there is no x^3 term)
3. The leading term is 2x^4: False
4. The constant term is 6: False (the constant term is 24)
5. The resulting polynomial has a degree of 4: True
6. The x has a coefficient of 1: False
Therefore, the correct statements are:
5. The resulting polynomial has a degree of 4.