Let f(x)=2x^2+3 and g(x)=−x^2+6x+2 . Which statements below are true for f(x)⋅g(x) ? Select all that apply. A) The x^3 term has a coefficient of 12 The x cubed term has a coefficient of 12 The resulting polynomial has degree 4
B)The resulting polynomial has degree 4
C)The x has a coefficient of 1 The x has a coefficient of 1
D)The x^2 term has a coefficient of −1 The x squared term has a coefficient of negative 1
E)The constant term is 6
F)The constant term is 6 The leading term is 2x^4
The function f(x)⋅g(x) is found by multiplying the two polynomials together:
f(x)⋅g(x) = (2x^2 + 3)(-x^2 + 6x + 2)
To determine the degree of the resulting polynomial, we need to look at the highest power of x.
The highest power of x will be obtained when we multiply the term with the highest power from f(x) (x^2) by the term with the highest power from g(x) (-x^2).
Therefore, the resulting polynomial will have a degree of (2 + 2) = 4.
So statement B) is true: The resulting polynomial has degree 4.
We can also determine the coefficients of the x^3 and x^2 terms by multiplying the corresponding terms from f(x) and g(x) together.
The coefficient of the x^3 term will be (2)(-1) = -2.
The coefficient of the x^2 term will be (2)(6) + (3)(-1) = 9.
Therefore, neither statement A) nor D) is true.
The constant term is found by multiplying the constant terms from f(x) and g(x) together.
The constant term will be (3)(2) = 6.
Therefore, statement E) is true: The constant term is 6.
The leading term is determined by multiplying the leading terms from f(x) and g(x) together.
The leading term will be (2x^2)(-x^2) = -2x^4.
Therefore, statement F) is false: The leading term is -2x^4.
In summary, the correct statements are:
B) The resulting polynomial has degree 4.
E) The constant term is 6.