Let f(x)=2x^2+3 and g(x) = -x^2 + 6x+2. Which statements below are true for f(x) x g(x)? Select all that apply.
a) The x^2 term has a coefficient of -1
b) The x^3 term as a coefficient of 12
c) The constant term is 6
d) the leading term is 2x^4
e) The resulting polynomial has degree 4
f) The x has a coefficient of 1
The product of two polynomials is found by multiplying each term of the first polynomial by each term of the second polynomial.
f(x) = 2x^2 + 3
g(x) = -x^2 + 6x + 2
To find the product f(x) x g(x), we use the distributive property:
f(x) x g(x) = (2x^2 + 3)(-x^2 + 6x + 2)
= (2x^2)(-x^2) + (2x^2)(6x) + (2x^2)(2) + (3)(-x^2) + (3)(6x) + (3)(2)
= -2x^4 + 12x^3 + 4x^2 - 3x^2 + 18x + 6
= -2x^4 + 12x^3 + x^2 + 18x + 6
Now, we can analyze each statement:
a) The x^2 term has a coefficient of -1
False. The x^2 term in the product is 1, not -1.
b) The x^3 term has a coefficient of 12
True. The x^3 term in the product is 12x^3.
c) The constant term is 6
True. The constant term in the product is 6.
d) The leading term is 2x^4
False. The leading term in the product is -2x^4.
e) The resulting polynomial has degree 4
True. The highest power of x in the product is x^4, making the degree 4.
f) The x term has a coefficient of 1
False. The x term in the product has a coefficient of 18, not 1.
The correct statements are:
b) The x^3 term has a coefficient of 12
c) The constant term is 6
e) The resulting polynomial has degree 4