Let f(x)=2x2+3 and g(x)=−x2+6x+2 . Which statements below are true for f(x)⋅g(x) ? Select all that apply. The x has a coefficient of 1 The resulting polynomial has degree 4 The constant term is 6 The leading term is 2 x to the 4th power The x3 term has a coefficient of 12 The x2 term has a coefficient of −1
To find the expression for f(x)⋅g(x), we need to multiply the two polynomials:
f(x)⋅g(x) = (2x^2 + 3)(-x^2 + 6x + 2)
Expanding this expression, we have:
f(x)⋅g(x) = -2x^4 + 12x^3 + 4x^2 - 3x^2 + 18x + 6x - 6
Simplifying further, we get:
f(x)⋅g(x) = -2x^4 + 12x^3 + x^2 + 24x - 6
Now, let's examine the given statements:
1. The x has a coefficient of 1.
This statement is not true. The coefficient of x in f(x)⋅g(x) is 24, not 1.
2. The resulting polynomial has degree 4.
This statement is true. The highest power of x in f(x)⋅g(x) is x^4.
3. The constant term is 6.
This statement is true. The constant term in f(x)⋅g(x) is -6.
4. The leading term is 2x to the 4th power.
This statement is not true. The leading term in f(x)⋅g(x) is -2x^4.
5. The x^3 term has a coefficient of 12.
This statement is true. The coefficient of x^3 in f(x)⋅g(x) is 12.
6. The x^2 term has a coefficient of −1.
This statement is not true. The coefficient of x^2 in f(x)⋅g(x) is 1, not -1.
Therefore, the statements that are true for f(x)⋅g(x) are: "The resulting polynomial has degree 4," "The constant term is 6," and "The x^3 term has a coefficient of 12."