Question

the polynomial function f is given by f(x)=(x - 4)(3x - 1)^2. which of the following descriptions of f is true? a f is a polynomial of degree 2 with a leading coefficient of 3. b f is a polynomial of degree 2 with a leading coefficient of 9. c f is a polynomial of degree 3 with a leading coefficient of 3. d f is a polynomial of degree 3 with a leading coefficient of 9.

Explanation:

Step1: Expand the polynomial

First, expand \((3x - 1)^2\) using \((a - b)^2=a^{2}-2ab + b^{2}\), where \(a = 3x\) and \(b = 1\). So \((3x-1)^2=(3x)^{2}-2\times3x\times1 + 1^{2}=9x^{2}-6x + 1\). Then \(f(x)=(x - 4)(9x^{2}-6x + 1)=x(9x^{2}-6x + 1)-4(9x^{2}-6x + 1)=9x^{3}-6x^{2}+x-36x^{2}+24x - 4=9x^{3}-42x^{2}+25x - 4\).

Step2: Determine degree and leading - coefficient

The degree of a polynomial is the highest power of the variable. Here, the highest power of \(x\) is 3, so the degree is 3. The leading - coefficient is the coefficient of the term with the highest power of the variable. For the term \(9x^{3}\), the leading - coefficient is 9.

Answer:

D. \(f\) is a polynomial of degree 3 with a leading coefficient of 9