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 $(3x - 1)^2$

Using the formula $(a - b)^2=a^{2}-2ab + b^{2}$, where $a = 3x$ and $b = 1$, we have $(3x - 1)^2=(3x)^{2}-2\times3x\times1+1^{2}=9x^{2}-6x + 1$.

Step2: Multiply by $(x - 4)$

$f(x)=(x - 4)(9x^{2}-6x + 1)=x(9x^{2}-6x + 1)-4(9x^{2}-6x + 1)=9x^{3}-6x^{2}+x-36x^{2}+24x - 4=9x^{3}-42x^{2}+25x - 4$.

Step3: 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, which is 9.

Answer:

D. $f$ is a polynomial of degree 3 with a leading coefficient of 9.