Chapter 9, Problem 15STP

a.

To determine

To calculate: the degree of the function

Answer to Problem 15STP

The degree of the function is $4$

Explanation of Solution

Given information: the polynomial function is $f\left(x\right)=3x^4+19x^3+7x^2-11x-2$

The degree of the function is the highest degree that a variable has

So, the degree of the polynomial $f\left(x\right)=3x^4+19x^3+7x^2-11x-2$ is $4$

b.

To determine

To calculate: the leading coefficient of the function

Answer to Problem 15STP

The leading coefficient of the function is $3$

Explanation of Solution

Given information: the polynomial function is $f\left(x\right)=3x^4+19x^3+7x^2-11x-2$

The leading coefficient of a function is the number that the highest degree variable is multiplied

So, the leading coefficient of the function is $3$

c.

To determine

To calculate : the value of $f\left(1\right),f\left(-2\right),f\left(2a\right)$

Answer to Problem 15STP

The value of $f\left(1\right)$ is $16$ , $f\left(-2\right)$ is $-60$ and $f\left(2a\right)$ is $44a^4+152a^3+28a^2-22a-2$

Explanation of Solution

Given information: the polynomial function is $f\left(x\right)=3x^4+19x^3+7x^2-11x-2$

Calculation:

To plug $1$ in for each $x$

  $\begin{array}{l}f\left(x\right)=3x^4+19x^3+7x^2-11x-2\\ f\left(1\right)=3{\left(1\right)}^4+19{\left(1\right)}^3+7{\left(1\right)}^2-11\left(1\right)-2\\ f\left(1\right)=3\cdot 1+19\cdot 1+7\cdot 1-11\cdot 1-2\\ f\left(1\right)=3+19+7-11-2\\ f\left(1\right)=16\end{array}$

To plug $-2$ in for each $x$

  $\begin{array}{l}f\left(x\right)=3x^4+19x^3+7x^2-11x-2\\ f\left(-2\right)=3{\left(-2\right)}^4+19{\left(-2\right)}^3+7{\left(-2\right)}^2-11\left(-2\right)-2\\ f\left(-2\right)=3\cdot 16+19\cdot \left(-8\right)+7\cdot 4-11\cdot \left(-2\right)-2\\ f\left(-2\right)=44-152+28+22-2\\ f\left(-2\right)=-60\end{array}$

To plug $2a$ in for each $x$

  $\begin{array}{l}f\left(x\right)=3x^4+19x^3+7x^2-11x-2\\ f\left(2a\right)=3{\left(2a\right)}^4+19{\left(2a\right)}^3+7{\left(2a\right)}^2-11\left(2a\right)-2\\ f\left(2a\right)=3\cdot \left(16a^4\right)+19\cdot \left(8a^3\right)+7\cdot \left(4a^2\right)-11\cdot \left(2a\right)-2\\ f\left(2a\right)=44a^4+152a^3+28a^2-22a-2\end{array}$