Chapter 2.5, Problem 24E

To determine

To find : the rational zeros of the given function

Answer to Problem 24E

The rational zeros of the function are: $\boxed{\boxed{None}}$

Explanation of Solution

Given information : $g\left(x\right)=x^3+8x^2+12x+18$

Concept Involved:

The Rational Zero Test : The Rational Zero Test relates the possible rational zeros of a polynomial (havinginteger coefficients) to the leading coefficient and to the constant term of the polynomial.

If the polynomial $f(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+...+a_2{x}^2+a_{1}x+a_0$ has integer coefficients, then every rational zero of $f$ has the formRational zero = $p/q$ where p and q have no common factors other than 1, $p$ is a factor of the constant term $a_0$ and $q$ is a factor of the leading coefficient $a_n$ .

To use the Rational Zero Test, you should first list all rational numbers whosenumerators are factors of the constant term and whose denominators are factors of theleading coefficient.

Possible rational zeros: $\frac{Factorsof\text{constant}term}{Factorsofleadingcoefficient}$

Synthetic Division (for a Cubic Polynomial): To divide $a{x}^3+b{x}^2+cx+d$ by $x-k$ , use this pattern.

$\begin{array}{l}\begin{array}{l}k\\ \end{array}|\stackrel{Coefficientofdividend}{\overbrace{\underset{\_}{\begin{array}{cccc}\boxed{a}& b& c& d\\ \downarrow & \underset{\downarrow }{\boxed{ka}}& \underset{\downarrow }{\boxed{}}& \underset{\downarrow }{\boxed{}}\end{array}}}}\\ \begin{array}{ccccc}& \nearrow & \nearrow & \nearrow & \end{array}\\ \underset{Coefficientofquotient}{\underbrace{\begin{array}{ccc}\boxed{a}& \boxed{}& \boxed{}\end{array}}}\begin{array}{cc}\boxed{r}& \leftarrow \text{Remainder}\end{array}\end{array}$

In case when we have a polynomial with a missing term, insert placeholders with zero coefficients for missing powers of the variable. Vertical pattern: Add terms in columns Diagonal pattern: Multiply results by k . This algorithm for synthetic division works only for divisors of the form x - k . Remember that $x+k=x-\left(-k\right)$.

The Division Algorithm : If $f\left(x\right)$ and $d\left(x\right)$ are polynomials such that $d\left(x\right)\ne 0$ , and the degree of $d\left(x\right)$ isless than or equal to the degree of $f\left(x\right)$ , then there exist unique polynomials $q\left(x\right)$ and $r\left(x\right)$ such that $\underset{\stackrel{\uparrow }{{\displaystyle Dividend}}}{{\displaystyle \text{f (x)}}}\text{ = }\underset{\stackrel{\uparrow }{{\displaystyle Divisor}}}{{\displaystyle \text{d(x)}}}\cdot \underset{\stackrel{\uparrow }{{\displaystyle Quotient}}}{{\displaystyle \text{q(x)}}}+\underset{\stackrel{\uparrow }{{\displaystyle Remainder}}}{{\displaystyle \text{ r (x)}}}$ where $r\left(x\right)=0$ or the degree of $r\left(x\right)$ is less than the degree of $d\left(x\right)$ . If theremainder $r\left(x\right)$ is zero, then $d\left(x\right)$ divides evenly into $f\left(x\right)$ and k can be called as zero of the polynomial.

Calculation:

Identifying the constant and leading coefficient of the given function

  $g\left(x\right)=x^3+8x^2+12x+18$

Constant is 18

Leading coefficient is 1

Listing the factors of constant 18: $\pm 1,\pm 2,\pm 3,\pm 6,\pm 9,\pm 18$

Listing the factors of Leading Coefficient 1: $\pm 1$

The possible rational zeros of the function are:

  $\pm 1,\pm 2,\pm 3,\pm 6,\pm 9,\pm 18$

Possible zero	Synthetic division	Is it a zero of polynomial?
$x=-1$	$\begin{array}{l}\begin{array}{l}-1\\ \end{array}|\underset{\_}{\begin{array}{c}1\\ \end{array}\begin{array}{c}8\\ -1\end{array}\begin{array}{c}12\\ -7\end{array}\begin{array}{c}18\\ -5\end{array}}\\ 175\overline{)13}\leftarrow Remainder\end{array}$	No
$x=1$	$\begin{array}{l}\begin{array}{l}1\\ \end{array}|\underset{\_}{\begin{array}{c}1\\ \end{array}\begin{array}{c}8\\ 1\end{array}\begin{array}{c}12\\ 9\end{array}\begin{array}{c}18\\ 21\end{array}}\\ 1921\overline{)39}\leftarrow Remainder\end{array}$	No
$x=-2$	$\begin{array}{l}\begin{array}{l}-2\\ \end{array}|\underset{\_}{\begin{array}{c}1\\ \end{array}\begin{array}{c}8\\ -2\end{array}\begin{array}{c}12\\ -12\end{array}\begin{array}{c}18\\ 0\end{array}}\\ 160\overline{)18}\leftarrow Remainder\end{array}$	No
$x=2$	$\begin{array}{l}\begin{array}{l}2\\ \end{array}|\underset{\_}{\begin{array}{c}1\\ \end{array}\begin{array}{c}8\\ 2\end{array}\begin{array}{c}12\\ 20\end{array}\begin{array}{c}18\\ 64\end{array}}\\ 11032\overline{)82}\leftarrow Remainder\end{array}$	No
$x=-3$	$\begin{array}{l}\begin{array}{l}-3\\ \end{array}|\underset{\_}{\begin{array}{c}1\\ \end{array}\begin{array}{c}8\\ -3\end{array}\begin{array}{c}12\\ -15\end{array}\begin{array}{c}18\\ 9\end{array}}\\ 15-3\overline{)27}\leftarrow Remainder\end{array}$	No

Possible zero	Synthetic division	Is it a zero of polynomial?
$x=3$	$\begin{array}{l}\begin{array}{l}3\\ \end{array}|\underset{\_}{\begin{array}{c}1\\ \end{array}\begin{array}{c}8\\ 3\end{array}\begin{array}{c}12\\ 33\end{array}\begin{array}{c}18\\ 135\end{array}}\\ 11145\overline{)153}\leftarrow Remainder\end{array}$	No
$x=-6$	$\begin{array}{l}\begin{array}{l}-6\\ \end{array}|\underset{\_}{\begin{array}{c}1\\ \end{array}\begin{array}{c}8\\ -6\end{array}\begin{array}{c}12\\ -12\end{array}\begin{array}{c}18\\ 0\end{array}}\\ 120\overline{)18}\leftarrow Remainder\end{array}$	No
$x=6$	$\begin{array}{l}\begin{array}{l}6\\ \end{array}|\underset{\_}{\begin{array}{c}1\\ \end{array}\begin{array}{c}8\\ 6\end{array}\begin{array}{c}12\\ 84\end{array}\begin{array}{c}18\\ 576\end{array}}\\ 11496\overline{)594}\leftarrow Remainder\end{array}$	No
$x=-9$	$\begin{array}{l}\begin{array}{l}-9\\ \end{array}|\underset{\_}{\begin{array}{c}1\\ \end{array}\begin{array}{c}8\\ -9\end{array}\begin{array}{c}12\\ 9\end{array}\begin{array}{c}18\\ -189\end{array}}\\ 1-121\overline{)-171}\leftarrow Remainder\end{array}$	No
$x=9$	$\begin{array}{l}\begin{array}{l}9\\ \end{array}|\underset{\_}{\begin{array}{c}1\\ \end{array}\begin{array}{c}8\\ 9\end{array}\begin{array}{c}12\\ 153\end{array}\begin{array}{c}18\\ 1485\end{array}}\\ 117165\overline{)1503}\leftarrow Remainder\end{array}$	No
$x=-18$	$\begin{array}{l}\begin{array}{l}-18\\ \end{array}|\underset{\_}{\begin{array}{c}1\\ \end{array}\begin{array}{c}8\\ -18\end{array}\begin{array}{c}12\\ 180\end{array}\begin{array}{c}18\\ -3456\end{array}}\\ 1-10192\overline{)-3438}\leftarrow Remainder\end{array}$	No
$x=18$	$\begin{array}{l}\begin{array}{l}18\\ \end{array}|\underset{\_}{\begin{array}{c}1\\ \end{array}\begin{array}{c}8\\ 18\end{array}\begin{array}{c}12\\ 468\end{array}\begin{array}{c}18\\ 8640\end{array}}\\ 126480\overline{)8658}\leftarrow Remainder\end{array}$	No

Conclusion:

The possible rational zeros of the function are: $\pm 1,\pm 2,\pm 3,\pm 6,\pm 9,\pm 18$

The actual rational zeros of the function are: $None$