Question

answer the following questions about the equation below.
6x³ + 37x² - 34x + 7 = 0
(a) list all rational roots that are possible according to the rational zero theorem. choose the correct answer below.
a. ±1,±7
b. ±1,±2,±3,±6,±1/7,±2/7,±3/7,±6/7
c. ±1,±7,±1/2,±7/2,±1/3,±7/3,±1/6,±7/6
d. ±1,±2,±3,±6
(b) use synthetic division to test several possible rational roots in order to identify one actual root.
one rational root of the given equation is 1/2.
(simplify your answer.)
(c) use the root from part (b) to solve the equation.
the solution set is {}.
(simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)

Explanation:

Step1: Recall Rational Zero Theorem

For a polynomial $a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_1x + a_0=0$, the possible rational roots are of the form $\frac{p}{q}$, where $p$ is a factor of the constant term $a_0$ and $q$ is a factor of the leading - coefficient $a_n$. For the equation $6x^3 + 37x^2-34x + 7 = 0$, $a_0 = 7$ and $a_n=6$. The factors of $7$ are $\pm1,\pm7$ and the factors of $6$ are $\pm1,\pm2,\pm3,\pm6$. So the possible rational roots are $\pm1,\pm7,\pm\frac{1}{2},\pm\frac{7}{2},\pm\frac{1}{3},\pm\frac{7}{3},\pm\frac{1}{6},\pm\frac{7}{6}$.

Step2: Use synthetic division with the root $\frac{1}{2}$

Set up synthetic division for $6x^3 + 37x^2-34x + 7$ with $x=\frac{1}{2}$. The coefficients are $6,37,-34,7$.
\[

$$\begin{array}{c|cccc} \frac{1}{2}&6&37&-34&7\\ & &3&20& - 7\\ \hline &6&40& - 14&0 \end{array}$$

\]
The quotient is $6x^2+40x - 14$.

Step3: Solve the quadratic equation

Set $6x^2+40x - 14 = 0$. Divide through by $2$ to get $3x^2+20x - 7=0$. Use the quadratic formula $x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}$ for the quadratic equation $ax^2+bx + c = 0$. Here, $a = 3$, $b = 20$, $c=-7$.
\[

$$\begin{align*} x&=\frac{-20\pm\sqrt{20^2-4\times3\times(-7)}}{2\times3}\\ &=\frac{-20\pm\sqrt{400 + 84}}{6}\\ &=\frac{-20\pm\sqrt{484}}{6}\\ &=\frac{-20\pm22}{6} \end{align*}$$

\]
We get $x_1=\frac{-20 + 22}{6}=\frac{2}{6}=\frac{1}{3}$ and $x_2=\frac{-20 - 22}{6}=\frac{-42}{6}=-7$.

Answer:

$\frac{1}{2},\frac{1}{3},-7$