Question

answer the following questions about the equation below.
x^3 - 37x + 6 = 0
(a) list all rational roots that are possible according to the rational zero theorem. choose the correct answer below.
a. 1,2,3,6
b. ±1
c. ±1,±2,±3,±6
d. ±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 . (simplify your answer.)

Explanation:

Step1: Recall Rational Zero Theorem

The Rational Zero Theorem states that if a polynomial equation $a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_1x + a_0=0$ has integer coefficients, then the possible rational zeros 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 $x^3-37x + 6=0$, $a_n = 1$ and $a_0=6$. The factors of $6$ are $\pm1,\pm2,\pm3,\pm6$ and the factors of $1$ is $\pm1$. So the possible rational roots are $\pm1,\pm2,\pm3,\pm6$.

Step2: Use synthetic division to test possible roots

Let's start testing the possible rational roots $\pm1,\pm2,\pm3,\pm6$ using synthetic division.
Test $x = 6$:
Set up the synthetic - division:

6 |  1  0 -37  6
   |     6  36 -6
   |----------------
   |  1  6  -1  0

Since the remainder is $0$, $x = 6$ is a root of the polynomial $x^3-37x + 6=0$.

Answer:

(a) C. $\pm1,\pm2,\pm3,\pm6$
(b) $6$