Determine which factors of $$ h(x) = x^{3} - 9x + 8 $$ can be identified using the Factor Theorem.

Question

UpStudy AI · Accepted Answer

To determine which factors of $$ h(x) = x^{3} - 9x + 8 $$ can be identified using the Factor Theorem, we need to find the values of $$ x $$ for which $$ h(x) = 0 $$ using the Factor Theorem.

The Factor Theorem states that if $$ a $$ is a factor of a polynomial $$ f(x) $$, then $$ f(a) = 0 $$.

Given polynomial: $$ h(x) = x^{3} - 9x + 8 $$

To find the factors using the Factor Theorem, we need to find the values of $$ x $$ for which $$ h(x) = 0 $$.

Let's find the factors of $$ h(x) = x^{3} - 9x + 8 $$ using the Factor Theorem.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$x^{3}-9x+8=0$$
- step1: Factor the expression:
  $$\left(x-1ight)\left(x^{2}+x-8ight)=0$$
- step2: Separate into possible cases:
  $$\begin{align}&x-1=0\&x^{2}+x-8=0\end{align}$$
- step3: Solve the equation:
  $$\begin{align}&x=1\&x=\frac{-1+\sqrt{33}}{2}\&x=-\frac{1+\sqrt{33}}{2}\end{align}$$
- step4: Rewrite:
  $$x_{1}=-\frac{1+\sqrt{33}}{2},x_{2}=1,x_{3}=\frac{-1+\sqrt{33}}{2}$$
The factors of $$ h(x) = x^{3} - 9x + 8 $$ that can be identified using the Factor Theorem are:

1. $$ x = -\frac{1+\sqrt{33}}{2} $$
2. $$ x = 1 $$
3. $$ x = \frac{-1+\sqrt{33}}{2} $$

These are the values of $$ x $$ for which $$ h(x) = 0 $$ using the Factor Theorem.
The factors of $$ h(x) = x^{3} - 9x + 8 $$ identified using the Factor Theorem are $$ x = -\frac{1+\sqrt{33}}{2} $$, $$ x = 1 $$, and $$ x = \frac{-1+\sqrt{33}}{2} $$.

Determine which factors of \( h(x) = x^{3} - 9x + 8 \) can be identified using the Factor Theorem.

Real Tutor Solution

Answer

Solution

Extra Insights

Related Questions

Latest Algebra Questions