Sketch the graph of the polynomial function. Use synthetic division and the remainder
theorem to find the zeros.

Question

Sketch the graph of the polynomial function. Use synthetic division and the remainder theorem to find the zeros. $$ f(x)=x^{3}-43 x+42 $$

UpStudy AI · Accepted Answer

To sketch the graph of the polynomial function $$ f(x) = x^{3} - 43x + 42 $$, we can use the information obtained from synthetic division and the remainder theorem to find the zeros of the function.

First, let's find the zeros of the function using synthetic division and the remainder theorem:

1. Synthetic Division:
   - Divide the polynomial $$ f(x) = x^{3} - 43x + 42 $$ by $$ x - 1 $$ to find the first zero.
   - The result of the synthetic division will give us the quotient and the remainder.

2. Remainder Theorem:
   - Use the remainder theorem to find the zeros of the function by substituting the zeros into the polynomial and setting it equal to zero.

After finding the zeros, we can use them to sketch the graph of the polynomial function. The zeros will be the x-intercepts of the graph.

Let's proceed with the synthetic division and the remainder theorem to find the zeros of the function.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$x^{3}-43x+42=0$$
- step1: Factor the expression:
  $$\left(x-6ight)\left(x-1ight)\left(x+7ight)=0$$
- step2: Separate into possible cases:
  $$\begin{align}&x-6=0\&x-1=0\&x+7=0\end{align}$$
- step3: Solve the equation:
  $$\begin{align}&x=6\&x=1\&x=-7\end{align}$$
- step4: Rewrite:
  $$x_{1}=-7,x_{2}=1,x_{3}=6$$
The zeros of the polynomial function $$ f(x) = x^{3} - 43x + 42 $$ are $$ x = -7, x = 1, $$ and $$ x = 6 $$.

Now, let's sketch the graph of the polynomial function using the zeros we found.
The zeros of the polynomial function $$ f(x) = x^{3} - 43x + 42 $$ are $$ x = -7, x = 1, $$ and $$ x = 6 $$.

Sketch the graph of the polynomial function. Use synthetic division and the remainder
theorem to find the zeros.

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