Using synthetic division, determine whether the numbers are zeros of the polynom
function.

Ask by Boone Bolton. in the United States

Mar 25,2025

Question

Using synthetic division, determine whether the numbers are zeros of the polynom function. $$ -1,1 ; h(x)=x^{4}+2 x^{3}+3 x^{2}-4 x-6 $$

UpStudy AI · Accepted Answer

To determine whether the numbers $$-1$$ and $$1$$ are zeros of the polynomial function $$h(x) = x^{4} + 2x^{3} + 3x^{2} - 4x - 6$$, we will use synthetic division for each value.

### Step 1: Synthetic Division with $$-1$$

1. Write down the coefficients of the polynomial: $$1, 2, 3, -4, -6$$.
2. Set up synthetic division with $$-1$$.

The synthetic division process is as follows:

- Bring down the leading coefficient (1).
- Multiply by $$-1$$ and add to the next coefficient.
- Repeat this process for all coefficients.

Let's perform the synthetic division for $$-1$$:

$$
\begin{array}{r|rrrrr}
-1 & 1 & 2 & 3 & -4 & -6 \
   &   & -1 & -1 & -2 & 6 \
\hline
   & 1 & 1 & 2 & -6 & 0 \
\end{array}
$$

The remainder is $$0$$, which means $$-1$$ is a zero of the polynomial.

### Step 2: Synthetic Division with $$1$$

Now, let's perform synthetic division for $$1$$:

$$
\begin{array}{r|rrrrr}
1 & 1 & 2 & 3 & -4 & -6 \
  &   & 1 & 3 & 6 & 2 \
\hline
  & 1 & 3 & 6 & 2 & -4 \
\end{array}
$$

The remainder is $$-4$$, which means $$1$$ is not a zero of the polynomial.

### Conclusion

- $$-1$$ is a zero of the polynomial $$h(x)$$.
- $$1$$ is not a zero of the polynomial $$h(x)$$.
-1 is a zero of the polynomial, but 1 is not.

Using synthetic division, determine whether the numbers are zeros of the polynom
function.

Upstudy AI Solution

Answer

Solution

Step 1: Synthetic Division with

Step 2: Synthetic Division with

Conclusion

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