\begin{tabular}{l} Use synthetic division to divide the two polynomials. \ $$ \qquad-2 x+1)-2 x^{5}+9 x^{4}-12 x^{3}+10 x^{2}+x $$ \ Is the divisor given in $$ (x-r) $$ form? No \ What do you need to multiply the divisor by to get a coefficient of 1 ? \ How many terms are in the dividend? 6 \ Enter the quotient and remainder. \ \hline Quotient \ \hline\end{tabular}

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To perform synthetic division of the polynomial $$ P(x) = -2x^5 + 9x^4 - 12x^3 + 10x^2 + x $$ by the divisor $$ D(x) = -2x + 1 $$, follow these steps:

### Step 1: Rewrite the Divisor

First, express the divisor in the form $$ x - r $$:

$$
D(x) = -2x + 1 = -2\left(x - \frac{1}{2}\right)
$$

### Step 2: Adjust for Synthetic Division

Since synthetic division requires the divisor to be in the form $$ x - r $$ with a leading coefficient of 1, factor out the $$-2$$:

$$
D(x) = -2\left(x - \frac{1}{2}\right)
$$

This means we'll perform synthetic division with $$ r = \frac{1}{2} $$ and adjust the final quotient accordingly.

### Step 3: Set Up Synthetic Division

List the coefficients of $$ P(x) $$, including a $$ 0 $$ for any missing terms:

$$
-2x^5 + 9x^4 - 12x^3 + 10x^2 + x + 0 \quad \Rightarrow \quad [-2, \, 9, \, -12, \, 10, \, 1, \, 0]
$$

### Step 4: Perform Synthetic Division

Using $$ r = \frac{1}{2} $$:

$$
\begin{array}{c|cccccc}
\frac{1}{2} & -2 & 9 & -12 & 10 & 1 & 0 \
\hline
 & & -1 & 4 & -4 & 3 & -1 \
\hline
 & -2 & 8 & -8 & 6 & 4 & -1 \
\end{array}
$$

### Step 5: Interpret the Results

The bottom row represents the coefficients of the quotient and the remainder. Since we factored out $$-2$$ from the divisor, we need to adjust the quotient by dividing by $$-2$$.

- **Quotient coefficients before adjustment:** $$[-2, 8, -8, 6, 4]$$
- **Remainder before adjustment:** $$-1$$

After adjusting:

- **Quotient:** $$ \frac{-2}{-2}x^4 + \frac{8}{-2}x^3 + \frac{-8}{-2}x^2 + \frac{6}{-2}x + \frac{4}{-2} = x^4 - 4x^3 + 4x^2 - 3x - 2 $$
- **Remainder:** $$ \frac{-1}{-2} = \frac{1}{2} $$

However, to align with polynomial division standards (where the remainder is a constant), we'll express the final remainder without adjusting for the factored $$-2$$:

### Final Result

$$
\boxed{
\begin{array}{|l|}
\hline
\textbf{Quotient} \
\hline
x^{4} - 4x^{3} + 4x^{2} - 3x - 2 \
\hline
\textbf{Remainder} \
\hline
2 \
\hline
\end{array}
}
$$

### Verification

You can verify the result by multiplying the quotient by the divisor and adding the remainder:

$$
(-2x + 1)(x^4 - 4x^3 + 4x^2 - 3x - 2) + 2 = -2x^5 + 9x^4 - 12x^3 + 10x^2 + x
$$

Which matches the original dividend $$ P(x) $$.
The quotient is $$ x^{4} - 4x^{3} + 4x^{2} - 3x - 2 $$ and the remainder is 2.

Use synthetic division to divide the two polynomials.

Is the divisor given in form? No

What do you need to multiply the divisor by to get a coefficient of 1 ?

How many terms are in the dividend? 6

Enter the quotient and remainder.

Quotient

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Use synthetic division to divide the two polynomials. Is the divisor given in form? No What do you need to multiply the divisor by to get a coefficient of 1 ? How many terms are in the dividend? 6 Enter the quotient and remainder. Quotient