# How do you use synthetic substitution to find p(-4) for p(x)=3x^3-2x^2+6x-4?

Aug 5, 2015

$\textcolor{red}{p \left(- 4\right) = - 252}$

#### Explanation:

$p \left(x\right) = 3 {x}^{3} - 2 {x}^{2} + 6 x - 4$

The Remainder Theorem states that when we divide a polynomial $f \left(x\right)$ by $x - c$ the remainder $R$ equals $f \left(c\right)$.

We use synthetic substitution to divide $f \left(x\right)$ by $x - c$, where $c = - 4$.

Step 1. Write only the coefficients of $x$ in the dividend inside an upside-down division symbol.

$| 3 \text{ "-2" " "6" " " } \textcolor{w h i t e}{1} - 4$
$| \textcolor{w h i t e}{1}$
stackrel("—————————————)

Step 2. Put the divisor at the left.

$\textcolor{red}{- 4} | 3 \text{ "-2" " "6" " " } \textcolor{w h i t e}{1} - 4$
$\text{ "color(white)(1)|" }$
" "" "stackrel("—————————————)

Step 3. Drop the first coefficient of the dividend below the division symbol.

$- 4 | 3 \text{ "-2" " "6" " " } \textcolor{w h i t e}{1} - 4$
$\text{ } \textcolor{w h i t e}{1} | \textcolor{w h i t e}{1}$
" "" "stackrel("—————————————)
$\text{ "" } \textcolor{red}{3}$

Step 4. Multiply the drop-down by the divisor, and put the result in the next column.

$- 4 | 3 \text{ "-2" " "6" " " } \textcolor{w h i t e}{1} - 4$
$\text{ "color(white)(1)|" " } \textcolor{w h i t e}{1} \textcolor{red}{- 12}$
" "" "stackrel("—————————————)
$\text{ "" } 3$

Step 5. Add down the column.

$- 4 | 3 \text{ "-2" " "6" " " } \textcolor{w h i t e}{1} - 4$
$\text{ "color(white)(1)|" " } - 12$
" "" "stackrel("—————————————)
$\text{ "" "3" } \textcolor{w h i t e}{1} \textcolor{red}{- 14}$

Step 6. Repeat Steps 4 and 5 until you can go no farther.

$- 4 | 3 \text{ "-2" "color(white)(1)6" " } \textcolor{w h i t e}{1} - 4$
$\text{ "color(white)(1)|" " " " "-12" " 56" } \textcolor{w h i t e}{1} - 248$
" "" "stackrel("—————————————)
$\text{ "" "3" "-14color(white)(1)62" } \textcolor{red}{- 252}$

The remainder is $- 252$, so $p \left(- 4\right) = - 252$.

Check:

$3 {x}^{3} - 2 {x}^{2} + 6 x - 4 = 3 {\left(- 4\right)}^{3} - 2 {\left(- 4\right)}^{2} + 6 \left(- 4\right) - 4 = 3 \left(- 64\right) - 2 \left(16\right) - 24 - 4 = - 192 - 32 - 28 = - 252$