How do you use synthetic substitution to evaluate the polynomial p(x)=x^3-4x^2+4x-5 for x=4?

Aug 5, 2015

$\textcolor{red}{p \left(4\right) = 11}$

Explanation:

$p \left(x\right) = {x}^{3} - 4 {x}^{2} + 4 x - 5$

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.

$| 1 \text{ "-4" " "4" " " } - 5$
$| \textcolor{w h i t e}{1}$
stackrel("—————————————)

Step 2. Put the divisor at the left.

$\textcolor{red}{4} | 1 \text{ "-4" " "4" " " } - 5$
$\textcolor{w h i t e}{1} | \textcolor{w h i t e}{1}$
" "stackrel("—————————————)

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

$4 | 1 \text{ "-4" " "4" " " } - 5$
$\textcolor{w h i t e}{1} | \text{ "" } \textcolor{w h i t e}{1}$
" "stackrel("—————————————)
$\text{ } \textcolor{w h i t e}{1} \textcolor{red}{1}$

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

$4 | 1 \text{ "-4" " "4" " " } - 5$
$\textcolor{w h i t e}{1} | \text{ "" } \textcolor{w h i t e}{1} \textcolor{red}{4}$
" "stackrel("—————————————)
$\text{ } \textcolor{w h i t e}{1} 1$

Step 5. Add down the column.

$4 | 1 \text{ "-4" " "4" " " } - 5$
$\textcolor{w h i t e}{1} | \text{ "" } \textcolor{w h i t e}{1} 4$
" "stackrel("—————————————)
$\text{ "color(white)(1)1" "" } \textcolor{red}{0}$

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

$4 | 1 \text{ "-4" " "4" " " } - 5$
$\textcolor{w h i t e}{1} | \text{ "" "color(white)(1)4" "0" "" } 16$
" "stackrel("—————————————)
$\text{ "color(white)(1)1" "" "0" "4" "" } \textcolor{red}{11}$

The remainder is $11$, so $p \left(4\right) = 11$.

Check:

$p \left(x\right) = {x}^{3} - 4 {x}^{2} + 4 x - 5$

$p \left(4\right) = {4}^{3} - 4 {\left(4\right)}^{2} + 4 \left(4\right) - 5 = 64 - 4 \left(16\right) + 16 - 5 = 64 - 64 - 11 = 11$