How do you use synthetic substitution to find g(-5) and g(4) for g(x)= 9x^4 - 8x^2 + 9x - 5?

1 Answer
Oct 4, 2015

g(-5)=color(blue)(5375); color(white)(X)g(4)=color(white)(X)color(blue)(2207)

Explanation:

$g \left(x\right) = 9 {x}^{4} - 8 {x}^{2} + 9 x - 5$

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

So, we use synthetic substitution to divide $g \left(x\right)$ by $x - c$, where $c = - 5$.

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

$\textcolor{w h i t e}{X l l} | 9 \textcolor{w h i t e}{X X l} 0 \textcolor{w h i t e}{I l} - 8 \textcolor{w h i t e}{X X X l l} 9 \textcolor{w h i t e}{X l l} - 5$
$\textcolor{w h i t e}{X X} |$
color(white)(XX)stackrel("———————————————————)

Step 2. Put the divisor $\left(- 5\right)$ at the left.

$\textcolor{red}{- 5} | 9 \textcolor{w h i t e}{X X l} 0 \textcolor{w h i t e}{I l} - 8 \textcolor{w h i t e}{X X X l l} 9 \textcolor{w h i t e}{X l l} - 5$
$\textcolor{w h i t e}{X X} |$
color(white)(XX)stackrel("———————————————————)

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

$- 5 | 9 \textcolor{w h i t e}{X X l} 0 \textcolor{w h i t e}{I l} - 8 \textcolor{w h i t e}{X X X l l} 9 \textcolor{w h i t e}{X l l} - 5$
$\textcolor{w h i t e}{X X} |$
color(white)(XX)stackrel("———————————————————)
$\textcolor{w h i t e}{X l l} | \textcolor{red}{9}$

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

$- 5 | 9 \textcolor{w h i t e}{X X l} 0 \textcolor{w h i t e}{I l} - 8 \textcolor{w h i t e}{X X X l l} 9 \textcolor{w h i t e}{X l l} - 5$
$\textcolor{w h i t e}{X l l} | \textcolor{w h i t e}{X l} \textcolor{red}{- 45}$
color(white)(XX)stackrel("———————————————————)
$\textcolor{w h i t e}{X l l} | 9$

Step 5. Add down the column.

$- 5 | 9 \textcolor{w h i t e}{X X l} 0 \textcolor{w h i t e}{| l} - 8 \textcolor{w h i t e}{X X X l l} 9 \textcolor{w h i t e}{X l l} - 5$
$\textcolor{w h i t e}{X l l} | \textcolor{w h i t e}{l l} - 45$
color(white)(XX)stackrel("———————————————————)
$\textcolor{w h i t e}{X l l} | 9 \textcolor{w h i t e}{l l} \textcolor{red}{- 45}$

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

$- 5 | 9 \textcolor{w h i t e}{X X l} 0 \textcolor{w h i t e}{I l} - 8 \textcolor{w h i t e}{X X X l l} 9 \textcolor{w h i t e}{X l l} - 5$
$\textcolor{w h i t e}{X l l} | \textcolor{w h i t e}{l l} - 45 \text{ } 225 \textcolor{w h i t e}{l} - 1085 \textcolor{w h i t e}{X l} 5380$
color(white)(XX)stackrel("———————————————————)
$\textcolor{w h i t e}{X l l} | 9 \textcolor{w h i t e}{l} - 45 \text{ "217color(white)(l)-1076" } \textcolor{red}{5375}$

$g \left(- 5\right) = 5375$

Check:

$g \left(x\right) = 9 {x}^{4} - 8 {x}^{2} + 9 x - 5$

$g \left(- 5\right) = 9 {\left(- 5\right)}^{4} - 8 {\left(- 5\right)}^{2} + 9 \left(- 5\right) - 5 = 9 \left(625\right) - 8 \left(25\right) - 45 - 5 = 5625 - 200 - 50 = 5375$

It works!

Now use the same method to show that

$g \left(4\right) = 2207$