# How do you use synthetic substitution to find x=3 for f(x)=-3x^3+7x^2-4x+8?

Sep 29, 2015

-22

#### Explanation:

f(3) would be $- 3 {\left(3\right)}^{3} + 7 {\left(3\right)}^{2} - 4 \left(3\right) + 8$

=-81 +63 -12+8

=-22

Sep 29, 2015

$- 22$ (using synthetic substitution)

#### Explanation:

Refer to the details below the diagram for an explanation of each step:

$\textcolor{red}{\text{[1]}}$
Write substitution value (in this case $3$) in a box to left side and the coefficients of the (canonically formed) polynomial to the right (in this case: $\left(- 3 {x}^{3} + 7 {x}^{2} - 4 x + 8\right) \rightarrow \left(- 3 + 7 - 4 + 8\right)$);
leave space for another line of numbers under this, then draw a vertical line.

$\textcolor{red}{\text{[2]}}$
Copy the first (highest order) coefficient to below the line.

$\textcolor{red}{\text{[3]}}$
Multiply the substitution value and the left-most value under the line and write the product under the second coefficient.

$\textcolor{red}{\text{[4]}}$
Add the two values in the second coefficient position and write the sum below the line.

$\textcolor{red}{\text{[5]}}$
Multiply the substitution value and the second value under the line and write the product under the third coefficient.

$\textcolor{red}{\text{[6]}}$
Add the two values in the third coefficient position and write the sum below the line.

$\textcolor{red}{\text{[7]}}$
Multiply the substitution value and the third value under the line and write the product under the fourth coefficient.

$\textcolor{red}{\text{[8]}}$
Add the two values in the fourth coefficient position and write the sum below the line.

This last sum is the value of the expression at the substitution value.