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

Jul 9, 2015

$f \left(2\right) = - 11$ (using synthetic substitution, below)

#### Explanation:

Rewrite $f \left(x\right)$ so all powers of $x$ are represented (in descending power sequence).
$\textcolor{w h i t e}{\text{XXXX}}$$f \left(x\right) = {x}^{4} - 4 {x}^{3} + 2 {x}^{2} + 0 x - 3$

$\textcolor{red}{\text{[A]}}$$\textcolor{w h i t e}{\text{XXXX}}$Write the substitution value ($2$) separated by a vertical bar from a row of the from the coefficients as noted above.
$\textcolor{w h i t e}{\text{XXXXXX}}$Draw a vertical line below the coefficients, allowing enough space to write another row of numbers between the coefficients and the line.

$\textcolor{red}{\text{[B]}}$$\textcolor{w h i t e}{\text{XXXX}}$Copy the first coefficient to a position below the line.

$\textcolor{red}{\text{[C]}}$$\textcolor{w h i t e}{\text{XXXX}}$Multiply the substitution value and the number just written below the line and record the product above the line under the second coefficient.

$\textcolor{red}{\text{[D]}}$$\textcolor{w h i t e}{\text{XXXX}}$Add the product just developed and the second coefficient and write the sum below the line.

$\textcolor{red}{\text{[E]}}$$\textcolor{w h i t e}{\text{XXXX}}$Multiply the substitution value and the number just written below the line and record the product below the line.

$\textcolor{red}{\text{[F] to [I]}}$$\textcolor{w h i t e}{\text{XXXX}}$Continue this process until all coefficients have been processed.

$\textcolor{red}{\text{[I]}}$The solution is the last value written below the line.