Chapter 3 - Polynomial and Rational Functions - 3.2 Synthetic Division - 3.2 Exercises - Page 325: 9

$5x^{3}+2x-3$

When dividing a polynomial $f(x)$ with $(x-k)$ We set up syntehetic division $\quad \text{divisor } )\overline{\text{ dividend }}$, by placing $k$ in place of the divisor, and listing ALL coefficients of $f(x)$ (including the zeros ), starting from highest power of x. $k=-1$ $ \begin{array}{lllllll} & & -- & -- & -- & -- & --\\ -1 & ) & 5 & 5 & 2 & -1 & -3\\ & & & & & & \\ & & -- & -- & -- & -- & --\\ & & & & & & \\ & & & & & & \end{array}$ We are ready. Bring down the leading coefficient, $5$ Multiply $(-1)(5)=-5$ and place it in the next free slot of the middle row. $ \begin{array}{lllllll} & & -- & -- & -- & -- & --\\ -1 & ) & 5 & 5 & 2 & -1 & -3\\ & & & -5 & & & \\ & & -- & -- & -- & -- & --\\ & & 5 & & & & \\ & & & & & & \end{array}$ Add $5+(-5)=0$ and place $0$ in the bottom row. Multiply $(-1)(0)=0$ and place it in the next free slot of the middle row. $ \begin{array}{lllllll} & & -- & -- & -- & -- & --\\ -1 & ) & 5 & 5 & 2 & -1 & -3\\ & & & -5 & 0 & & \\ & & -- & -- & -- & -- & --\\ & & 5 & 0 & & & \\ & & & & & & \end{array}$ Add $2+(0) =2$ and place $2$ in the bottom row. Multiply $(-1)(2)=-2$ and place it in the next free slot of the middle row. $ \begin{array}{lllllll} & & -- & -- & -- & -- & --\\ -1 & ) & 5 & 5 & 2 & -1 & -3\\ & & & -5 & 0 & -2 & \\ & & -- & -- & -- & -- & --\\ & & 5 & 0 & 2 & & \\ & & & & & & \end{array}$ Add $-1+(-2)=-3$ and place it in the bottom row. Multiply $(-1)(-3)=3$ and place it in the next free slot of the middle row. $ \begin{array}{lllllll} & & -- & -- & -- & -- & --\\ -1 & ) & 5 & 5 & 2 & -1 & -3\\ & & & -5 & 0 & -2 & 3\\ & & -- & -- & -- & -- & --\\ & & 5 & 0 & 2 & -3 & \\ & & & & & & \end{array}$ Add $-3+(3)=0$ and place it in the bottom row. $ \begin{array}{lllllll} & & -- & -- & -- & -- & --\\ -1 & ) & 5 & 5 & 2 & -1 & -3\\ & & & -5 & 0 & -2 & 3\\ & & -- & -- & -- & -- & --\\ & & 5 & 0 & 2 & -3 & \fbox{$0$}\\ & & & & & & \end{array}$ Interpret the result: The last number of the bottom row represents the remainder, $0.$ The rest of the botom row holds coefficients of the quotient, which has a degree of one less than $f(x):\quad q(x)=5x^{3}+2x-3$ So, $\displaystyle \quad \frac{f(x)}{x-k}=q(x)+\frac{r}{x-k}$ $\displaystyle \frac{5x^{4}+5x^{3}+2x^{2}-x-3}{x+1}=\quad 5x^{3}+2x-3+\frac{0}{x+1}\quad =5x^{3}+2x-3$