Chapter 3 - Polynomial and Rational Functions - 3.2 Synthetic Division - 3.2 Exercises - Page 325: 10

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

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. $k=2$ $ \begin{array}{lllllll} & & -- & -- & -- & -- & --\\ 2 & ) & 2 & -1 & -7 & 7 & -10\\ & & & & & & \\ & & -- & -- & -- & -- & --\\ & & & & & & \\ & & & & & & \end{array}$ We are ready. Bring down the leading coefficient, $2$ Multiply $(2)(2)=4$ and place it in the next free slot of the middle row. $ \begin{array}{llllllll} & & -- & -- & -- & -- & -- & \\ 2 & ) & 2 & -1 & -7 & 7 & -10 & \\ & & & 4 & & & & \\ & & -- & -- & -- & -- & -- & \\ & & 2 & & & & & \\ & & & & & & & \end{array}$ Add $-1+4=3$ and place $3$ in the bottom row. Multiply $(2)(3)=6$ and place it in the next free slot of the middle row. $ \begin{array}{llllllll} & & -- & -- & -- & -- & -- & \\ 2 & ) & 2 & -1 & -7 & 7 & -10 & \\ & & & 4 & 6 & & & \\ & & -- & -- & -- & -- & -- & \\ & & 2 & 3 & & & & \\ & & & & & & & \end{array}$ Add $-7+(6) =-1$ and place $-1$ in the bottom row. Multiply $(2)(-1)=-2$ and place it in the next free slot of the middle row. $ \begin{array}{llllllll} & & -- & -- & -- & -- & -- & \\ 2 & ) & 2 & -1 & -7 & 7 & -10 & \\ & & & 4 & 6 & -2 & & \\ & & -- & -- & -- & -- & -- & \\ & & 2 & 3 & -1 & & & \\ & & & & & & & \end{array}$ Add $7+(-2)=5$ and place it in the bottom row. Multiply $(2)(5)=10$ and place it in the next free slot of the middle row. $ \begin{array}{llllllll} & & -- & -- & -- & -- & -- & \\ 2 & ) & 2 & -1 & -7 & 7 & -10 & \\ & & & 4 & 6 & -2 & 10 & \\ & & -- & -- & -- & -- & -- & \\ & & 2 & 3 & -1 & 5 & & \\ & & & & & & & \end{array}$ Add $-10+(10)=0$ and place it in the bottom row. $ \begin{array}{llllllll} & & -- & -- & -- & -- & -- & \\ 2 & ) & 2 & -1 & -7 & 7 & -10 & \\ & & & 4 & 6 & -2 & 10 & \\ & & -- & -- & -- & -- & -- & \\ & & 2 & 3 & -1 & 5 & \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 $f(x):\quad q(x)=2x^{3}+3x^{2}-x+5$ So, $\displaystyle \quad \frac{f(x)}{x-k}=q(x)+\frac{r}{x-k}$ $\displaystyle \frac{2x^{4}-x^{3}-7x^{2}+7x-10}{x-2}=\quad 2x^{3}+3x^{2}-x+5+\frac{0}{x-2}\quad $ $=2x^{3}+3x^{2}-x+5$