Chapter 3 - Polynomial and Rational Functions - 3.2 Synthetic Division - 3.2 Exercises - Page 325: 11

$x^{3}+2x+1$

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. $k=-4$ $ \begin{array}{lllllll} & & -- & -- & -- & -- & --\\ -4 & ) & 1 & 4 & 2 & 9 & 4\\ & & & & & & \\ & & -- & -- & -- & -- & --\\ & & & & & & \\ & & & & & & \end{array}$ We are ready. Bring down the leading coefficient, $1$ Multiply $(-4)(1)=-4$ and place it in the next free slot of the middle row. $ \begin{array}{llllllll} & & -- & -- & -- & -- & -- & \\ -4 & ) & 1 & 4 & 2 & 9 & 4 & \\ & & & -4 & & & & \\ & & -- & -- & -- & -- & -- & \\ & & 1 & & & & & \\ & & & & & & & \end{array}$ Add $4+(-4)=0$ and place $0$ in the bottom row. Multiply $(-4)(0)=0$ and place it in the next free slot of the middle row. $ \begin{array}{llllllll} & & -- & -- & -- & -- & -- & \\ -4 & ) & 1 & 4 & 2 & 9 & 4 & \\ & & & -4 & 0 & & & \\ & & -- & -- & -- & -- & -- & \\ & & 1 & 0 & & & & \\ & & & & & & & \end{array}$ Add $2+(0) =2$ and place $2$ in the bottom row. Multiply $(-4)(2)=-8$ and place it in the next free slot of the middle row. $ \begin{array}{llllllll} & & -- & -- & -- & -- & -- & \\ -4 & ) & 1 & 4 & 2 & 9 & 4 & \\ & & & -4 & 0 & -8 & & \\ & & -- & -- & -- & -- & -- & \\ & & 1 & 0 & 2 & & & \\ & & & & & & & \end{array}$ Add $9+(-8)=1$ and place it in the bottom row. Multiply $(-4)(1)=-4$ and place it in the next free slot of the middle row. $ \begin{array}{llllllll} & & -- & -- & -- & -- & -- & \\ -4 & ) & 1 & 4 & 2 & 9 & 4 & \\ & & & -4 & 0 & -8 & -4 & \\ & & -- & -- & -- & -- & -- & \\ & & 1 & 0 & 2 & 1 & & \\ & & & & & & & \end{array}$ Add $4+(-4)=0$ and place it in the bottom row. $ \begin{array}{llllllll} & & -- & -- & -- & -- & -- & \\ -4 & ) & 1 & 4 & 2 & 9 & 4 & \\ & & & -4 & 0 & -8 & -4 & \\ & & -- & -- & -- & -- & -- & \\ & & 1 & 0 & 2 & 1 & \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 $f(x):\quad q(x)=x^{3}+2x+1$ So, $\displaystyle \quad \frac{f(x)}{x-k}=q(x)+\frac{r}{x-k}$ $\displaystyle \frac{x^{4}+4x^{3}+2x^{2}+9x+4}{x+4}=\quad x^{3}+2x+1+\frac{0}{x+4}\quad $ $=x^{3}+2x+1$