Synthetic division

To divide a polynomial $f(x)$ by $x k$, you can use synthetic division. Write $k$ and the coefficients of $f(x)$ on the first line. Bring down the first coefficient, then repeatedly multiply the newest quotient coefficient by $k$ and add it to the next coefficient of $f(x)$. The final result is the remainder.

Example 1: To divide $2x^3 3x^2 5x + 7$ by $x + 1$, use $k= 1$. The quotient is $2x^2 5x$ and the remainder is $7$. \n $$ \begin{array}{c|cccc} 1 & 2 & 3 & 5 & 7 \\ & & 2 & 5 & 0 \\ \hline & 2 & 5 & 0 & 7 \end{array} $$ \n Example 2: To divide $x^3 + 4x^2 9$ by $x 2$, use $k=2$ and a placeholder for the missing $x$ term. The quotient is $x^2 + 6x + 12$ and the remainder is $15$. \n $$ \begin{array}{c|cccc} 2 & 1 & 4 & 0 & 9 \\ & & 2 & 12 & 24 \\ \hline & 1 & 6 & 12 & 15 \end{array} $$.

Think of synthetic division as a cheat code for polynomial division, but only when you're dividing by a simple term like $x k$. It strips away all the variables, letting you work just with the numbers in a neat little grid. It's a fast paced rhythm of multiplying and adding that makes complex division problems much cleaner and quicker to solve.