3.02 Synthetic and long division
Long division with polynomials
We have already discussed how to divide polynomials using factoring and laws of exponents. The video below explains how to divide polynomials using long division by linking the process to elementary long division.
In algebra, we can divide one polynomial by another polynomial of the same or lower degree, using a process similar to long division. You can do it without a calculator because it breaks a complex-looking problem up into smaller ones.
Worked examples
Question 1
Simplify: $\frac{3x^3+2x^2-6}{x+2}$3x3+2x2−6x+2 using long division
The dividend: $3x^3+2x^2-6$3x3+2x2−6
The divisor: $x+2$x+2
Think: It's helpful to write the dividend like this: $3x^3+2x^2+0x-6$3x3+2x2+0x−6.
Do: Then we can use long division to solve the problem.
1. Divide the first term of the dividend by the greatest term of the divisor (meaning the one with the greatest power of $x$x, which in this case is $x$x). Place the result above the bar ($\frac{3x^3}{x}=3x^2$3x3x=3x2).
2. Multiply the divisor by the number you just wrote above the top line. Write the result under the first two terms of the dividend. ($3x^2\times\left(x+2\right)=3x^3+6x^2$3x2×(x+2)=3x3+6x2).
3. Subtract these terms from the from those in the original dividend, making sure you pay attention to the positive and negative signs ($3x^3+2x^2-\left(3x^3+6x^2\right)=-4x^2+0x$3x3+2x2−(3x3+6x2)=−4x2+0x).
4. Repeat the previous three steps, except this time use the two terms that have just been written as the dividend (ie. divide $\frac{-4x^2+0x}{x+2}$−4x2+0xx+2)
. 
5. Repeat step 4. Keep going until there is nothing to "pull down".
The answer can be written: $3x^2-4x+8$3x2−4x+8 with remainder $-22$−22 or $3x^2-4x+8-\frac{22}{x+2}$3x2−4x+8−22x+2
Reflect: This means that $3x^3+2x^2-6=\left(x+2\right)\left(3x^2-4x+8\right)-22$3x3+2x2−6=(x+2)(3x2−4x+8)−22
Practice questions
Question 2
Consider the following division:
$\left(3x^3-15x^2+2x-10\right)\div\left(x-5\right)$(3x3−15x2+2x−10)÷(x−5)
Fill in the gaps to complete the long division process below.
$\editable{}$ $+$+ $0x+$0x+ $\editable{}$ $x-5$x−5 $3x^3$3x3 $-$−$15x^2$15x2 $+$+$2x$2x $-$−$10$10 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ State the quotient and remainder when $3x^3-15x^2+2x-10$3x3−15x2+2x−10 is divided by $x-5$x−5.
Quotient = $\editable{}$
Remainder = $\editable{}$
Question 3
Consider the division $\left(x^2-6x+1\right)\div\left(x+3\right)$(x2−6x+1)÷(x+3).
What term needs to be brought down to move onto the next step in the algorithm?
$x$x $x$x $+$+ $3$3 $x^2$x2 $-$− $6x$6x $+$+ $1$1 $x^2$x2 $+$+ $3x$3x $-$− $9x$9x What is the remainder of this division?
$x$x $-$− $9$9 $x$x $+$+ $3$3 $x^2$x2 $-$− $6x$6x $+$+ $1$1 $x^2$x2 $+$+ $3x$3x $-$− $9x$9x $+$+ $1$1 $-$− $9x$9x $-$− $27$27 Without including the remainder, what is the quotient of this division?
Rewrite $x^2-6x+1$x2−6x+1 in terms of the divisor, the quotient and the remainder.
$x^2-6x+1$x2−6x+1$=$=$\left(x+\editable{}\right)\left(x-\editable{}\right)+\editable{}$(x+)(x−)+
Synthetic division
To save time and space, and minimize error, an efficient method of polynomial division emerges, known as synthetic division. Synthetic division can only be used when the divisor is a linear expression.
Worked example
question 4
To see how it works we divide $3x^3+2x^2-6$3x3+2x2−6 by the linear divisor $x+2$x+2 .
Setting up the synthetic table: Top row contains the coefficients of the numerator, include any terms with coefficient of $0$0: $3x^3+2x^2+0x-6$3x3+2x2+0x−6
| 3 | 2 | 0 | -6 | |
|---|---|---|---|---|
| add | ||||
| quotient |
| $3$3 | $2$2 | $0$0 | $-6$−6 | |
|---|---|---|---|---|
| $-2$−2 | ||||
| add | ||||
| quotient |
Take the first coefficient and carry it down, unchanged:
| $3$3 | $2$2 | $0$0 | $-6$−6 | |
|---|---|---|---|---|
| $-2$−2 | ||||
| add | ||||
| quotient | $3$3 |
Multiply the carry down value by the result of the linear divisor: $-2\times3$−2×3
| $3$3 | $2$2 | $0$0 | $-6$−6 | |
|---|---|---|---|---|
| $-2$−2 | ||||
| add | $-2\times3=-6$−2×3=−6 | |||
| quotient | $3$3 |
Add this value to the coefficient in that column and repeat the process:
| $3$3 | $2$2 | $0$0 | $-6$−6 | |
|---|---|---|---|---|
| $-2$−2 | ||||
| add | $-2\times3=-6$−2×3=−6 | $-2\times\left(-4\right)=8$−2×(−4)=8 | $-2\times8=-16$−2×8=−16 | |
| quotient | $3$3 | $2+\left(-6\right)=-4$2+(−6)=−4 | $8$8 | $-22$−22 |
The last row are the coefficients of the answer in degree order (starting with one degree less than the dividend) with the last value being the remainder.
The answer can be written: $3x^2-4x+8-\frac{22}{x+2}$3x2−4x+8−22x+2or $3x^2-4x+8$3x2−4x+8 with remainder -22
QUESTION 3
Using synthetic division to find the quotient and remainder when $x^3-4x^2-5$x3−4x2−5 is divided by $x-3$x−3
Think: To set-up the synthetic table, find the coefficients of the dividend and the result of the divisor set equal to zero.
Do: Create the synthetic table
| $1$1 | $-4$−4 | $0$0 | $-5$−5 | |
|---|---|---|---|---|
| $3$3 | ||||
| add | $3$3 | $-3$−3 | $-9$−9 | |
| quotient | $1$1 | $-1$−1 | $-3$−3 | $-14$−14 |
Answer: $x^2-x-3-\frac{14}{x-3}$x2−x−3−14x−3 or $x^2-x-3$x2−x−3with remainder $-14$−14
Practice question
Question 5
Consider the division $\frac{x^3+2x^2-29x-30}{x+6}$x3+2x2−29x−30x+6.
Perform the division using synthetic division.
$\editable{}$ $1$1 $2$2 $-29$−29 $-30$−30 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Hence rewrite the result of the division in terms of the quotient and remainder.
Question 6
Consider the division $\frac{x^4+9x^3+17x^2-13x+10}{x+5}$x4+9x3+17x2−13x+10x+5.
Perform the division using synthetic division.
$\editable{}$ $1$1 $9$9 $17$17 $-13$−13 $10$10 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Hence rewrite the result of the division in terms of the quotient and remainder.