Synthetic division

Synthetic division is an algorithm for dividing polynomials (usually when the divisor is linear and monic (that is, of the form $x-a$ for some number $a$ )). It is generally used as an alternative for other methods, such as the often more tedious polynomial long division.

A Brief Note

When learning a mathematical procedure such as this one, many practice problems are absolutely essential. This article provides a few exercises, but more will likely be necessary for long-term retention of this information. Fortunately, it is quite easy to come up with two polynomials you would like to divide in this manner.

Linear Factors

Procedure

Suppose we have some polynomial $P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ that we would like to divide by some factor $x-a$ . Then, we set the $a$ off to the side and arrange the coefficients of $P(x)$ as follows while taking care not to omit coefficients equal to $0$ and keeping negative signs: $[asy] draw((0,0)--(1,0)--(1,1)); draw((2,-2.5)--(11,-2.5)); draw((9.5,1)--(9.5,-3.5),dotted); label("$a$", (1,0), NW); label("$a_n$", (2,0), NE); label("$a_{n-1}$", (4,0), NE); label("$\cdots$", (6,0), NE); label("$a_1$", (8,0), NE); label("$a_0$", (10,0), NE); [/asy]$

For this explanation, we will evaluate $\frac{2x^4-6x^3+7x^2+5x-22}{x-2}$ .

We begin by bringing down the first coefficient:

$[asy] draw((0,0)--(1,0)--(1,1)); draw((2,-2.5)--(12,-2.5)); draw((9.5,1)--(9.5,-3.5),dotted); label("$2$", (1,0), NW); label("$2$", (2,0), NE); label("$-6$", (3.5,0), NE); label("$7$", (6,0), NE); label("$5$", (8,0), NE); label("$-22$", (10,0), NE); label("$2$", (2,-3.25), E); [/asy]$

Now, we multiply the coefficient we just brought down (namely, $2$ ) by the number we put in the top left (which also equals $2$ ), and then place this product (which equals $4$ ) into the next column above the bar:

$[asy] draw((0,0)--(1,0)--(1,1)); draw((2,-2.5)--(12,-2.5)); draw((9.5,1)--(9.5,-3.5),dotted); label("$2$", (1,0), NW); label("$2$", (2,0), NE); label("$-6$", (3.5,0), NE); label("$7$", (6,0), NE); label("$5$", (8,0), NE); label("$-22$", (10,0), NE); label("$4$", (4, -1.5), E); label("$2$", (2,-3.25), E); [/asy]$

Then, add this product to the coefficient above it:

$[asy] draw((0,0)--(1,0)--(1,1)); draw((2,-2.5)--(12,-2.5)); draw((9.5,1)--(9.5,-3.5),dotted); label("$2$", (1,0), NW); label("$2$", (2,0), NE); label("$-6$", (3.5,0), NE); label("$7$", (6,0), NE); label("$5$", (8,0), NE); label("$-22$", (10,0), NE); label("$4$", (4, -1.5), E); label("$2$", (2,-3.25), E); label("$-2$", (3.5,-3.25), E); [/asy]$

Repeat this process (as shown below) until all rows have been filled: $[asy] draw((0,0)--(1,0)--(1,1)); draw((2,-2.5)--(12,-2.5)); draw((9.5,1)--(9.5,-3.5),dotted); label("$2$", (1,0), NW); label("$2$", (2,0), NE); label("$-6$", (3.5,0), NE); label("$7$", (6,0), NE); label("$5$", (8,0), NE); label("$-22$", (10,0), NE); label("$4$", (4, -1.5), E); label("$-4$", (5.5, -1.5), E); label("$6$", (8,-1.5), E); label("$22$", (10.25,-1.5), E); label("$2$", (2,-3.25), E); label("$-2$", (3.5,-3.25), E); label("$3$", (6, -3.25), E); label("$11$", (7.75, -3.25), E); label("$0$", (10.25, -3.25), E); [/asy]$

Now, the numbers to the left of the dotted line represent the coefficients of the quotient polynomial (which has a degree one less than that of the dividend), and the rightmost term represents the remainder (which is multiplied by $\tfrac1{x-a}$ in the final result).

So, we use the coefficients we derived above to deduce our final answer: $\frac{2x^4-6x^3+7x^2+5x-22}{x-2}=2x^3-2x^2+3x+11+\frac0{x-2}=\boxed{2x^3-2x^2+3x+11}.$

Another Example

Now, we will evaluate $\frac{x^3-4x+3}{x+5}$ . We arrange the coefficients as follows while making sure not to forget the $x^2$ term, which has a coefficient of $0$ :

$[asy] draw((0,0)--(1,0)--(1,1)); draw((2,-2.5)--(9,-2.5)); draw((7.5,1)--(7.5,-3.5),dotted); label("$-5$", (1,0), NW); label("$1$", (2,0), NE); label("$0$", (4,0), NE); label("$-4$", (5.5,0), NE); label("$3$", (8,0), NE); [/asy]$

Now, we can follow the above procedure to find our desired answer: $[asy] draw((0,0)--(1,0)--(1,1)); draw((2,-2.5)--(9,-2.5)); draw((7.5,1)--(7.5,-3.5),dotted); label("$-5$", (1,0), NW); label("$1$", (2,0), NE); label("$0$", (4,0), NE); label("$-4$", (5.5,0), NE); label("$3$", (8,0), NE); label("$-5$", (3.5, -1.5), E); label("$25$", (5.75, -1.5), E); label("$-105$", (7.25, -1.5), E); label("$1$", (2,-3.25), E); label("$-5$", (3.5, -3.25), E); label("$21$", (5.75, -3.25), E); label("$-102$", (7.25, -3.25), E); [/asy]$

From these numbers, we can deduce our final answer: $\frac{x^3-4x+3}{x+5}=\boxed{x^2-5x+21-\frac{102}{x+5}}.$

Non-monic Factors

If the divisor of polynomial $P(x)$ is linear and non-monic (i.e. of the form $ax-b$ for some constants $a$ and $b$ ), then we can factor out the linear term coefficient to reduce this fraction to a constant times a fraction in which the divisor is monic, as follows: $\frac{P(x)}{ax-b}=\frac{P(x)}{a(x-\tfrac ba)}=\frac1a\cdot\frac{P(x)}{x-\tfrac ba}.$

For example, $\frac{x^3+2x+1}{3x-6}=\frac13\cdot\frac{x^3+2x+1}{x-2}.$ We can now evaluate this expression via the procedure described above.

Connection to Remainder Theorem

The Polynomial Remainder Theorem states that the remainder when dividing $P(x)$ by $x-a$ equals $P(a)$ . This fact can be used to evaluate values of $P(a)$ often more efficiently than plugging in $a$ manually. For example, suppose $P(x)=x^6-2x^5+3x^2-80x+1$ , and we desire to find $P(3)$ . We could plug in $3$ and calculate this manually, or we could use synthetic division, where we divide $P(x)$ by $x-3$ and find the remainder:

$[asy] size(7cm); draw((0,0)--(1,0)--(1,1)); draw((2,-2.5)--(15,-2.5)); draw((13.5,1)--(13.5,-3.5),dotted); label("$3$", (1,0), NW); label("$1$", (2,0), NE); label("$-2$", (3.5,0), NE); label("$0$", (6,0), NE); label("$0$", (8,0), NE); label("$3$", (10,0), NE); label("$-80$", (11.25,0), NE); label("$1$", (14,0), NE); label("$3$", (4, -1.5), E); label("$3$", (6, -1.5), E); label("$9$", (8, -1.5), E); label("$27$", (9.75, -1.5), E); label("$90$", (11.75, -1.5), E); label("$30$", (13.75, -1.5), E); label("$1$", (2,-3.25), E); label("$1$", (4, -3.25), E); label("$3$", (6, -3.25), E); label("$9$", (8, -3.25), E); label("$30$", (9.75, -3.25), E); label("$10$", (11.75, -3.25), E); label("$31$", (13.75, -3.25), E); [/asy]$

Thus, because our remainder equals $31$ , $P(3)=\boxed{31}$ by the Remainder Theorem.

Exercises

Show that $\frac{3x^5-2x^2+3x+8}{2x+2}=\frac12(3x^4-3x^3+3x^2-5x+8)$ .
Evaluate $\frac{z^4-3z^3+6z-4}{z-3}$ .
Evaluate $\frac{(y+3)^2-5}{y-4}$ .
Suppose $P(x)=3x^4+2x^3-80x^2-10x+9$ . Without plugging into this function, find $P(5)$ . Then, check your answer by plugging in $5$ manually.

Non-Linear Factors

Quadratic Factors

Suppose we have some polynomial $P(x)$ that we would like to divide by $x^2-bx-c$ . Then, we can place $b$ and $c$ along a diagonal along the side and arrange the coefficients of $P(x)$ as usual, as seen in the example below.

Now, we will evaluate $\frac{x^4-6x^3-40x^2+52x+6}{x^2+4x-8}$ after first performing the setup described above. Notice how $4$ and $-8$ become $-4$ and $8,$ respectively, and how they are placed on the diagonal:

$[asy] draw((-1,-1)--(0,-1)--(0,0)--(1,0)--(1,1)); draw((2,-3.5)--(11,-3.5)); draw((7.5,1)--(7.5,-4.5),dotted); label("$8$", (1,0), NW); label("$-4$", (0,-1), NW); label("$1$", (2,0), NE); label("$-6$", (3.5,0), NE); label("$-40$", (5.25,0), NE); label("$52$", (7.75,0), NE); label("$6$", (10,0), NE); [/asy]$

Notice that the dotted line (which separates the quotient from the remainder) now has two terms to its right, because the remainder of dividing a polynomial by a factor of degree $2$ will have degree $2-1=1$ , and so will be of the form $Ax+B$ for constants $A$ and $B$ .

Now, as we did for linear factors, we drop down the first coefficient: $[asy] draw((-1,-1)--(0,-1)--(0,0)--(1,0)--(1,1)); draw((2,-3.5)--(11,-3.5)); draw((7.5,1)--(7.5,-4.5),dotted); label("$8$", (1,0), NW); label("$-4$", (0,-1), NW); label("$1$", (2,0), NE); label("$-6$", (3.5,0), NE); label("$-40$", (5.25,0), NE); label("$52$", (7.75,0), NE); label("$6$", (10,0), NE); label("$1$", (2,-4.25), E); [/asy]$

Now, we multiply this first term below the bar by the diagonal (which, because this term equals $1$ , keeps the diagonal as it is) and then place this multiplied diagonal in the second and third columns:

$[asy] draw((-1,-1)--(0,-1)--(0,0)--(1,0)--(1,1)); draw((2,-3.5)--(11,-3.5)); draw((7.5,1)--(7.5,-4.5),dotted); label("$8$", (1,0), NW); label("$-4$", (0,-1), NW); label("$1$", (2,0), NE); label("$-6$", (3.5,0), NE); label("$-40$", (5.25,0), NE); label("$52$", (7.75,0), NE); label("$6$", (10,0), NE); label("$8$", (6, -1.25), E); label("$-4$", (3.5, -2.5), E); label("$1$", (2,-4.25), E); [/asy]$

Now, we add up all the numbers in the second column and place the resulting sum below the bar:

$[asy] draw((-1,-1)--(0,-1)--(0,0)--(1,0)--(1,1)); draw((2,-3.5)--(11,-3.5)); draw((7.5,1)--(7.5,-4.5),dotted); label("$8$", (1,0), NW); label("$-4$", (0,-1), NW); label("$1$", (2,0), NE); label("$-6$", (3.5,0), NE); label("$-40$", (5.25,0), NE); label("$52$", (7.75,0), NE); label("$6$", (10,0), NE); label("$8$", (6, -1), E); label("$-4$", (3.5, -2.5), E); label("$1$", (2,-4.25), E); label("$-10$", (3.25, -4.25), E); [/asy]$

Now, we can multiply the diagonal by this second term below the bar, place this diagonal on the next two columns, add up the next column, and repeat until we have found all of the coefficients of the desired quotient and remainder. Note that once numbers are placed on the last column, no new multiplied diagonals should be placed:

$[asy] size(7cm); draw((-1,-1)--(0,-1)--(0,0)--(1,0)--(1,1)); draw((2,-3.5)--(11,-3.5)); draw((7.25,1)--(7.25,-4.5),dotted); label("$8$", (1,0), NW); label("$-4$", (0,-1), NW); label("$1$", (2,0), NE); label("$-6$", (3.5,0), NE); label("$-40$", (5.25,0), NE); label("$52$", (7.75,0), NE); label("$6$", (10,0), NE); label("$8$", (6, -1), E); label("$-80$", (7.25, -1), E); label("$64$", (9.75, -1), E); label("$-4$", (3.5, -2.5), E); label("$40$", (5.75,-2.5), E); label("$-32$", (7.25, -2.5), E); label("$1$", (2,-4.25), E); label("$-10$", (3.25, -4.25), E); label("$8$", (6,-4.25), E); label("$-60$", (7.25, -4.25), E); label("$70$", (9.75, -4.25), E); [/asy]$

Note that because we are dividing a polynomial of degree $4$ by one of degree $2$ , the quotient will have degree $4-2=2$ .

Now, we can use these coefficients we derived to deduce our final answer: $\frac{x^4-6x^3-40x^2+52x+6}{x^2+4x-8}=\boxed{x^2-10x+8+\frac{-60x+70}{x^2+4x-8}}.$

Non-monic Factors

Just like with linear factors, we can factor out the leading coefficient from the factor to transform the fraction into a constant factor times a fraction with a monic factor. For example: $\frac{x^3-4x+5}{4x^2+8x-12}=\frac14\cdot\frac{x^3-4x+5}{x^2+2x-3}.$

Higher Degree Factors

For factors that are cubic or of higher degree, perform the same process with the opposites of the non-leading coefficients lined up in a diagonal from left to right. Once you cannot place any more full diagonals, do not place any more numbers, just as we stopped in the quadratic example above.

Problems

Show that $\frac{2z^4+3}{2z^2-6z+8}=\frac12\left(2z^2+6z+10+\frac{6z-37}{z^2-3z+4}\right)$ .
Evaluate $\frac{x^5-3x^3+6x^2-20x+12}{x^2-x+2}$ .
Evaluate $\frac{(y-8)^3+3y^4+3y^2}{y^2+2y-1}$ .

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Synthetic division

Contents

A Brief Note

Linear Factors

Procedure

Another Example

Non-monic Factors

Connection to Remainder Theorem

Exercises

Non-Linear Factors

Quadratic Factors

Non-monic Factors

Higher Degree Factors

Problems

See Also