Dividing algebraic expressions

Fac­tori­sa­tion makes divi­sion easy. To divide one expres­sion by another, fac­torise both and can­cel the fac­tors they share, exactly as you sim­plify $\displaystyle \frac{12}{18}$ by can­celling the com­mon fac­tor $6$. This les­son works through four kinds of divi­sion, from a mono­mial divided by a mono­mial up to one poly­no­mial divided by another, and ends with long divi­sion for the cases where fac­toris­ing does not show the answer straight away.

1. A mono­mial divided by a mono­mial#

Write each mono­mial as a prod­uct of fac­tors, then can­cel the com­mon ones.

Exam­ple 1#

Divide $24x^{3}y^{2}$ by $6xy$.

$$\displaystyle \frac{24x^{3}y^{2}}{6xy} = \frac{4 \times 6 \times x \times x^{2} \times y \times y}{6 \times x \times y} = 4x^{2}y$$

For the num­bers, divide as usual: $24 \div 6 = 4$. For each let­ter, sub­tract the pow­ers: $x^{3} \div x = x^{2}$ and $y^{2} \div y = y$.

Exam­ple 2#

Divide $-42x^{2}y^{3}z$ by $14xyz$.

$$\displaystyle \frac{-42x^{2}y^{3}z}{14xyz} = -3xy^{2}$$

Here $-42 \div 14 = -3$, $x^{2} \div x = x$, $y^{3} \div y = y^{2}$ and $z \div z = 1$.

2. A poly­no­mial divided by a mono­mial#

Divide every term of the poly­no­mial by the mono­mial. Or, which comes to the same thing, take the mono­mial out as a com­mon fac­tor and can­cel it.

Exam­ple 3#

Divide $12x^{3} - 8x^{2} + 4x$ by $4x$.

$$\displaystyle \frac{12x^{3} - 8x^{2} + 4x}{4x} = \frac{12x^{3}}{4x} - \frac{8x^{2}}{4x} + \frac{4x}{4x} = 3x^{2} - 2x + 1$$

The last term, $\displaystyle \frac{4x}{4x}$, is $1$, not $0$.

Exam­ple 4#

Divide $24x^{2}yz + 24xy^{2}z + 24xyz^{2}$ by $8xyz$.

Fac­torise the top first: every term con­tains $24xyz$.

$$\displaystyle \frac{24x^{2}yz + 24xy^{2}z + 24xyz^{2}}{8xyz} = \frac{24xyz(x + y + z)}{8xyz} = 3(x + y + z)$$

3. A poly­no­mial divided by a poly­no­mial, by fac­toris­ing#

Fac­torise the div­i­dend (the expres­sion being divided). If the divi­sor is one of its fac­tors, can­cel it, and what remains is the answer.

Exam­ple 5#

Divide $x^{2} + 7x + 12$ by $x + 3$.

$$\displaystyle \frac{x^{2} + 7x + 12}{x + 3} = \frac{(x + 3)(x + 4)}{x + 3} = x + 4$$

Exam­ple 6#

Divide $10a^{2} + 37a + 33$ by $2a + 3$.

Split the mid­dle term: prod­uct $10 \times 33 = 330$, sum $37$, so the num­bers are $22$ and $15$.

$$\begin{aligned} 10a^{2} + 37a + 33 &= 10a^{2} + 22a + 15a + 33 \\ &= 2a(5a + 11) + 3(5a + 11) \\ &= (5a + 11)(2a + 3) \end{aligned}$$

So

$$\displaystyle \frac{10a^{2} + 37a + 33}{2a + 3} = \frac{(5a + 11)(2a + 3)}{2a + 3} = 5a + 11$$

Exam­ple 7#

Divide $y^{4} - 16$ by $y^{2} + 4$.

$$\displaystyle \frac{y^{4} - 16}{y^{2} + 4} = \frac{(y^{2} + 4)(y^{2} - 4)}{y^{2} + 4} = y^{2} - 4$$

4. Long divi­sion#

When the divi­sor is not obvi­ously a fac­tor, or does not divide exactly, use long divi­sion, just as with num­bers. Write both expres­sions in descend­ing pow­ers of $x$, then repeat four steps:

1. Divide the first term of what is left by the first term of the divi­sor.
2. Mul­ti­ply the whole divi­sor by that result.
3. Sub­tract it from what is left.
4. Bring down the next term, and repeat until what is left has a lower power than the divi­sor.

Exam­ple 8#

Divide $2x^{2} + 7x + 6$ by $x + 2$.

• Divide: $2x^{2} \div x = 2x$. This is the first term of the quo­tient.
• Mul­ti­ply: $2x(x + 2) = 2x^{2} + 4x$.
• Sub­tract: $(2x^{2} + 7x) - (2x^{2} + 4x) = 3x$. Bring down $+6$ to get $3x + 6$.
• Divide: $3x \div x = 3$. This is the next term of the quo­tient.
• Mul­ti­ply: $3(x + 2) = 3x + 6$.
• Sub­tract: $(3x + 6) - (3x + 6) = 0$.

The remain­der is $0$, so the quo­tient is $2x + 3$, and $2x^{2} + 7x + 6 = (x + 2)(2x + 3)$.

Exam­ple 9: a remain­der#

Divide $x^{3} + 2x^{2} - 5x + 4$ by $x - 1$.

• $x^{3} \div x = x^{2}$; $x^{2}(x - 1) = x^{3} - x^{2}$; sub­tract­ing leaves $3x^{2}$; bring down to get $3x^{2} - 5x$.
• $3x^{2} \div x = 3x$; $3x(x - 1) = 3x^{2} - 3x$; sub­tract­ing leaves $-2x$; bring down to get $-2x + 4$.
• $-2x \div x = -2$; $-2(x - 1) = -2x + 2$; sub­tract­ing leaves $2$.

The quo­tient is $x^{2} + 3x - 2$ and the remain­der is $2$. As with num­bers,

$$\text{dividend} = \text{divisor} \times \text{quotient} + \text{remainder}$$

$$x^{3} + 2x^{2} - 5x + 4 = (x - 1)(x^{2} + 3x - 2) + 2$$

The short cut for a divi­sor x minus a num­ber#

When the divi­sor has the form $x - a$, there is a quicker lay­out, called syn­thetic divi­sion. Write down the coef­fi­cients of the div­i­dend, and put $a$ on the left. Bring the first coef­fi­cient down. Then, over and over: mul­ti­ply the last num­ber writ­ten below the line by $a$, write the result under the next coef­fi­cient, and add.

For Exam­ple 9, the coef­fi­cients are $1, 2, -5, 4$ and $a = 1$:

The bot­tom row gives the quo­tien­t's coef­fi­cients, $1, 3, -2$, that is $x^{2} + 3x - 2$, and the last num­ber, $2$, is the remain­der. The same answer as long divi­sion, in three lines.

Prac­tice#

1. Divide $36a^{3}b^{2}$ by $9ab$.
2. Divide $15x^{4} - 10x^{3} + 5x^{2}$ by $5x^{2}$.
3. Divide $8p^{3}q - 12p^{2}q^{2} + 4pq^{3}$ by $4pq$.
4. Divide $x^{2} + 9x + 20$ by $x + 4$.
5. Divide $6y^{2} - 7y - 3$ by $2y - 3$.
6. Divide $m^{4} - 81$ by $m - 3$.
7. Divide $27x^{3} - 8$ by $3x - 2$.
8. Use long divi­sion to divide $3x^{2} + 11x + 10$ by $x + 2$.
9. Divide $2x^{3} - 3x^{2} + 4x - 5$ by $x - 2$, and give the quo­tient and the remain­der.

Answers#