3.5 - Division of Expressions

Before studying this topic you may wish to review common fractions and the division of numbers.

Suppose that A and B are any two expressions. Dividing A by B means setting up the quotient as

(This quotient is called an algebraic fraction.) Then you actually carry out the division.

Note that in the first form,

, (which is the preferred form), no brackets are shown but they are implied. The reason that we put brackets around A and B is that they are expressions, not just numbers, and the division is supposed to apply to whatever A and B may contain. (We want the division to be at the end in the order of operations.)

How the division is actually carried out and what the result is, depends on whether A and B are monomials, multinomials or polynomials:

Case I: Numerator A and denominator B are both monomials. We merely reduce the algebraic fraction to lowest terms. The result of the division is a monomial.
Case II: Numerator A is a multinomial and denominator B is a monomial. We place each term of the numerator over the denominator. The result of the division is a multinomial.
Case III: Numerator A and denominator B are both polynomials and the degree of A is higher than the degree of B. This is called an improper rational algebraic fraction. We divide the denominator into the numerator using long division (or synthetic division). The result of the division is a polynomial with possibly a proper rational algebraic fraction remainder.

Case I: Dividing a monomial by a monomial

In this case carrying out the division means just simplifying or reducing the resulting algebraic fraction to lowest terms. Specifically:

Get the sign using the rules for signs.
Reduce the coefficient to lowest terms.
Cancel identical factors that appear in both the numerator and denominator.
Combine exponentials having the same base using the division property of exponentials.

Example: Divide A = 6 b d by B = −9 a c

Reduce the coefficient 6/9 to lowest terms.

Notice that the − sign is put either in front of the result or in front of the numerator; never in front of the denominator.

Example: Divide A = −2 x³ y by B = −8 x⁻² z

The two − signs are replaced by a + sign which we don’t have to display. The coefficient reduces to (positive) ¼. The numerator contains other factors so the 1 in the numerator can be omitted.

The factors with base x are like factors. They are combined using the properties of exponents.

Example: Divide A = −2 x³ by B = 6 x³ z

The coefficient reduces to −1/3. The identical factors of x³ in the numerator and denominator cancel. The numerator contains no other factors so this time the 1 must remain.

Again the − sign is put in front.

Example: Divide B = 6 x³ z by A = −2 x

After carrying out all the simplifications, the denominator equals 1, so we don’t have to display it. Thus the result is an ordinary expression, not an algebraic fraction.

Algebra Coach Exercises

Case II: Dividing a multinomial by a monomial

In this case each term of the multinomial is divided by the monomial like this:

and then each of the resulting terms is simplified as in Case I above.

This method is actually a consequence of replacing a division by a multiplication by a reciprocal and then using the distributive law to distribute the reciprocal onto each term of the multinomial, like this:

Note that the Algebra Coach does not divide a multinomial by a monomial when you click the Simplify button. You must use the Distribute button to do that. The reason is that the single fraction form of the expression is considered to be simpler than the multiple fraction form.

Example: Divide A = −2 x⁴ z² − 3 x z by B = 6 x³ z

Divide each term of the multinomial by the monomial.

Simplify each term using the division property of exponents.

Example: Divide A = x² − 2 x − 5 by B = −2 x

Divide each term of the multinomial by the monomial. Notice how the signs are reversed (just like when distributing a negative).

Simplify each term using the division property of exponents.

Warning: A common error is to try to simplify a monomial divided by a multinomial. However there is no simplification possible for this.

Algebra Coach Exercises

Case III: Dividing a polynomial by a polynomial

An algebraic fraction whose numerator and denominator are both polynomials in the same variable is called a rational algebraic fraction. If the degree of the numerator polynomial is higher than the degree of the denominator polynomial then it is called an improper rational algebraic fraction. Just like long division can be used to convert an improper fraction to a mixed fraction (click here to review that topic), so long division can be used to convert an improper rational algebraic fraction into a polynomial plus a proper rational algebraic fraction (the analog of a mixed fraction).

Example: Divide 2 x² + 2 x − 3 by x − 2.

Carry out the following steps:

Set up the algebraic fraction in long division format, namely .
We will call x − 2 the divisor and 2 x² + 2 x − 3 the dividend. Divide the first term of the dividend, namely 2 x², by the first term of the divisor, namely x. The result, 2 x, is written above the symbol, in line with the other x’s. It will be the first term of the quotient. Multiply the divisor by the first term of the quotient and write the result below the dividend and subtract it from the dividend, like this:
This step has shown that .
We are not done yet. We repeat the previous step. The divisor is again x − 2 but the remainder from the previous step, namely 6 x − 3, becomes the new dividend. Divide the first term of the dividend, namely 6 x, by the first term of the divisor, namely x. The result, 6, is written above the symbol, in line with the other constants. It will be the second term of the quotient. Multiply the divisor by the second term of the quotient and write the result below the dividend and subtract it from the dividend, like this:
This step has shown that:
Since the remainder, 9, is of lower degree than the divisor, x − 2, this is our final “mixed fraction” result.

Notes:

Click here to compare this long division with that of an improper fraction.
When setting up the long division format the divisor and dividend must both be written in descending powers of the variable. Also if terms of either are missing they must be included with zero coefficients. For example if the dividend is 2 x − 3 x³ then it must be rewritten as −3 x³ + 0 x² + 2 x + 0.

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