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How are Rational Expressions Simplified?

When you are simplifying a rational expression, you often use polynomial long division. In that case, the answer to the polynomial long division is the simplification you’re looking for. There are two important cases to make a note of:

1.
The case where the polynomial long division is solvable and the expression can be simplified.
2.
The case where the polynomial long division is not solvable, and simplifying the expression is not possible.

Rule

Simplifying Rational Expressions

Rational expressions are factorized by doing one of these things:

1.
You factorize the numerator and denominator by themselves and cancel common factors.

This method is often helpful when the degree of the numerator is the same as or lower than the degree of the denominator.

2.
You perform polynomial long division, where the answer to the division is the simplified expression.

This method is helpful when the degree of the numerator is higher than the degree of the denominator.

Below are examples illustrating this.

The Expression Can Be Simplified

If the degree of the numerator is higher than the degree in the denominator, you will want to use polynomial long division. If the division is solvable without a remainder, you have simplified your expression. The result of the division is the simplified expression. If the polynomial long division is not solvable, the expression can’t be simplified.

If the degree of the numerator is lower than the degree of the denominator, you should try to factorize the numerator and the denominator by themselves. A clever way to do this is to find the zeros of the quadratic expression and check whether they are zeros of the other polynomial as well.

Example 1

Simplify the expression x3 + 2x2 − 5x − 6 x − 2

You can see that the numerator is of a higher degree than the denominator. That means you should proceed with polynomial long division:

Polynomial long division of x^3+2x^2-5x-6 divided by x-2

You get no remainder, and the simplified expression is

x3 + 2x2 − 5x − 6 x − 2 = x2 + 4x + 3.

Example 2

Write the expression

x2 + x − 6 x3 − 2x2 − x + 2

as simply as possible

The degree of the numerator is lower than the degree of the denominator, which means you should try to factorize the numerator and the denominator by themselves.

You start with the numerator, which is a quadratic expression. Through inspection or the quadratic formula, you find the zeros to be x = −3 and x = 2, which means the factorization is (x + 3) (x − 2).

Next, you need to factorize the cubic expression in the denominator. Now is a good time to check whether any of the zeros you found for the numerator are zeros of the denominator as well.

You check x = −3 first:

= (−3) 3 − 2 (−3) 2 −(−3) + 2 = −27 − 18 + 3 + 2 = −40 ≠0.

(−3) 3 − 2 (−3) 2 −(−3) + 2 = −27 − 18 + 3 + 2 = −40 ≠0.

x = −3 is not a zero of the denominator.

Then you check x = 2:

(2) 3 − 2 (2) 2 −(2) + 2 = 8 − 8 − 2 + 2 = 0

x = 2 is a zero of the denominator, meaning that one of its factors is (x − 2). You can use this to carry out polynomial long division (x3 − 2x2 − x + 2) ÷(x − 2):

Polynomial long division of x^3-2x^2-x+2 divided by x-2

You can now see that

= (x3 − 2x2 − x + 2) = (x − 2) (x2 − 1) = (x − 2) (x − 1) (x + 1)

(x3 − 2x2 − x + 2) = (x − 2) (x2 − 1) = (x − 2) (x − 1) (x + 1)

The factorization ends up looking like this:

x2 + x −6 x3 −2x2 −x + 2 = (x + 3) (x −2) (x −1) (x −2) (x + 1) = x + 3 (x −1) (x + 1)

x2 + x − 6 x3 − 2x2 − x + 2 = (x + 3) (x − 2) (x − 1) (x − 2) (x + 1) = x + 3 (x − 1) (x + 1)

Example 3

Simplify the expression x2 + x − 6 x3 − x2 − 2x

x2 + x − 6 x3 − x2 − 2x = (x − 2) (x + 3) x (x2 − x − 2) = (x − 2) (x + 3) x(x − 2) (x + 1) = x + 3 x (x + 1)

The Expression Cannot Be Simplified

Theory

Why a Remainder Makes Simplification Impossible

To simplify a rational expression, you need to have a common factor between the numerator and the denominator. When the division is not solvable, the numerator and denominator have no common factors, which means there are no common factors to cancel!

Example 4

Simplify the expression 4x2 + x + 6 x − 2

As the numerator is of a higher degree than the denominator, you begin with polynomial long division:

Polynomial long division of 4x^2+x+6 divided by x-2

You get a remainder of 24, which means you can’t simplify the expression any further.

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