# Simplify a Rational Expression

How to simplify a rational expression: 2 examples and their solutions.

## Example(x - 5)/(x^{2} - 7x + 10)

Factor the denominator

x^{2} - 7x + 10.

Find a pair of numbers

whose product is the constant term +10

and whose sum is the coefficient of the middle term -7.

-2⋅(-5) = +10

-2 - 5 = -7

So x^{2} - 7x + 10

= (x - 2)(x - 5).

Factor a Quadratic Trinomial

x^{2} - 7x + 10

= (x - 2)(x - 5)

So (given) = (x - 5) / [(x - 2)(x - 5)].

Cancel the common factor (x - 5).

Then 1/(x - 2).

So

1/(x - 2)

is the answer.

## Example(2x^{2} + x - 1)/(x^{2} - 2x - 3)

Factor the numberator

2x^{2} + x - 1.

Find a pair of numbers

whose product is the last term, -1,

and whose sum satisfies

[number 1] + [number 2]⋅[first term's coefficient, 2] = +1.

-1⋅1 = -1

-1 + 1⋅2 = -1 + 2 = +1

So 2x^{2} + x - 1

= (2x - 1)(x + 1).

Factor a Quadratic Trinomial: Example 5

Factor the denominator

x^{2} - 2x - 3.

Find a pair of numbers

whose product is the constant term -3

and whose sum is the coefficient of the middle term -2.

-3⋅1 = -3

-3 + 1 = -2

So x^{2} - 2x - 3

= (x - 3)(x + 1).

2x^{2} + x - 1

= (2x - 1)(x + 1)

x^{2} - 2x - 3

= (x - 3)(x + 1)

So (given) = [(2x - 1)(x + 1)] / [(x - 3)(x + 1)].

Cancel the common factor (x + 1).

Then (2x - 1)/(x - 3).

So

(2x - 1)/(x - 3)

is the answer.