Simplify a Rational Expression
How to simplify a rational expression: 2 examples and their solutions.
Example 1
Example
Solution
Factor the denominator
x2 - 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 x2 - 7x + 10
= (x - 2)(x - 5).
Factor a Quadratic Trinomial
x2 - 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 2
Example
Solution
Factor the numberator
2x2 + 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 2x2 + x - 1
= (2x - 1)(x + 1).
Factor a Quadratic Trinomial: a ≠ 1
Factor the denominator
x2 - 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 x2 - 2x - 3
= (x - 3)(x + 1).
2x2 + x - 1
= (2x - 1)(x + 1)
x2 - 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.