Rational Expressions and Rational Equations

Example 5. Simplify  .

Solution . Factoring the numerator , we can first take out a factor of 4.

We then factor x²-3x-10 with the quadratic formula . We could also do this by observation
and finding the factors of -10 which sum to -3. Identifying the coefficients

a = 1, b = -3, c = -10.

Computing the discriminant,

We then obtain the factors from the zeros .

or

Thus, the numerator factors as

In factoring the denominator, we notice that we have a common factor of 2.

Factor of 15 which sum to -8 are -3 and -5,

(-3)(-5) = 15 and - 3 + (-5) = -8.

This means that x²-8x+15 will have factors x-3 and x-5. Thus, factoring the denominator,
we have

We are now able to simplify the expression .

Thus, we have

Example 6. Simplify .

Solution. Combining like terms in the numerator , we have

We then need factors of -12 which sum to 1. We notice that 4 and -3 do the trick,

(4)(-3) = -12 and - 3 + 4 = 1.

Thus, x² + x - 12 has factors x - 3 and x + 4. Our ex pression then simplifies as

Example 7. Simplify .

Solution. In the numerator we notice that we have a common factor of 3, so that

Factoring the numerator using the Quadratic Formula , we have for the coefficients

The discriminant is then

The factors are then given by

and

The numerator then factors as

Likewise, factoring the denominator, we have for coefficients

Computing the discriminant, we have

The factors are then

and

Thus, the denominator factors as

Simplifying our rational expression gives

Thus, our rational expression simplifies as