# Algebra II : Factoring Rational Expressions

## Example Questions

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### Example Question #38 : Binomials

Simplify:

Explanation:

If we factors the denominator we get

Hence the rational expression becomes equal to

which is equal to

### Example Question #1 : Factoring Rational Expressions

Simplify.

The expression cannot be simplified.

Explanation:

a. Simplify the numerator and denominator separately by pulling out common factors.

b. Reduce if possible.

c. Factor the trinomial in the numerator.

d. Cancel common factors between the numerator and the denominator.

### Example Question #2 : Factoring Rational Expressions

Transform the following equation from standard into vertex form:

Explanation:

To take this standard form equation and transform it into vertex form, we need to complete the square. That can be done as follows:

We will complete the square on . In this case, our  in our soon-to-be  is . We therefore want our , so

Since we are adding  on the right side (as we are completing the square inside the parentheses), we need to add  on the left side as well. Our equation therefore becomes:

### Example Question #3 : Factoring Rational Expressions

Evaluate the following expression:

Explanation:

When we multiply expressions with exponents, we need to keep in mind some rules:

Divided variables subtract exponents.

Variables raised to a power multiply exponents.

Therefore, when we mulitiply the two fractions, we obtain:

### Example Question #7 : How To Factor An Equation

Simplify:

Explanation:

First factor the numerator. We need two numbers with a sum of 3 and a product of 2. The numbers 1 and 2 satisfy these conditions:

Now, look to see if there are any common factors that will cancel:

The  in the numerator and denominator cancel, leaving .

### Example Question #4 : Factoring Rational Expressions

Simplify this rational expression:

Explanation:

To see what can be simplified, factor the quadratic equations.

Cancel out like terms:

Combine terms:

### Example Question #5 : Factoring Rational Expressions

Simplify this rational expression:

None of these.

Explanation:

To simplify it is best to completely factor all polynomials:

Now cancel like terms:

Combine like terms:

### Example Question #6 : Factoring Rational Expressions

Factor and simplify this rational expression:

None of these.

Explanation:

Completely factor all polynomials:

Cancel like terms:

### Example Question #7 : Factoring Rational Expressions

Factor .

Explanation:

In the beginning, we can treat this as two separate problems, and factor the numerator and the denominator independently:

After we've factored them, we can put the factored equations back into the original problem:

From here, we can cancel the  from the top and the bottom, leaving:

### Example Question #8 : Factoring Rational Expressions

Factor:

Explanation:

Factor a two out in the numerator.

Factor the trinomial.

Factor the denominator.

Divide the terms.