Algebra II : Factoring Rational Expressions

Example Questions

Example Question #1 : Factoring Rational Expressions

Simplify:

Possible Answers:

Correct answer:

Explanation:

If we factors the denominator we get

Hence the rational expression becomes equal to

which is equal to

Example Question #2 : Factoring Rational Expressions

Simplify.

Possible Answers:

The expression cannot be simplified.

Correct answer:

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:

Possible Answers:

Correct answer:

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:

Our final answer is therefore

Example Question #4 : Factoring Rational Expressions

Evaluate the following expression:

Possible Answers:

Correct answer:

Explanation:

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

Multiplied variables add exponents.

Divided variables subtract exponents.

Variables raised to a power multiply exponents.

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

Our final answer is therefore

Example Question #8 : How To Factor An Equation

Simplify:

Possible Answers:

Correct answer:

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 #3 : Factoring Rational Expressions

Simplify this rational expression:

Possible Answers:

None of the other answers.

Correct answer:

Explanation:

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

Cancel out like terms:

Combine terms:

Example Question #7 : Factoring Rational Expressions

Factor and simplify this rational expression:

Possible Answers:

None of these.

Correct answer:

Explanation:

Completely factor all polynomials:

Cancel like terms:

Example Question #8 : Factoring Rational Expressions

Factor .

Possible Answers:

Correct answer:

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 #9 : Factoring Rational Expressions

Factor:

Possible Answers:

Correct answer:

Explanation:

Factor a two out in the numerator.

Factor the trinomial.

Factor the denominator.

Divide the terms.

The answer is:

Example Question #4 : Factoring Rational Expressions

Simplify to simplest terms.

Possible Answers:

Correct answer:

Explanation:

The correct answer is . The numerator and denominator can both be factored to simpler terms:

The  terms will cancel out. Leaving . While this is an answer choice, it can be simplified further. Factoring out a  from the denominator will allow the  terms to cancel out leaving .