Simplify Rational Expressions

How do we solve this problem?​

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​Correct Answer:

1. ​Combine Like Terms

   Not necessary for this problem

2. Factor out polynomials​

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3. Cancel Out Common Factors​

   We can cancel ou​t (x-2), you will then have the correct answer

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How do we solve this problem?​

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​Correct Answer:

1. ​Combine Like Terms

   This is not necessary for this problem

2. Factor Polynomials​

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3. Cancel out Common Factors​

We can cancel out (n+5), giving us the correct answer.​

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Multiple Choice

Factor:

$$x^2-7x+10$$  

• $$\left(x+5\right)\left(x+2\right)$$  

• $$\left(x-5\right)\left(x+2\right)$$  

• $$\left(x-5\right)\left(x-2\right)$$  

• $$\left(x+5\right)\left(x-2\right)$$  

Multiple Choice

Simplify the Expression

Multiple Choice

Simplify the Expression

Multiple Choice

Simplify

$$\frac{\left(2x-2x^2\right)}{\left(x-1\right)}$$  

• $$-2x$$  
  

• $$\frac{-1}{2x}$$  

• $$2x$$  
  

• $$\frac{1}{2x}$$  
  

Multiple Choice

Factor completely:

$$2x^2-8x$$  

• $$2x\left(x-4\right)$$  

• Not Factorable

• $$2x\left(2x-8\right)$$  

• $$2\left(x^2-4x\right)$$  

Reviewing Simplifying Rational Expressions by Factoring

Multiple Choice

Simplify

$$\frac{\left(2x-2x^2\right)}{\left(x-1\right)}$$  

• $$-2x$$  
  

• $$\frac{-1}{2x}$$  

• $$2x$$  
  

• $$\frac{1}{2x}$$  
  

Multiple Choice

Factor:

$$4x^2-9$$  

• $$\left(2x+3\right)\left(2x-3\right)$$  

• $$\left(2x+3\right)\left(2x+3\right)$$  

• $$\left(2x-3\right)\left(2x-3\right)$$  

• $$\left(4x+9\right)\left(4x-9\right)$$  

Multiple Choice

What expression is equivalent to  $$\frac{\left(x-5\right)}{\left(x-5\right)\left(x-4\right)}$$  

• $$\frac{1}{x-4}$$

• $$x-4$$

• $$\frac{1}{-9x-4}$$

• $$-9x-4$$

Multiple Choice

What is the simplified version of $$\frac{\left(x+4\right)\left(x+1\right)}{\left(x-5\right)\left(x+1\right)}$$

• $$\frac{x+4}{x+1}$$

• $$x+1$$

• $$\frac{x+4}{x-5}$$

• $$\frac{x+1}{x-5}$$

We will simplify rational expressions and state their restrictions.
I will summarize how to simplify rational expressions and determine their restrictions.

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​Lesson 11 Simplifying Rational Expressions

Multiple Choice

What is the correct factored form? $$\frac{x^2+5x+4}{x^{2\ }-4x-5}$$

• Factor:

  $$\frac{\left(x+1\right)\left(x+4\right)}{\left(x-5\right)\left(x+1\right)}$$

• Factor:

  $$\frac{\left(x+2\right)\left(x+2\right)}{\left(x-5\right)\left(x-1\right)}$$

• Factor:

  $$\frac{\left(x+2\right)\left(x-2\right)}{\left(x+5\right)\left(x-5\right)}$$

• Factor:

  $$\frac{\left(x+4\right)\left(x-1\right)}{\left(x-4\right)\left(x-1\right)}$$

Multiple Choice

What is the simplified version of $$\frac{\left(x+4\right)\left(x+1\right)}{\left(x-5\right)\left(x+1\right)}$$

• $$\frac{x+4}{x+1}$$

• $$x+1$$

• $$\frac{x+4}{x-5}$$

• $$\frac{x+1}{x-5}$$

Multiple Choice

What is the correct factored form? $$\frac{x^2+5x}{x^{2\ }-25}$$

• Factor:

  $$\frac{x\left(x+5\right)}{\left(x-5\right)\left(x-5\right)}$$

• Factor:

  $$\frac{\left(x+5\right)\left(x-5\right)}{\left(x-5\right)\left(x+5\right)}$$

• Factor:

  $$\frac{x\left(x+5\right)}{\left(x+5\right)\left(x+5\right)}$$

• Factor:

  $$\frac{x\left(x+5\right)}{\left(x-5\right)\left(x+5\right)}$$

Multiple Choice

What is the simplified version of $$\frac{x\left(x+5\right)}{\left(x-5\right)\left(x+5\right)}$$

• $$\frac{x}{x+5}$$

• $$x+5$$

• $$x$$

• $$\frac{x}{x-5}$$

Adding/Subtracting Rational Expressions

• Exact same procedure as adding/subtracting fractions

• Must have a common denominator!

• Once you have a common denominator, you add/subtract the numerators

Example of adding rational expressions with unlike denominators

Subtracting Rational Expressions!

• Same as adding rational expressions, but one extra thing!

• DISTRIBUTE THE NEGATIVE ALL THE WAY THROUGH

Multiple Choice

Find the sum:

$$\frac{1}{x}+\frac{9}{x}=$$  

• $$\frac{10}{x}$$  

• $$\frac{10}{2x}$$  

• $$\frac{5}{x}$$  

• $$\frac{10}{x^2}$$  

Multiple Choice

Find the difference:

$$\frac{2x}{x^3-2}-\frac{1}{x^3-2}$$  

• $$\frac{2x-1}{x^3-2}$$  

• $$\frac{2x+1}{x^3-2}$$  

• $$\frac{2x-1}{2x^3-4}$$  

• $$\frac{2x+1}{2x^3-4}$$  

Multiple Choice

Find the sum:

$$\frac{4}{6}+\frac{1}{8}=$$  

• $$\frac{19}{24}$$  

• $$\frac{19}{48}$$  

• $$\frac{5}{14}$$  

• $$\frac{1}{2}$$  

Multiple Choice

Find the sum:

$$\frac{6k}{k+2}+\frac{3}{k+5}=$$  

Multiple Choice

Hint: Factor first

Multiplying Rational Expressions

• Factor first (if possible)

• Cancel/Cross out common terms

• Multiply across

Dividing Rational Expressions

• Similar to multiplying rational expressions with one extra step in the beginning

• Can't divide by fractions so we have to flip it. (Keep Change Flip)

• Then we follow the steps to multiply rational expressions!

Multiple Choice

Simplify completely.

$$\frac{v^2-v-42}{v+1}\cdot\frac{1}{v+6}$$  

• $$6v$$  

• $$\frac{v-7}{v+1}$$  

• $$\frac{v+7}{3}$$  

• $$v+4$$  

Multiple Choice

Simplify completely.

$$\frac{a+3}{a+6}\cdot\frac{6a+36}{a+3}$$  

• $$6$$  

• $$\frac{8\left(a+7\right)}{a+9}$$  

• $$\frac{4a^2}{\left(a-2\right)\left(a+6\right)}$$  

• $$\frac{1}{8a^2}$$  

Multiple Choice

Simplify completely.

$$\frac{n-5}{n^2-4n-5}\div\frac{1}{6n}$$  

• $$\frac{8}{n+2}$$  

• $$\frac{n-4}{8}$$  

• $$\left(n+10\right)\left(n-7\right)$$  

• $$\frac{6n}{n+1}$$  

Multiple Choice

Simplify completely.

$$\frac{1}{a-4}\div\frac{a-6}{5a^2-20a}$$  

• $$\frac{2a\left(a+7\right)}{a+3}$$  

• $$\frac{5a}{a-6}$$  

• $$a-7$$  

• $$\frac{a+7}{a+3}$$