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Simplifying Rational Expressions
(more challenging problems)
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Rational expressions are an integral part of Algebra 2.
Rational expressions can be expressed in simplified forms, used to perform arithmetic operations, involved in solving challenging equations, investigated to determine functional behavior, and used to model real-world problems.

Remember that rational expressions are basically "fractions with polynomial parts".
Algebraically, they extend the rules for fractions to apply to variables, they focus on factoring, and they always avoid division by zero.

As we move forward, the simplification of rational expressions will become more useful,
and a simplified form will often be considered the "more desirable" form of the expression.

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Let's take a look at some more challenging simplifications.

ex1
Explain if the simplification shown below is true, and whether there are any restrictions placed on the value of m:    ratsimp244

Answer:
This simplification is true since a value divided by itself equals one.

Yes, there are restrictions on m.
m ≠ -6 and m ≠ 3. Notice that when the second fraction is inverted, (x - 3) becomes a denominator which cannot be zero.

ratsimp5

m ≠ -6 and m ≠ 3.

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ex3
   

Answer:
• Write the given rational expression as the difference of two cubes.
• Recall: (a3 - b3) = (a - b)(a2 + ab + b2)

• Factor (-1) from the denominator and commute.

• Reduce factors.

• Deal with negative one.

• Express answer with (-1) times the entire expression, or distribute the (-1) across all of the terms.

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ex4

Answer:
• Factor out (x + 6) in the numerator.
Distributive property in reverse.

• Reduce factors.

• Combine like terms in the numerator.

• Express final result.

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ex4

Answer:
• This rational expression is the difference of two cubes divided by the difference of two squares.

• Recall: (a3 - b3) = (a - b)(a2 + ab + b2)
(a2 - b2) = (a - b)(a + b)

• Factor numerator and denominator using the know formulas (stated above).

• Reduce factors (a - b).

• Factor numerator again.

• Reduce factors (a + b).

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ex4

Answer:
• Factor by grouping, both numerator and denominator. Look for possible shared factors.

• Reduce factors (a - b).

• Express final result.


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