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Rational Expressions Simplifying

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Simplifying Rational Expressions The objective is to be able to simplify a rational expression.

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Polynomial – The sum or difference of monomials. Rational expression – A fraction whose numerator and denominator are polynomials. Domain of a rational expression – the set of all real numbers except those for which the denominator is zero. Reduced form – a rational expression in which the numerator and denominator have no factors in common.

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Divide out the common factors Factor the numerator and denominator and then divide the common factors

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Dividing Out Common Factors Step 1 – Identify any factors which are common to both the numerator and the denominator. The numerator and denominator have a common factor. The common factor is the five.

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Dividing Out Common Factors Step 2 – Divide out the common factors. The fives can be divided since 5/5 = 1 The x remains in the numerator. The (x-7) remains in the denominator

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Factoring the Numerator and Denominator Factor the numerator. Factor the denominator. Divide out the common factors. Write in simplified form.

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Factoring Step 1: Look for common factors to both terms in the numerator. 3 is a factor of both 3 and 9. X is a factor of both x 2 and x. Step 2: Factor the numerator. 33 12 3 xx x ()

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Factoring Step 3: Look for common factors to the terms in the denominator and factor. The denominator only has one term. The 12 and x 3 can be factored. The 12 can be factored into 3 x 4. The x 3 can be written as x x 2. 33 34 2 xx xx ()

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Divide and Simplify Step 4: Divide out the common factors. In this case, the common factors divide to become 1. Step 5: Write in simplified form. x x 3 4 2

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You Try It Simplify the following rational expressions.

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Problem 1 Divide out the common factors. Write in simplified form. 3 8 x z

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Problem 2 Factor the numerator and denominator Divide out the common factors. Write in simplified form. You Try It

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Problem 3 Factor the numerator and denominator Divide out the common factors. Write in simplified form. You Try It

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Problem 4 You Try It Factor the numerator and denominator Divide out the common factors. Write in simplified form.

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Problem 5 You Try It Factor the numerator and denominator Divide out the common factors. Write in simplified form.

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Restrictions on Rational Expressions For what value of x is undefined? It is undefined for any value of “x” which makes the denominator zero. The restriction is that x cannot equal 5.

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YOU TRY IT What are the excluded values of the variables for the following rational expressions?

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Problem 1 Solution y 0 z 0

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Problem 2 Solution 2x - 12 = 0 2x - 12 + 12 = 0 + 12 2x = 12 2x 2 = 12 2 x = 6 ANSWER x 6

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Problem 3 Solution C 2 + 2C - 8 = 0 (C-2)(C+4) = 0 C-2 = 0 or C + 4 = 0 C-2+2 = 0 + 2 C + 4 - 4 = 0 - 4 C = 2 or C = -4 Answer C 2 C -4

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Assignment Problems 24 - 31

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