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§ 6.3 Complex Rational Expressions

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**Woe is me, for I am complex.**

Simplifying Complex Fractions Complex rational expressions, also called complex fractions, have numerators or denominators containing one or more fractions. Woe is me, for I am complex. I am currently suffering from… a feeling of complexity Blitzer, Intermediate Algebra, 5e – Slide #2 Section 6.3

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**Complex Rational Expressions**

Simplifying a Complex Rational Expression by Multiplying by One in the Form T 1) Find the LCD of all rational expressions within the complex rational expression. 2) Multiply both the numerator and the denominator of the complex rational expression by this LCD. 3) Use the distributive property and multiply each term in the numerator and denominator by this LCD. Simplify. No fractional expressions should remain. 4) If possible, factor and simplify. Blitzer, Intermediate Algebra, 5e – Slide #3 Section 6.3

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**Simplifying Complex Fractions**

EXAMPLE Simplify: SOLUTION The denominators in the complex rational expression are 5 and x. The LCD is 5x. Multiply both the numerator and the denominator of the complex rational expression by 5x. Multiply the numerator and denominator by 5x. Blitzer, Intermediate Algebra, 5e – Slide #4 Section 6.3

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**Simplifying Complex Fractions**

CONTINUED Use the distributive property. Divide out common factors. Simplify. Factor and simplify. Blitzer, Intermediate Algebra, 5e – Slide #5 Section 6.3

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**Simplifying Complex Fractions**

CONTINUED Simplify. Simplify. Blitzer, Intermediate Algebra, 5e – Slide #6 Section 6.3

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**Simplifying Complex Fractions**

EXAMPLE Simplify: SOLUTION The denominators in the complex rational expression are x + 6 and x. The LCD is (x + 6)x. Multiply both the numerator and the denominator of the complex rational expression by (x + 6)x. Multiply the numerator and denominator by (x + 6)x. Blitzer, Intermediate Algebra, 5e – Slide #7 Section 6.3

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**Simplifying Complex Fractions**

CONTINUED Use the distributive property. Divide out common factors. Simplify. Simplify. Blitzer, Intermediate Algebra, 5e – Slide #8 Section 6.3

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**Simplifying Complex Fractions**

CONTINUED Subtract. Factor and simplify. Simplify. Blitzer, Intermediate Algebra, 5e – Slide #9 Section 6.3

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**Simplifying a Complex Rational Expression by Dividing**

Simplifying Complex Fractions Simplifying a Complex Rational Expression by Dividing 1) If necessary, add or subtract to get a single rational expression in the numerator. 2) If necessary, add or subtract to get a single rational expression in the denominator. 3) Perform the division indicated by the main fraction bar: Invert the denominator of the complex rational expression and multiply. 4) If possible, simplify. Blitzer, Intermediate Algebra, 5e – Slide #10 Section 6.3

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**Simplifying Complex Fractions**

EXAMPLE Simplify: SOLUTION 1) Subtract to get a single rational expression in the numerator. Blitzer, Intermediate Algebra, 5e – Slide #11 Section 6.3

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**Simplifying Complex Fractions**

CONTINUED 2) Add to get a single rational expression in the denominator. Blitzer, Intermediate Algebra, 5e – Slide #12 Section 6.3

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**Simplifying Complex Fractions**

CONTINUED 3) & 4) Perform the division indicated by the main fraction bar: Invert and multiply. If possible, simplify. Blitzer, Intermediate Algebra, 5e – Slide #13 Section 6.3

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**Simplifying Complex Fractions**

Important to Remember: Complex rational expressions have numerators or denominators containing one or more fractions. Complex rational expressions can be simplified by one of two methods presented in this section: (a) multiplying the numerator and denominator by the LCD (b) obtaining single expressions in the numerator and denominator and then dividing, using the definition of division for fractions Blitzer, Intermediate Algebra, 5e – Slide #14 Section 6.3

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