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Operations with Algebraic Fractions

Tutoring and Learning Centre, George Brown College 2014 www.georgebrown.ca/tlc

Operations with algebraic fractions follow the

same rules as operations with fractions:

1. Algebraic fractions can be added or subtracted ONLY if they have the SAME

DENOMINATOR (a common denominator).

To find a common denominator, find the

least common multiple of the denominators of all algebraic fractions being added or subtracted.

2. When multiplying algebraic fractions,

multiply the numerator by the numerator and denominator by denominator.

3. When dividing algebraic fractions, multiply by the reciprocal. The reciprocal is the

multiplicative inverse of a number. For a fraction , the reciprocal is IMPORTANT: For algebraic fractions containing trinomials in the form ax 2 + bx + c, try to factor the trinomials. When the trinomials are factored it is much easier to see what the common denominator is. Furthermore, some of the factors may cancel out if they appear both in the numerator and in the denominator of the answer. Example 1:

Simplify.

b(b - 4) + 2b b 2 a Step 1: To add the algebraic fractions, find a common denominator.

The least common multiple of b

2 and a is b

2a. Thus, the common denominator is b

2 a. Multiply the numerator AND the denominator of the first algebraic fraction, b(b - 4) by a.

Simplify.

b 2 Multiply the numerator AND the denominator of the second algebraic fraction , 2b by b2

Simplify.

a [b(b - 4)](a) + 2b(b 2 b 2 (a) a(b 2 = ab(b -

4) + 2b3

ab 2 ab 2 Step 2: Add the numerators of the algebraic fractions. Simplify.

Operations with Algebraic Fractions

Tutoring and Learning Centre, George Brown College 2014 www.georgebrown.ca/tlc

Keep the denominator the same.

ab(b - 4) + 2b 3 ab 2 = ab 2 - 4ab +2b 3 ab 2 = 2b 3 + ab 2 - 4ab ab 2

Example

2:

Simplify.

5x - 3 - 1

2x + 9 x + 8

Step 1

: To subtract algebraic fractions, find a common denominator.

The common denominator is (2x + 9)(x + 8).

Multiply the numerator AND denominator of the first algebraic fraction by (x + 8). Multiply the numerator AND denominator of the second algebraic fraction by (2x + 9). (5x - 3)(x + 8) - 1(2x + 9) (2x + 9 )(x + 8) (x + 8)(2x + 9) Step 2: Subtract the numerators of the algebraic fractions. Simplify by expanding and collecting like terms.

Keep the denominator the same.

(5x - 3)(x + 8) -1(2x + 9) (2x + 9 )(x + 8) = (5x - 3)(x + 8) -1(2x + 9) (2x + 9 )(x + 8) = 5x 2 + 40x -3x -24 -2x - 9 (2x + 9 )(x + 8) = 5x 2 + 35x - 33 (2x + 9 )(x + 8) Step 3: Try to factor the trinomial in the numerator. In this case, the trinomial is not factorable.

Final answer is

5x 2 + 35x - 33 (2x + 9 )(x + 8)

Operations with Algebraic Fractions

Tutoring and Learning Centre, George Brown College 2014 www.georgebrown.ca/tlc

Example

3:

Simplify.

x 2 - 4x - 21 ÷ x + 5 x 2 - 6x - 7 x + 1 Step 1: Factor the trinomials in the numerator and denominator of the first algebraic fraction. x 2 - 4x - 21 = (x - 7)(x + 3) x 2 - 6x - 7 = (x - 7)(x + 1)

Rewrite the trinomials with their factors.

(x - 7)(x + 3) ÷ x + 5 (x - 7)(x + 1) x + 1 Step 2: Multiply the first algebraic fraction by the reciprocal of the second algebraic fraction. (x - 7)(x + 3) · x + 1 (x - 7)(x + 1) x + 5 Step 3: Multiply the numerator by the numerator. Multiply the denominator by the denominator. (x - 7)(x + 3)(x + 1) (x - 7)(x + 1)(x + 5) Step 4: Cancel out any factors common to the numerator and the denominator. These factors cancel out because anything divided by itself equals 1. (x - 7)(x + 3)(x + 1) (x - 7)(x + 1)(x + 5)

Final answer is x + 3

x + 5

Practice Questions:

1. Simplify.

a) 2x 2 + x - 6 c) 4x 2 + 2x · 3x - 1 2x 2 - x - 3 x(2x - 1) 2x + 1 b) x - 1 - 3x d) x ÷ 10x x 2 -3x + 2 x 2 - 1 x 2 - x x 2 + x - 2

Answers:

1. a) x + 2 b) -2x 2 + 6x - 1 c) 6x - 2 d) x + 2 x + 1 x 3 - 2x 2 - x + 2 2x - 1 10xquotesdbs_dbs47.pdfusesText_47

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