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  • Holt McDougal Algebra 2

    Adding and Subtracting

    Rational Expressions

    Adding and Subtracting

    Rational Expressions

    Holt Algebra 2

    Warm Up

    Lesson Presentation

    Lesson Quiz

    Holt McDougal Algebra 2

  • Holt McDougal Algebra 2

    Adding and Subtracting

    Rational Expressions

    Warm UpAdd or subtract.

    1.

    2.

    3.

    4.

    x 0

    1324

    x 1, x 1

    Simplify. Identify any x-values for which the expression is undefined.

    1315

    1112

    38

    715

    25

    +

    x 1x2 1

    1x + 1

    13

    x64x9

    12x3

  • Holt McDougal Algebra 2

    Adding and Subtracting

    Rational Expressions

    Add and subtract rational expressions.

    Simplify complex fractions.

    Objectives

  • Holt McDougal Algebra 2

    Adding and Subtracting

    Rational Expressions

    complex fraction

    Vocabulary

  • Holt McDougal Algebra 2

    Adding and Subtracting

    Rational Expressions

    Adding and subtracting rational expressions is similar to adding and subtracting fractions. To add or subtract rational expressions with like denominators, add or subtract the numerators and use the same denominator.

  • Holt McDougal Algebra 2

    Adding and Subtracting

    Rational Expressions

    Add or subtract. Identify any x-values for which the expression is undefined.

    Example 1A: Adding and Subtracting Rational

    Expressions with Like Denominators

    Add the numerators.

    The expression is undefined at x = 4 because this value makes x + 4 equal 0.

    x 3x + 4

    +x 2x + 4

    x 3 +x + 4

    x 2

    2x 5x + 4

    Combine like terms.

  • Holt McDougal Algebra 2

    Adding and Subtracting

    Rational Expressions

    Add or subtract. Identify any x-values for which the expression is undefined.

    Example 1B: Adding and Subtracting Rational

    Expressions with Like Denominators

    Subtract the numerators.

    There is no real value of x for which x2 + 1 = 0; the expression is always defined.

    3x 4x2 + 1

    6x + 1x2 + 1

    3x 4 x2 + 1

    (6x + 1)

    3x 5x2 + 1 Combine like terms.

    Distribute the negative sign.3x 4

    x2 + 16x 1

  • Holt McDougal Algebra 2

    Adding and Subtracting

    Rational Expressions

    Check It Out! Example 1a

    Add or subtract. Identify any x-values for which the expression is undefined.

    Add the numerators.

    6x + 5x2 3

    +3x 1x2 3

    6x + 5 +x2 3

    3x 1

    9x + 4x2 3

    Combine like terms.

    The expression is undefined at x = because this value makes x2 3 equal 0.

  • Holt McDougal Algebra 2

    Adding and Subtracting

    Rational Expressions

    Check It Out! Example 1b

    Add or subtract. Identify any x-values for which the expression is undefined.

    Subtract the numerators.

    3x2 53x 1

    2x2 3x 2

    3x 1

    3x 13x2 5 (2x2 3x 2)

    x2 + 3x 33x 1 Combine like terms.

    Distribute the negative sign.3x2 5

    3x 12x2 + 3x + 2

    The expression is undefined at x = because this value makes 3x 1 equal 0.

    13

  • Holt McDougal Algebra 2

    Adding and Subtracting

    Rational Expressions

    To add or subtract rational expressions with unlike denominators, first find the least common denominator (LCD). The LCD is the least common multiple of the polynomials in the denominators.

  • Holt McDougal Algebra 2

    Adding and Subtracting

    Rational Expressions

    Find the least common multiple for each pair.

    Example 2: Finding the Least Common Multiple of

    Polynomials

    A. 4x2y3 and 6x4y5

    4x2y3 = 2 2 x2 y3

    6x4y5 = 3 2 x4 y5

    The LCM is 2 2 3 x4 y5, or 12x4y5.

    B. x2 2x 3 and x2 x 6

    x2 2x 3 = (x 3)(x + 1)

    x2 x 6 = (x 3)(x + 2)

    The LCM is (x 3)(x + 1)(x + 2).

  • Holt McDougal Algebra 2

    Adding and Subtracting

    Rational Expressions

    Check It Out! Example 2

    Find the least common multiple for each pair.

    a. 4x3y7 and 3x5y4

    4x3y7 = 2 2 x3 y7

    3x5y4 = 3 x5 y4

    The LCM is 2 2 3 x5 y7, or 12x5y7.

    b. x2 4 and x2 + 5x + 6

    x2 4 = (x 2)(x + 2)

    x2 + 5x + 6 = (x + 2)(x + 3)

    The LCM is (x 2)(x + 2)(x + 3).

  • Holt McDougal Algebra 2

    Adding and Subtracting

    Rational Expressions

    To add rational expressions with unlike denominators, rewrite both expressions with the LCD. This process is similar to adding fractions.

  • Holt McDougal Algebra 2

    Adding and Subtracting

    Rational Expressions

    Add. Identify any x-values for which the expression is undefined.

    Example 3A: Adding Rational Expressions

    x 3x2 + 3x 4

    +2x

    x + 4

    x 3(x + 4)(x 1)

    +2x

    x + 4Factor the denominators.

    The LCD is (x + 4)(x 1), so multiply by . 2x

    x + 4

    x 1x 1

    x 3(x + 4)(x 1)

    +2x

    x + 4x 1x 1

  • Holt McDougal Algebra 2

    Adding and Subtracting

    Rational Expressions

    x 3 + 2x(x 1)(x + 4)(x 1)

    Add the numerators.

    2x2 x 3(x + 4)(x 1)

    Simplify the numerator.

    Example 3A Continued

    Add. Identify any x-values for which the expression is undefined.

    Write the sum in

    factored or expanded

    form.

    2x2 x 3(x + 4)(x 1)

    2x2 x 3x2 + 3x 4

    or

    The expression is undefined at x = 4 and x = 1 because these values of x make the factors (x + 4) and (x 1) equal 0.

  • Holt McDougal Algebra 2

    Adding and Subtracting

    Rational Expressions

    Add. Identify any x-values for which the expression is undefined.

    x x + 2

    +8

    x2 4

    x x + 2

    +8

    (x + 2)(x 2)Factor the denominator.

    The LCD is (x + 2)(x 2), so multiply by . x

    x + 2

    x 2x 2

    Example 3B: Adding Rational Expressions

    x 2x 2

    x x + 2

    +8

    (x + 2)(x 2)

    x(x 2) + (8) (x + 2)(x 2)

    Add the numerators.

  • Holt McDougal Algebra 2

    Adding and Subtracting

    Rational Expressions

    x2 2x 8 (x + 2)(x 2)

    Write the numerator in

    standard form.

    Add. Identify any x-values for which the expression is undefined.

    Example 3B Continued

    (x + 2)(x 4)(x + 2)(x 2)

    Factor the numerator.

    x 4x 2

    Divide out common

    factors.

    The expression is undefined at x = 2 and x = 2 because these values of x make the factors (x + 2) and (x 2) equal 0.

  • Holt McDougal Algebra 2

    Adding and Subtracting

    Rational Expressions

    Check It Out! Example 3a

    Add. Identify any x-values for which the expression is undefined.

    3x 2x 2

    +3x 23x 3

    Factor the denominators.

    The LCD is 6(x 1), so

    multiply by 3 and

    by 2.

    3x 2(x 1)

    3x 23(x 1)

    3x 2(x 1)

    +3x 2

    3(x 1)

    33

    3x 2(x 1)

    +3x 2

    3(x 1)22

  • Holt McDougal Algebra 2

    Adding and Subtracting

    Rational Expressions

    Check It Out! Example 3a Continued

    Add. Identify any x-values for which the expression is undefined.

    15x 46(x 1)

    Simplify the numerator.

    The expression is undefined at x = 1 because this value of x make the factor (x 1) equal 0.

    Add the numerators.9x + 6x 4

    6(x 1)

  • Holt McDougal Algebra 2

    Adding and Subtracting

    Rational Expressions

    Add. Identify any x-values for which the expression is undefined.

    2x + 6x2 + 6x + 9

    +x

    x + 3

    Factor the denominators.

    Check It Out! Example 3b

    2x + 6(x + 3)(x + 3)

    +x

    x + 3

    The LCD is (x + 3)(x + 3),

    so multiply by .x

    (x + 3)(x + 3) (x + 3)

    2x + 6(x + 3)(x + 3)

    +x

    x + 3x + 3x + 3

    Add the numerators.x2 + 3x + 2x + 6

    (x + 3)(x + 3)

  • Holt McDougal Algebra 2

    Adding and Subtracting

    Rational Expressions

    Check It Out! Example 3b Continued

    Add. Identify any x-values for which the expression is undefined.

    Write the numerator in

    standard form.

    (x + 3)(x + 2)(x + 3)(x + 3)

    Factor the numerator.

    x + 2x + 3

    Divide out common

    factors.

    The expression is undefined at x = 3 because this value of x make the factors (x + 3) and (x + 3) equal 0.

    x2 + 5x + 6 (x + 3)(x + 3)

  • Holt McDougal Algebra 2

    Adding and Subtracting

    Rational Expressions

    Example 4: Subtracting Rational Expressions

    Factor the denominators.

    Subtract . Identify any x-

    values for which the expression is undefined.

    2x2 30

    x2 9

    x + 3

    x + 5

    2x2 30

    (x 3)(x + 3)

    x + 3

    x + 5

    The LCD is (x 3)(x + 3),

    so multiply by .x + 5 x + 3

    (x 3) (x 3)

    2x2 30

    (x 3)(x + 3)

    x + 3

    x + 5 x 3x 3

    2x2 30 (x + 5)(x 3)(x 3)(x + 3)

    Subtract the numerators.

    2x2 30 (x2 + 2x 15)(x 3)(x + 3)

    Multiply the binomials in

    the numerator.

  • Holt McDougal Algebra 2

    Adding and Subtracting

    Rational Expressions

    Example 4 Continued

    Subtract . Identify any x-

    values for which the expression is undefined.

    2x2 30

    x2 9

    x + 3

    x + 5

    2x2 30 x2 2x + 15(x 3)(x + 3)

    Distribute the negative sign.

    x2 2x 15(x 3)(x + 3)

    Write the numerator in

    st