Analysis of Functions: Rational Expressions

Rational Expressions and Domain Restrictions

  • Definition of a Rational Expression: A rational expression is defined as the quotient of two polynomials.

  • Definition of Domain: The set of real numbers for which an expression is defined is called the domain of the expression.

  • Domain Restriction Principle: Division by zero is undefined. Consequently, any real number that causes the denominator of a rational expression to equal zero must be excluded from its domain.

  • Domain Exclusion Examples:

    • Example 1: Finding excluded numbers for 7x+5\frac{7}{x+5}

      • Set denominator equal to zero: x+5=0x + 5 = 0

      • Solve for xx: x=−5x = -5

      • Excluded value: x=−5x = -5

    • Example 2: Finding excluded numbers for xx2−36\frac{x}{x^2-36}

      • Set denominator equal to zero: x2−36=0x^2 - 36 = 0

      • Factor denominator: (x−6)(x+6)=0(x - 6)(x + 6) = 0

      • Solve for xx: x=6x = 6 or x=−6x = -6

      • Excluded values: x=6x = 6 and x=−6x = -6

    • Example 3: Finding excluded numbers for x4\frac{x}{4}

      • The denominator is the constant 44, which is never equal to zero.

      • Excluded values: None (the domain consists of all real numbers).

Simplifying Rational Expressions

  • Procedure for Simplifying Rational Expressions:

    1. Factor the numerator and the denominator completely.

    2. Divide both the numerator and the denominator by any common factors.

  • Numerical Simplification Examples:

    • Example 1: Simplifying 2035\frac{20}{35}

      • Factor numerator and denominator: 4×57×5\frac{4 \times 5}{7 \times 5}

      • Divide out common factor 55: 47\frac{4}{7}

  • Algebraic Simplification Examples:

    • Example 4: Simplifying x3+3x2x+3\frac{x^3 + 3x^2}{x + 3}

      • Factor numerator: x2(x+3)x^2(x + 3)

      • Divide by common factor (x+3)(x + 3): x2x^2

    • Example 5: Simplifying x2−1x2+2x+1\frac{x^2 - 1}{x^2 + 2x + 1}

      • Factor numerator and denominator: (x−1)(x+1)(x+1)2\frac{(x - 1)(x + 1)}{(x + 1)^2}

      • Divide by common factor (x+1)(x + 1): x−1x+1\frac{x - 1}{x + 1}

    • Example 6: Simplifying 3y36y4−3y3\frac{3y^3}{6y^4 - 3y^3}

      • Factor denominator: 3y3(2y−1)3y^3(2y - 1)

      • Divide by common factor 3y33y^3: 12y−1\frac{1}{2y - 1}

    • Example 7: Simplifying 5+xx+5\frac{5 + x}{x + 5}

      • Reorder numerator addition: 5+x=x+55 + x = x + 5

      • Divide out common factor (x+5)(x + 5): 11

    • Example 8: Simplifying 5−xx−5\frac{5 - x}{x - 5}

      • Factor −1-1 from numerator: −1(x−5)x−5\frac{-1(x - 5)}{x - 5}

      • Divide out common factor (x−5)(x - 5): −1-1

    • Example 9: Simplifying 3−3x2x2+x−2\frac{3 - 3x^2}{x^2 + x - 2}

      • Factor numerator: 3(1−x2)=3(1−x)(1+x)=−3(x−1)(x+1)3(1 - x^2) = 3(1 - x)(1 + x) = -3(x - 1)(x + 1)

      • Factor denominator: (x+2)(x−1)(x + 2)(x - 1)

      • Divide out common factor (x−1)(x - 1): −3(x+1)x+2\frac{-3(x + 1)}{x + 2}

    • Example 10: Simplifying x3+27x+3\frac{x^3 + 27}{x + 3}

      • Factor numerator using sum of cubes formula a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2): (x+3)(x2−3x+9)(x + 3)(x^2 - 3x + 9)

      • Divide out common factor (x+3)(x + 3): x2−3x+9x^2 - 3x + 9

    • Example 11: Simplifying 3x2+6x3−3x2+2x−6\frac{3x^2 + 6}{x^3 - 3x^2 + 2x - 6}

      • Factor numerator: 3(x2+2)3(x^2 + 2)

      • Factor denominator by grouping: x2(x−3)+2(x−3)=(x2+2)(x−3)x^2(x - 3) + 2(x - 3) = (x^2 + 2)(x - 3)

      • Divide out common factor (x2+2)(x^2 + 2): 3x−3\frac{3}{x - 3}

Multiplying Rational Expressions

  • Procedure for Multiplying Rational Expressions:

    1. Factor all numerators and denominators completely.

    2. Divide numerators and denominators by common factors.

    3. Multiply the remaining factors in the numerators and in the denominators.

  • Numerical Multiplication Example:

    • Multiplying 67×83\frac{6}{7} \times \frac{8}{3}

      • Divide numerator 66 and denominator 33 by 33: 27×81\frac{2}{7} \times \frac{8}{1}

      • Multiply remaining terms: 167\frac{16}{7}

  • Algebraic Multiplication Examples:

    • Example 12: Multiplying x+3x2−x−6⋅x2−4x2+6x+9\frac{x + 3}{x^2 - x - 6} \cdot \frac{x^2 - 4}{x^2 + 6x + 9}

      • Factor all terms: x+3(x−3)(x+2)⋅(x−2)(x+2)(x+3)2\frac{x + 3}{(x - 3)(x + 2)} \cdot \frac{(x - 2)(x + 2)}{(x + 3)^2}

      • Cancel common factors (x+3)(x + 3) and (x+2)(x + 2): x−2(x−3)(x+3)\frac{x - 2}{(x - 3)(x + 3)} or x−2x2−9\frac{x - 2}{x^2 - 9}

    • Example 13: Multiplying x3−115x2⋅−3x+3x2+x+1\frac{x^3 - 1}{15x^2} \cdot \frac{-3x + 3}{x^2 + x + 1}

      • Factor difference of cubes in numerator: x3−1=(x−1)(x2+x+1)x^3 - 1 = (x - 1)(x^2 + x + 1)

      • Factor second numerator: −3x+3=−3(x−1)-3x + 3 = -3(x - 1)

      • Set up product: (x−1)(x2+x+1)15x2⋅−3(x−1)x2+x+1\frac{(x - 1)(x^2 + x + 1)}{15x^2} \cdot \frac{-3(x - 1)}{x^2 + x + 1}

      • Cancel (x2+x+1)(x^2 + x + 1) and reduce −315\frac{-3}{15} to −15\frac{-1}{5}: −(x−1)25x2\frac{-(x - 1)^2}{5x^2}

Dividing Rational Expressions

  • Division Rule:     PQ÷RS=PQ⋅SR\frac{P}{Q} \div \frac{R}{S} = \frac{P}{Q} \cdot \frac{S}{R}

  • Verbatim Definition: The quotient of two rational expressions is the product of the first and the reciprocal of the second.

  • Algebraic Division Examples:

    • Example 14: Dividing x2−2x+1x3+x÷x2+x−23x2+3\frac{x^2 - 2x + 1}{x^3 + x} \div \frac{x^2 + x - 2}{3x^2 + 3}

      • Multiply by reciprocal: x2−2x+1x3+x⋅3x2+3x2+x−2\frac{x^2 - 2x + 1}{x^3 + x} \cdot \frac{3x^2 + 3}{x^2 + x - 2}

      • Factor terms completely: (x−1)2x(x2+1)⋅3(x2+1)(x+2)(x−1)\frac{(x - 1)^2}{x(x^2 + 1)} \cdot \frac{3(x^2 + 1)}{(x + 2)(x - 1)}

      • Cancel common factors (x−1)(x - 1) and (x2+1)(x^2 + 1): 3(x−1)x(x+2)\frac{3(x - 1)}{x(x + 2)}

    • Example 15: Dividing 12y35y2−5÷61−y\frac{12y^3}{5y^2 - 5} \div \frac{6}{1 - y}

      • Multiply by reciprocal: 12y35y2−5⋅1−y6\frac{12y^3}{5y^2 - 5} \cdot \frac{1 - y}{6}

      • Factor denominator: 5y2−5=5(y−1)(y+1)5y^2 - 5 = 5(y - 1)(y + 1)

      • Rewrite numerator term: (1−y)=−(y−1)(1 - y) = -(y - 1)

      • Substitute and cancel factors: 12y35(y−1)(y+1)⋅−(y−1)6=−2y35(y+1)\frac{12y^3}{5(y - 1)(y + 1)} \cdot \frac{-(y - 1)}{6} = \frac{-2y^3}{5(y + 1)}

  • Combined Operations Example:

    • Example 16: Performing operations for (x+3)2x2−9⋅2x−65x÷x2+7x+12x\frac{(x+3)^2}{x^2-9} \cdot \frac{2x-6}{5x} \div \frac{x^2+7x+12}{x}

      • Convert division to multiplication by reciprocal: (x+3)2x2−9⋅2x−65x⋅xx2+7x+12\frac{(x+3)^2}{x^2-9} \cdot \frac{2x-6}{5x} \cdot \frac{x}{x^2+7x+12}

      • Factor completely: (x+3)2(x−3)(x+3)⋅2(x−3)5x⋅x(x+3)(x+4)\frac{(x+3)^2}{(x-3)(x+3)} \cdot \frac{2(x-3)}{5x} \cdot \frac{x}{(x+3)(x+4)}

      • Cancel all common factors (x+3)2(x+3)^2, (x−3)(x-3), and xx: 25(x+4)\frac{2}{5(x+4)}

Adding and Subtracting Rational Expressions with the Same Denominator

  • General Rule: To add or subtract rational expressions with the same denominator, combine the numerators and keep the common denominator unchanged.

  • Examples:

    • Example 17: Subtracting x2−4xx2−x−6−x−6x2−x−6\frac{x^2 - 4x}{x^2 - x - 6} - \frac{x - 6}{x^2 - x - 6}

      • Combine numerators: (x2−4x)−(x−6)x2−x−6=x2−5x+6x2−x−6\frac{(x^2 - 4x) - (x - 6)}{x^2 - x - 6} = \frac{x^2 - 5x + 6}{x^2 - x - 6}

      • Factor numerator and denominator: (x−2)(x−3)(x−3)(x+2)\frac{(x - 2)(x - 3)}{(x - 3)(x + 2)}

      • Divide out common factor (x−3)(x - 3): x−2x+2\frac{x - 2}{x + 2}

    • Example 18: Subtracting xx+1−3x+2x+1\frac{x}{x+1} - \frac{3x+2}{x+1}

      • Combine numerators: x−(3x+2)x+1=−2x−2x+1\frac{x - (3x + 2)}{x + 1} = \frac{-2x - 2}{x + 1}

      • Factor numerator: −2(x+1)x+1\frac{-2(x + 1)}{x + 1}

      • Divide out common factor (x+1)(x + 1): −2-2

Adding and Subtracting Rational Expressions with Different Denominators

  • Procedure for Finding the Least Common Denominator (LCD):

    1. Factor each denominator completely.

    2. The LCD is the product of all unique factors, each raised to a power equal to the greatest number of times that the factor appears in any one factored denominator.

  • Example 19 (Finding LCD): Find the LCD of 3x2−6x+9\frac{3}{x^2 - 6x + 9} and 7x2−9\frac{7}{x^2 - 9}

    • Factor denominators: x2−6x+9=(x−3)2x^2 - 6x + 9 = (x - 3)^2 and x2−9=(x−3)(x+3)x^2 - 9 = (x - 3)(x + 3)

    • Identify unique factors with highest powers: (x−3)2(x - 3)^2 and (x+3)(x + 3)

    • LCD=(x−3)2(x+3)\text{LCD} = (x - 3)^2(x + 3)

  • Addition and Subtraction Examples:

    • Example 20: Subtracting 3x+1−3x\frac{3}{x+1} - \frac{3}{x}

      • LCD=x(x+1)\text{LCD} = x(x + 1)

      • Convert fractions: 3xx(x+1)−3(x+1)x(x+1)=3x−3x−3x(x+1)=−3x(x+1)\frac{3x}{x(x + 1)} - \frac{3(x + 1)}{x(x + 1)} = \frac{3x - 3x - 3}{x(x + 1)} = \frac{-3}{x(x + 1)}

    • Example 21: Adding x+5x−5+x−5x+5\frac{x+5}{x-5} + \frac{x-5}{x+5}

      • LCD=(x−5)(x+5)\text{LCD} = (x - 5)(x + 5)

      • Convert fractions: (x+5)2+(x−5)2(x−5)(x+5)\frac{(x + 5)^2 + (x - 5)^2}{(x - 5)(x + 5)}

      • Expand numerators: (x2+10x+25)+(x2−10x+25)(x−5)(x+5)=2x2+50(x−5)(x+5)\frac{(x^2 + 10x + 25) + (x^2 - 10x + 25)}{(x - 5)(x + 5)} = \frac{2x^2 + 50}{(x - 5)(x + 5)}

    • Example 22: Adding 32x+4+23x+6\frac{3}{2x+4} + \frac{2}{3x+6}

      • Factor denominators: 2(x+2)2(x + 2) and 3(x+2)3(x + 2)

      • LCD=6(x+2)\text{LCD} = 6(x + 2)

      • Convert fractions: 3(3)6(x+2)+2(2)6(x+2)=9+46(x+2)=136(x+2)\frac{3(3)}{6(x + 2)} + \frac{2(2)}{6(x + 2)} = \frac{9 + 4}{6(x + 2)} = \frac{13}{6(x + 2)}

    • Example 23: Subtracting xx2−2x−24−xx2−7x+6\frac{x}{x^2-2x-24} - \frac{x}{x^2-7x+6}

      • Factor denominators: (x−6)(x+4)(x - 6)(x + 4) and (x−6)(x−1)(x - 6)(x - 1)

      • LCD=(x−6)(x+4)(x−1)\text{LCD} = (x - 6)(x + 4)(x - 1)

      • Convert and combine numerators: x(x−1)−x(x+4)(x−6)(x+4)(x−1)=x2−x−x2−4x(x−6)(x+4)(x−1)=−5x(x−6)(x+4)(x−1)\frac{x(x - 1) - x(x + 4)}{(x - 6)(x + 4)(x - 1)} = \frac{x^2 - x - x^2 - 4x}{(x - 6)(x + 4)(x - 1)} = \frac{-5x}{(x - 6)(x + 4)(x - 1)}

    • Example 24: Performing operations for 6x2+17x−40x2+x−20−3x−4+5xx+5\frac{6x^2+17x-40}{x^2+x-20} - \frac{3}{x-4} + \frac{5x}{x+5}

      • Factor first denominator: x2+x−20=(x+5)(x−4)x^2 + x - 20 = (x + 5)(x - 4)

      • LCD=(x+5)(x−4)\text{LCD} = (x + 5)(x - 4)

      • Combine numerators: (6x2+17x−40)−3(x+5)+5x(x−4)(x+5)(x−4)\frac{(6x^2 + 17x - 40) - 3(x + 5) + 5x(x - 4)}{(x + 5)(x - 4)}

      • Expand: 6x2+17x−40−3x−15+5x2−20x(x+5)(x−4)\frac{6x^2 + 17x - 40 - 3x - 15 + 5x^2 - 20x}{(x + 5)(x - 4)}

      • Simplify numerator: 11x2−6x−55(x+5)(x−4)\frac{11x^2 - 6x - 55}{(x + 5)(x - 4)}

    • Example 25: Subtracting expressions with opposite sign denominators 3m−26m−6−86−m\frac{3m-26}{m-6} - \frac{8}{6-m}

      • Note that 6−m=−(m−6)6 - m = -(m - 6)

      • Change sign of second term: 3m−26m−6+8m−6\frac{3m - 26}{m - 6} + \frac{8}{m - 6}

      • Combine numerators: 3m−26+8m−6=3m−18m−6\frac{3m - 26 + 8}{m - 6} = \frac{3m - 18}{m - 6}

      • Factor numerator and simplify: 3(m−6)m−6=3\frac{3(m - 6)}{m - 6} = 3

Simplifying Complex Rational Expressions

  • Definition: A complex rational expression (complex fraction) contains one or more rational expressions in its numerator, denominator, or both.

  • Examples:

    • Example 26: Simplifying 1x+321x−13\frac{\frac{1}{x} + \frac{3}{2}}{\frac{1}{x} - \frac{1}{3}}

      • Multiply numerator and denominator by overall LCD 6x6x:

      • Numerator: 6x(1x+32)=6+9x6x\left(\frac{1}{x} + \frac{3}{2}\right) = 6 + 9x

      • Denominator: 6x(1x−13)=6−2x6x\left(\frac{1}{x} - \frac{1}{3}\right) = 6 - 2x

      • Result: 6+9x6−2x\frac{6 + 9x}{6 - 2x}

    • Example 27: Simplifying 1x+7−17x\frac{\frac{1}{x+7} - \frac{1}{7}}{x}

      • Simplify numerator: 7−(x+7)7(x+7)=−x7(x+7)\frac{7 - (x + 7)}{7(x + 7)} = \frac{-x}{7(x + 7)}

      • Divide by xx: −x7(x+7)⋅1x=−17(x+7)\frac{-x}{7(x + 7)} \cdot \frac{1}{x} = \frac{-1}{7(x + 7)}

    • Example 28: Simplifying 3x−2−4x+27x2−4\frac{\frac{3}{x-2} - \frac{4}{x+2}}{\frac{7}{x^2-4}}

      • Multiply top and bottom by (x2−4)=(x−2)(x+2)(x^2 - 4) = (x - 2)(x + 2):

      • Numerator: 3(x+2)−4(x−2)=3x+6−4x+8=−x+143(x + 2) - 4(x - 2) = 3x + 6 - 4x + 8 = -x + 14

      • Denominator: 77

      • Result: −x+147\frac{-x + 14}{7}

    • Example 29: Simplifying 1(x+h)2−1x2h\frac{\frac{1}{(x+h)^2} - \frac{1}{x^2}}{h}

      • Combine numerator fractions using common denominator x2(x+h)2x^2(x + h)^2: x2−(x+h)2x2(x+h)2\frac{x^2 - (x + h)^2}{x^2(x + h)^2}

      • Expand numerator: x2−(x2+2xh+h2)=−2xh−h2=−h(2x+h)x^2 - (x^2 + 2xh + h^2) = -2xh - h^2 = -h(2x + h)

      • Divide by hh: −h(2x+h)x2(x+h)2⋅h=−(2x+h)x2(x+h)2\frac{-h(2x + h)}{x^2(x + h)^2 \cdot h} = \frac{-(2x + h)}{x^2(x + h)^2}

    • Example 30: Simplifying 1+3x+25x−1\frac{1 + \frac{3}{x+2}}{\frac{5}{x-1}}

      • Combine numerator: x+2+3x+2=x+5x+2\frac{x + 2 + 3}{x + 2} = \frac{x + 5}{x + 2}

      • Multiply by reciprocal of denominator: x+5x+2⋅x−15=(x+5)(x−1)5(x+2)\frac{x + 5}{x + 2} \cdot \frac{x - 1}{5} = \frac{(x + 5)(x - 1)}{5(x + 2)}

    • Example 31: Simplifying y−1−(y+2)−15\frac{y^{-1} - (y+2)^{-1}}{5}

      • Rewrite using positive exponents: 1y−1y+25\frac{\frac{1}{y} - \frac{1}{y+2}}{5}

      • Simplify numerator: (y+2)−yy(y+2)=2y(y+2)\frac{(y + 2) - y}{y(y + 2)} = \frac{2}{y(y + 2)}

      • Divide by 55: 25y(y+2)\frac{2}{5y(y + 2)}

Practice Exercises

  • Example 32: Simplifying 1x+h−1xh\frac{\frac{1}{x+h} - \frac{1}{x}}{h}

    • Combine numerator fractions: x−(x+h)x(x+h)=−hx(x+h)\frac{x - (x + h)}{x(x + h)} = \frac{-h}{x(x + h)}

    • Divide by hh: −hx(x+h)⋅h=−1x(x+h)\frac{-h}{x(x + h) \cdot h} = \frac{-1}{x(x + h)}

  • Example 33: Simplifying y−1−(y+5)−15\frac{y^{-1} - (y+5)^{-1}}{5}

    • Rewrite with positive exponents: 1y−1y+55\frac{\frac{1}{y} - \frac{1}{y+5}}{5}

    • Combine numerator fractions: (y+5)−yy(y+5)=5y(y+5)\frac{(y + 5) - y}{y(y + 5)} = \frac{5}{y(y + 5)}

    • Divide by 55: 55y(y+5)=1y(y+5)\frac{5}{5y(y + 5)} = \frac{1}{y(y + 5)}

  • Example 34: Simplifying 2a−1+(2a)−1a−1+2a−2\frac{2a^{-1} + (2a)^{-1}}{a^{-1} + 2a^{-2}}

    • Rewrite all terms as fractions:

      • 2a−1=2a2a^{-1} = \frac{2}{a}

      • (2a)−1=12a(2a)^{-1} = \frac{1}{2a}

      • a−1=1aa^{-1} = \frac{1}{a}

      • 2a−2=2a22a^{-2} = \frac{2}{a^2}

    • Substitute fractions into expression: 2a+12a1a+2a2\frac{\frac{2}{a} + \frac{1}{2a}}{\frac{1}{a} + \frac{2}{a^2}}

    • Simplify numerator: 4+12a=52a\frac{4 + 1}{2a} = \frac{5}{2a}

    • Simplify denominator: a+2a2\frac{a + 2}{a^2}

    • Divide numerator by denominator: 52a⋅a2a+2=5a2(a+2)\frac{5}{2a} \cdot \frac{a^2}{a + 2} = \frac{5a}{2(a + 2)}