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Mathematics LibreTexts

8.3: Add and Subtract Rational Expressions with a Common Denominator

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    By the end of this section, you will be able to:
    • Add rational expressions with a common denominator
    • Subtract rational expressions with a common denominator
    • Add and subtract rational expressions whose denominators are opposites

    Before you get started, take this readiness quiz.

    If you miss a problem, go back to the section listed and review the material.

    1. Add: \(\frac{y}{3}+\frac{9}{3}\).
      If you missed this problem, review [link].
    2. Subtract: \(\frac{10}{x}−\frac{2}{x}\).
      If you missed this problem, review [link].
    3. Factor completely: \(8n^5−20n^3\).
      If you missed this problem, review [link].
    4. Factor completely: \(45a^3−5ab^2\).
      If you missed this problem, review [link].

    Add Rational Expressions with a Common Denominator

    What is the first step you take when you add numerical fractions? You check if they have a common denominator. If they do, you add the numerators and place the sum over the common denominator. If they do not have a common denominator, you find one before you add.

    It is the same with rational expressions. To add rational expressions, they must have a common denominator. When the denominators are the same, you add the numerators and place the sum over the common denominator.

    Definition: RATIONAL EXPRESSION ADDITION

    If p, q, and r are polynomials where \(r \ne 0\), then

    \(\frac{p}{r}+\frac{q}{r}=\frac{p+q}{r}\)

    To add rational expressions with a common denominator, add the numerators and place the sum over the common denominator.

    We will add two numerical fractions first, to remind us of how this is done.

    Example \(\PageIndex{1}\)

    Add: \(\frac{5}{18}+\frac{7}{18}\).

    Answer
    \(\frac{5}{18}+\frac{7}{18}\)
    The fractions have a common denominator, so add the numerators and place the sum over the common denominator. \(\frac{5+7}{18}\)
    Add in the numerator. \(\frac{12}{18}\)
    Factor the numerator and denominator to show the common factors. \(\frac{6·2}{6·3}\)
    Simplify. \(\frac{2}{3}\)

    Example \(\PageIndex{2}\)

    Add: \(\frac{7}{16}+\frac{5}{16}\).

    Answer

    \(\frac{3}{4}\)

    Example \(\PageIndex{3}\)

    Add: \(\frac{3}{10}+\frac{1}{10}\).

    Answer

    \(\frac{2}{5}\)

    Remember, we do not allow values that would make the denominator zero. What value of yy should be excluded in the next example?

    Example \(\PageIndex{4}\)

    Add: \(\frac{3y}{4y−3}+\frac{7}{4y−3}\).

    Answer
    \(\frac{3y}{4y−3}+\frac{7}{4y−3}\).
    The fractions have a common denominator, so add the numerators and place the sum over the common denominator. \(\frac{3y+7}{4y−3}\)
    The numerator and denominator cannot be factored. The fraction is simplified.

    Example \(\PageIndex{5}\)

    Add: \(\frac{5x}{2x+3}+\frac{2}{2x+3}\).

    Answer

    \(\frac{5x+2}{2x+3}\).

    Example \(\PageIndex{6}\)

    Add: \(\frac{x}{x−2}+\frac{1}{x−2}\).

    Answer

    \(\frac{x+1}{x−2}\)

    Example \(\PageIndex{7}\)

    Add: \(\frac{7x+12}{x+3}+\frac{x^2}{x+3}\).

    Answer
    \(\frac{7x+12}{x+3}+\frac{x^2}{x+3}\)
    The fractions have a common denominator, so add the numerators and place the sum over the common denominator. \(\frac{7x+12+x^2}{x+3}\)
    Write the degrees in descending order. \(\frac{x^2+7x+12}{x+3}\)
    Factor the numerator. \(\frac{(x+3)(x+4)}{x+3}\)
    Simplify. x+4

    Example \(\PageIndex{8}\)

    Add: \(\frac{9x+14}{x+7}+\frac{x^2}{x+7}\).

    Answer

    x+2

    Example \(\PageIndex{9}\)

    Add: \(\frac{x^2+8x}{x+5}+\frac{15}{x+5}\).

    Answer

    x+3

    Subtract Rational Expressions with a Common Denominator

    To subtract rational expressions, they must also have a common denominator. When the denominators are the same, you subtract the numerators and place the difference over the common denominator.

    Definition: RATIONAL EXPRESSION SUBTRACTION

    If p, q, and r are polynomials where \(r \ne 0\)

    \(\frac{p}{r}−\frac{q}{r}=\frac{p−q}{r}\)

    To subtract rational expressions, subtract the numerators and place the difference over the common denominator.

    We always simplify rational expressions. Be sure to factor, if possible, after you subtract the numerators so you can identify any common factors.

    Example \(\PageIndex{10}\)

    Subtract: \(\frac{n^2}{n−10}−\frac{100}{n−10}\).

    Answer
    \(\frac{n^2}{n−10}−\frac{100}{n−10}\)
    The fractions have a common denominator, so add the numerators and place the sum over the common denominator. \(\frac{n^2−100}{n−10}\)
    Factor the numerator. \(\frac{(n−10)(n+10)}{n−10}\)
    Simplify. n+10

    Example \(\PageIndex{11}\)

    Subtract: \(\frac{x^2}{x+3}−\frac{9}{x+3}\).

    Answer

    x−3

    Example \(\PageIndex{12}\)

    Subtract: \(\frac{4x^2}{2x−5}−\frac{25}{2x−5}\).

    Answer

    2x+5

    ​​​​​​​Be careful of the signs when you subtract a binomial!​​​​​​​

    Example \(\PageIndex{13}\)

    Subtract: \(\frac{y^2}{y−6}−\frac{2y+24}{y−6}\).

    Answer
    \(\frac{y^2}{y−6}−\frac{2y+24}{y−6}\)
    The fractions have a common denominator, so add the numerators and place the sum over the common denominator. \(\frac{y^2−(2y+24)}{y−6}\)
    Distribute the sign in the numerator. \(\frac{y^2−2y−24}{y−6}\)
    Factor the numerator. \(\frac{(y−6)(y+4)}{y−6}\)
    Simplify. y+4

    Example \(\PageIndex{14}\)

    Subtract: \(\frac{n^2}{n−4}−\frac{n+12}{n−4}\).

    Answer

    n+3

    Example \(\PageIndex{15}\)

    Subtract: \(\frac{y^2}{y−1}−\frac{9y−8}{y−1}\).

    Answer

    y−8

    Example \(\PageIndex{16}\)

    Subtract: \(\frac{5x^2−7x+3}{x^2−3x+18}−\frac{4x^2+x−9}{x^2−3x+18}\).

    Answer
    \(\frac{5x^2−7x+3}{x^2−3x+18}−\frac{4x^2+x−9}{x^2−3x+18}\)
    The fractions have a common denominator, so add the numerators and place the sum over the common denominator. \(\frac{5x^2−7x+3−(4x^2+x−9)}{x^2−3x+18}\)
    Distribute the sign in the numerator. \(\frac{5x^2−7x+3−4x^2−x+9}{x^2−3x+18}\)
    Combine like terms. \(\frac{x^2−8x+12}{x^2−3x+18}\)
    Factor the numerator and the denominator. \(\frac{(x−2)(x−6)}{(x+3)(x−6)}\)
    Simplify. \(\frac{x−2}{x+3}\)

    Example \(\PageIndex{17}\)

    Subtract: \(\frac{4x^2−11x+8}{x^2−3x+2}−\frac{3x^2+x−3}{x^2−3x+2}\).

    Answer

    \(\frac{x−11}{x−2}\)

    Example \(\PageIndex{18}\)

    Subtract: \(\frac{6x^2−x+20}{x^2−81}−\frac{5x^2+11x−7}{x^2−81}\).

    Answer

    \(\frac{x−3}{x+9}\)

    Add and Subtract Rational Expressions whose Denominators are Opposites

    When the denominators of two rational expressions are opposites, it is easy to get a common denominator. We just have to multiply one of the fractions by \(\frac{−1}{−1}\)

    Let’s see how this works.

    .
    Multiply the second fraction by \(\frac{−1}{−1}\). .
    The denominators are the same. .
    Simplify. .

    Example \(\PageIndex{19}\)

    Add: \(\frac{4u−1}{3u−1}+\frac{u}{1−3u}\).

    Answer
    .
    Multiply the second fraction by \(\frac{−1}{−1}\). .
    Simplify the second fraction. .
    The denominators are the same. Add the numerators. .
    Simplify. .
    Simplify. .

    Example \(\PageIndex{20}\)

    Add: \(\frac{8x−15}{2x−5}+\frac{2x}{5−2x}\).

    Answer

    3

    Example \(\PageIndex{21}\)

    Add: \(\frac{6y^2+7y−10}{4y−7}+\frac{2y^2+2y+11}{7−4y}\).

    Answer

    y+3

    Example \(\PageIndex{22}\)

    Subtract: \(\frac{m^2−6m}{m^2−1}−\frac{3m+2}{1−m^2}\).

    Answer
    .
    Multiply the second fraction by \(\frac{−1}{−1}\). .
    Simplify the second fraction. .
    The denominators are the same. Subtract the numerators. .
    Distribute. m2−6m+3m+2m2−1
    Combine like terms. .
    Factor the numerator and denominator. .
    Simplify by removing common factors. .
    Simplify. .

    Example \(\PageIndex{23}\)

    Subtract: \(\frac{y^2−5y}{y^2−4}−\frac{6y−6}{4−y^2}\).

    Answer

    \(\frac{y+3}{y+2}\)

    Example \(\PageIndex{24}\)

    Subtract: \(\frac{2n^2+8n−1}{n^2−1}−\frac{n^2−7n−1}{1−n^2}\).

    Answer

    \(\frac{3n−2}{n−1}\)

    Key Concepts

    • Rational Expression Addition

      • If p, q, and r are polynomials where \(r \ne 0\), then

        \(\frac{p}{r}+\frac{q}{r}=\frac{p+q}{r}\)

      • To add rational expressions with a common denominator, add the numerators and place the sum over the common denominator.
    • Rational Expression Subtraction

      • If p, q, and r are polynomials where \(r \ne 0\)

        \(\frac{p}{r}−\frac{q}{r}=\frac{p−q}{r}\)​​​​​​​

      • To subtract rational expressions, subtract the numerators and place the difference over the common denominator.

    Practice Makes Perfect

    Add Rational Expressions with a Common Denominator

    In the following exercises, add.

    Example \(\PageIndex{25}\)

    \(\frac{2}{15}+\frac{7}{15}\)

    Answer

    \(\frac{3}{5}\)

    Example \(\PageIndex{26}\)

    \(\frac{4}{21}+\frac{3}{21}\)​​​​​​​

    Example \(\PageIndex{27}\)

    \(\frac{7}{24}+\frac{11}{24}\)

    Answer

    \(\frac{3}{4}\)

    Example \(\PageIndex{28}\)

    \(\frac{7}{36}+\frac{13}{36}\)

    Example \(\PageIndex{29}\)

    \(\frac{3a}{a−b}+\frac{1}{a−b}\)

    Answer

    \(\frac{3a+1}{a+b}\)

    Example \(\PageIndex{30}\)

    \(\frac{3c}{4c−5}+\frac{5}{4c−5}\)

    Example \(\PageIndex{31}\)

    \(\frac{d}{d+8}+\frac{5}{d+8}\)

    Answer

    \(\frac{d+5}{d+8}\)

    Example \(\PageIndex{32}\)

    \(\frac{7m}{2m+n}+\frac{4}{2m+n}\)

    Example \(\PageIndex{33}\)

    \(\frac{p^2+10p}{p+2}+\frac{16}{p+2}\)

    Answer

    p+8

    Example \(\PageIndex{34}\)

    \(\frac{q^2+12q}{q+3}+\frac{27}{q+3}\)

    Example \(\PageIndex{35}\)

    \(\frac{2r^2}{2r−1}+\frac{15r−8}{2r−1}\)

    Answer

    r+8​​​​​​​

    Example \(\PageIndex{36}\)

    \(\frac{3s^2}{3s−2}+\frac{13s−10}{3s−2}\)

    Example \(\PageIndex{37}\)

    \(\frac{8t^2}{t+4}+\frac{32t}{t+4}\)

    Answer

    8t

    Example \(\PageIndex{38}\)

    \(\frac{6v^2}{v+5}+\frac{30v}{v+5}\)

    Example \(\PageIndex{39}\)

    \(\frac{2w^2}{w^2−16}+\frac{8w}{w^2−16}\)

    Answer

    \(\frac{2w}{w−4}\)

    Example \(\PageIndex{40}\)

    \(\frac{7x^2}{x^2−9}+\frac{21x}{x^2−9}\)

    ​​​​​​​Subtract Rational Expressions with a Common Denominator

    In the following exercises, subtract.

    Example \(\PageIndex{41}\)

    \(\frac{y^2}{y+8}−\frac{64}{y+8}\)

    Answer

    y−8​​​​​​​

    Example \(\PageIndex{42}\)

    \(\frac{z^2}{z+2}−\frac{4}{z+2}\)​​​​​​​

    Example \(\PageIndex{43}\)

    \(\frac{9a^2}{3a−7}−\frac{49}{3a−7}\)

    Answer

    3a+7

    Example \(\PageIndex{44}\)

    \(\frac{25b^2}{5b−6}−\frac{36}{5b−6}\)

    Example \(\PageIndex{45}\)

    \(\frac{c^2}{c−8}−\frac{6c+16}{c−8}\)

    Answer

    c+2

    Example \(\PageIndex{46}\)

    \(\frac{d^2}{d−9}−\frac{6d+27}{d−9}\)

    Example \(\PageIndex{47}\)

    \(\frac{3m^2}{6m−30}−\frac{21m−30}{6m−30}\)

    Answer

    \(\frac{m−2}{3}\)

    Example \(\PageIndex{48}\)

    \(\frac{2n^2}{4n−32}−\frac{30n−16}{4n−32}\)

    Example \(\PageIndex{49}\)

    \(\frac{6p^2+3p+4}{p^2+4p−5}−\frac{5p^2+p+7}{p^2+4p−5}\)

    Answer

    \(\frac{p+3}{p+5}\)

    Example \(\PageIndex{50}\)

    \(\frac{5q^2+3q−9}{q^2+6q+8}−\frac{4q^2+9q+7}{q^2+6q+8}\)

    Example \(\PageIndex{51}\)

    \(\frac{5r^2+7r−33}{r^2−49}−\frac{4r^2−5r−30}{r^2−49}\)

    Answer

    \(\frac{r+9}{r+7}\)

    Example \(\PageIndex{52}\)

    \(\frac{7t^2−t−4}{t^2−25}−\frac{6t^2+2t−1}{t^2−25}\)

    ​​​​​​​Add and Subtract Rational Expressions whose Denominators are Opposites

    In the following exercises, add.

    Example \(\PageIndex{53}\)

    \(\frac{10v^2}{v−1}+\frac{2v+4}{1−2v}\)

    Answer

    4​​​​​​​

    Example \(\PageIndex{54}\)

    \(\frac{20w}{5w−2}+\frac{5w+6}{2−5w}\)

    Example \(\PageIndex{55}\)

    \(\frac{10x^2+16x−7}{8x−3}+\frac{2x^2+3x−1}{3−8x}\)

    Answer

    x+2

    Example \(\PageIndex{56}\)

    \(\frac{6y^2+2y−11}{3y−7}+\frac{3y^2−3y+17}{7−3y}\)

    ​​​​​​​In the following exercises, subtract.

    Example \(\PageIndex{57}\)

    \(\frac{z^2+6z}{z^2−25}−\frac{3z+20}{25−z^2}\)

    Answer

    \(\frac{z+4}{z−5}\)

    Example \(\PageIndex{58}\)

    \(\frac{a^2+3a}{a^2−9}−\frac{3a−27}{9−a^2}\)

    Example \(\PageIndex{59}\)

    \(\frac{2b^2+30b−13}{b^2−49}−\frac{2b^2−5b−8}{49−b^2}\)

    Answer

    \(\frac{4b−3}{b−7}\)

    Example \(\PageIndex{60}\)

    \(\frac{c^2+5c−10}{c^2−16}−\frac{c^2−8c−10}{16−c^2}\)​​​​​​​

    Everyday Math

    Example \(\PageIndex{61}\)

    Sarah ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. If rr represents Sarah’s speed when she ran, then her running time is modeled by the expression \(\frac{8}{r}\) and her biking time is modeled by the expression \(\frac{24}{r+4}\). Add the rational expressions \(\frac{8}{r}+\frac{24}{r+4}\) to get an expression for the total amount of time Sarah ran and biked.

    Answer

    \(\frac{32r+32}{r(r+4)}\)

    Example \(\PageIndex{62}\)

    If Pete can paint a wall in pp hours, then in one hour he can paint \(\frac{1}{p}\) of the wall. It would take Penelope 3 hours longer than Pete to paint the wall, so in one hour she can paint \(\frac{1}{p+3}\) of the wall. Add the rational expressions \(\frac{1}{p}+\frac{1}{p+3}\) to get an expression for the part of the wall Pete and Penelope would paint in one hour if they worked together.

    Writing Exercises

    Example \(\PageIndex{63}\)

    Donald thinks that \(\frac{3}{x}+\frac{4}{x}\) is \(\frac{7}{2x}\). Is Donald correct? Explain.

    Example \(\PageIndex{64}\)

    Explain how you find the Least Common Denominator of \(x^2+5x+4\) and \(x^2−16\).​​​​​​​

    Self Check

    ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    The above image is a table with four columns and four rows. The first row is the header row. The first header is labeled “I can…”, the second “Confidently”, the third, “With some help”, and the fourth “No – I don’t get it!”. In the first column under “I can”, the next row reads “add rational expressions with a common denominator.”, the next row reads “subtract rational expressions with a common denominator.”, the next row reads, “add and subtract rational expressions whose denominators are opposites.”, the last row reads “What does this checklist tell you about your mastery of this section? What steps will you take to improve?” The remaining columns are blank.

    ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?