Linear Factors
Learning Objectives
- Decompose a rational expression with distinct linear factors
- Decompose a rational expression with repeated linear factors
[latex]\frac{2}{x - 3}+\frac{-1}{x+2}[/latex]
We would first need to find a common denominator,[latex]\left(x+2\right)\left(x - 3\right)[/latex].
Next, we would write each expression with this common denominator and find the sum of the terms.[latex]\begin{array}{l}\frac{2}{x - 3}\left(\frac{x+2}{x+2}\right)+\frac{-1}{x+2}\left(\frac{x - 3}{x - 3}\right)=\hfill \\ \text{ }\frac{2x+4-x+3}{\left(x+2\right)\left(x - 3\right)}=\frac{x+7}{{x}^{2}-x - 6}\hfill \end{array}[/latex]
Partial fraction decomposition is the reverse of this procedure. We would start with the solution and rewrite (decompose) it as the sum of two fractions.[latex]\underset{\begin{array}{l}\\ \text{Simplified sum}\end{array}}{\frac{x+7}{{x}^{2}-x - 6}}=\underset{\begin{array}{l}\\ \text{Partial fraction decomposition}\end{array}}{\frac{2}{x - 3}+\frac{-1}{x+2}}[/latex]
We will investigate rational expressions with linear factors and quadratic factors in the denominator where the degree of the numerator is less than the degree of the denominator. Regardless of the type of expression we are decomposing, the first and most important thing to do is factor the denominator. When the denominator of the simplified expression contains distinct linear factors, it is likely that each of the original rational expressions, which were added or subtracted, had one of the linear factors as the denominator. In other words, using the example above, the factors of [latex]{x}^{2}-x - 6[/latex] are [latex]\left(x - 3\right)\left(x+2\right)[/latex], the denominators of the decomposed rational expression. So we will rewrite the simplified form as the sum of individual fractions and use a variable for each numerator. Then, we will solve for each numerator using one of several methods available for partial fraction decomposition.A General Note: Partial Fraction Decomposition of [latex]\frac{P\left(x\right)}{Q\left(x\right)}:Q\left(x\right)[/latex] Has Nonrepeated Linear Factors
The partial fraction decomposition of [latex]\frac{P\left(x\right)}{Q\left(x\right)}[/latex] when [latex]Q\left(x\right)[/latex] has nonrepeated linear factors and the degree of [latex]P\left(x\right)[/latex] is less than the degree of [latex]Q\left(x\right)[/latex] is[latex]\frac{P\left(x\right)}{Q\left(x\right)}=\frac{{A}_{1}}{\left({a}_{1}x+{b}_{1}\right)}+\frac{{A}_{2}}{\left({a}_{2}x+{b}_{2}\right)}+\frac{{A}_{3}}{\left({a}_{3}x+{b}_{3}\right)}+\cdot \cdot \cdot +\frac{{A}_{n}}{\left({a}_{n}x+{b}_{n}\right)}[/latex].
How To: Given a rational expression with distinct linear factors in the denominator, decompose it.
- Use a variable for the original numerators, usually [latex]A,B,[/latex] or [latex]C[/latex], depending on the number of factors, placing each variable over a single factor. For the purpose of this definition, we use [latex]{A}_{n}[/latex] for each numerator
[latex]\frac{P\left(x\right)}{Q\left(x\right)}=\frac{{A}_{1}}{\left({a}_{1}x+{b}_{1}\right)}+\frac{{A}_{2}}{\left({a}_{2}x+{b}_{2}\right)}+\cdots \text{+}\frac{{A}_{n}}{\left({a}_{n}x+{b}_{n}\right)}[/latex]
- Multiply both sides of the equation by the common denominator to eliminate fractions.
- Expand the right side of the equation and collect like terms.
- Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators.
Example: Decomposing a Rational Expression with Distinct Linear Factors
Decompose the given rational expression with distinct linear factors.[latex]\frac{3x}{\left(x+2\right)\left(x - 1\right)}[/latex]
Answer: We will separate the denominator factors and give each numerator a symbolic label, like [latex]A,B\text{\hspace{0.17em},}[/latex] or [latex]C[/latex].
[latex]\frac{3x}{\left(x+2\right)\left(x - 1\right)}=\frac{A}{\left(x+2\right)}+\frac{B}{\left(x - 1\right)}[/latex]
Multiply both sides of the equation by the common denominator to eliminate the fractions:[latex]\left(x+2\right)\left(x - 1\right)\left[\frac{3x}{\left(x+2\right)\left(x - 1\right)}\right]=\overline{)\left(x+2\right)}\left(x - 1\right)\left[\frac{A}{\overline{)\left(x+2\right)}}\right]+\left(x+2\right)\overline{)\left(x - 1\right)}\left[\frac{B}{\overline{)\left(x - 1\right)}}\right][/latex]
The resulting equation is[latex]3x=A\left(x - 1\right)+B\left(x+2\right)[/latex]
Expand the right side of the equation and collect like terms.[latex]\begin{array}{l}3x=Ax-A+Bx+2B\\ 3x=\left(A+B\right)x-A+2B\end{array}[/latex]
Set up a system of equations associating corresponding coefficients.[latex]\begin{array}{l}3=A+B\\ 0=-A+2B\end{array}[/latex]
Add the two equations and solve for [latex]B[/latex].[latex]\begin{array}\text{ }3=A+B \\ 0=-A+2B \hfill& \\ \text{_____________} \\ 3=0+3B \hfill& \\ 1=B \hfill& \end{array}[/latex]
Substitute [latex]B=1[/latex] into one of the original equations in the system.[latex]\begin{array}{l}3=A+1\\ 2=A\end{array}[/latex]
Thus, the partial fraction decomposition is[latex]\frac{3x}{\left(x+2\right)\left(x - 1\right)}=\frac{2}{\left(x+2\right)}+\frac{1}{\left(x - 1\right)}[/latex]
Another method to use to solve for [latex]A[/latex] or [latex]B[/latex] is by considering the equation that resulted from eliminating the fractions and substituting a value for [latex]x[/latex] that will make either the A- or B-term equal 0. If we let [latex]x=1[/latex], the [latex]A-[/latex] term becomes 0 and we can simply solve for [latex]B[/latex].[latex]\begin{array}{l}\text{ }3x=A\left(x - 1\right)+B\left(x+2\right)\hfill \\ 3\left(1\right)=A\left[\left(1\right)-1\right]+B\left[\left(1\right)+2\right]\hfill \\ \text{ }3=0+3B\hfill \\ \text{ }1=B\hfill \end{array}[/latex]
Next, either substitute [latex]B=1[/latex] into the equation and solve for [latex]A[/latex], or make the B-term 0 by substituting [latex]x=-2[/latex] into the equation.[latex]\begin{array}{l}\text{ }3x=A\left(x - 1\right)+B\left(x+2\right)\hfill \\ \text{ }3\left(-2\right)=A\left[\left(-2\right)-1\right]+B\left[\left(-2\right)+2\right]\hfill \\ \text{ }-6=-3A+0\hfill \\ \text{ }\frac{-6}{-3}=A\hfill \\ \text{ 2}=A\hfill \end{array}[/latex]
We obtain the same values for [latex]A[/latex] and [latex]B[/latex] using either method, so the decompositions are the same using either method.[latex]\frac{3x}{\left(x+2\right)\left(x - 1\right)}=\frac{2}{\left(x+2\right)}+\frac{1}{\left(x - 1\right)}[/latex]
Although this method is not seen very often in textbooks, we present it here as an alternative that may make some partial fraction decompositions easier. It is known as the Heaviside method, named after Charles Heaviside, a pioneer in the study of electronics.Try It
Find the partial fraction decomposition of the following expression.[latex]\frac{x}{\left(x - 3\right)\left(x - 2\right)}[/latex]
Answer: [latex]\frac{3}{x - 3}-\frac{2}{x - 2}[/latex]
Decomposing P(x)/ Q(x), Where Q(x) Has Repeated Linear Factors
Some fractions we may come across are special cases that we can decompose into partial fractions with repeated linear factors. We must remember that we account for repeated factors by writing each factor in increasing powers.A General Note: Partial Fraction Decomposition of [latex]\frac{P\left(x\right)}{Q\left(x\right)}:Q\left(x\right)[/latex] Has Repeated Linear Factors
The partial fraction decomposition of [latex]\frac{P\left(x\right)}{Q\left(x\right)}[/latex], when [latex]Q\left(x\right)[/latex] has a repeated linear factor occurring [latex]n[/latex] times and the degree of [latex]P\left(x\right)[/latex] is less than the degree of [latex]Q\left(x\right)[/latex], is[latex]\frac{P\left(x\right)}{Q\left(x\right)}=\frac{{A}_{1}}{\left(ax+b\right)}+\frac{{A}_{2}}{{\left(ax+b\right)}^{2}}+\frac{{A}_{3}}{{\left(ax+b\right)}^{3}}+\cdot \cdot \cdot +\frac{{A}_{n}}{{\left(ax+b\right)}^{n}}[/latex]
Write the denominator powers in increasing order.How To: Given a rational expression with repeated linear factors, decompose it.
- Use a variable like [latex]A,B[/latex], or [latex]C[/latex] for the numerators and account for increasing powers of the denominators.
[latex]\frac{P\left(x\right)}{Q\left(x\right)}=\frac{{A}_{1}}{\left(ax+b\right)}+\frac{{A}_{2}}{{\left(ax+b\right)}^{2}}+ \text{. }\text{. }\text{. + }\frac{{A}_{n}}{{\left(ax+b\right)}^{n}}[/latex]
- Multiply both sides of the equation by the common denominator to eliminate fractions.
- Expand the right side of the equation and collect like terms.
- Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators.
Example: Decomposing with Repeated Linear Factors
Decompose the given rational expression with repeated linear factors.[latex]\frac{-{x}^{2}+2x+4}{{x}^{3}-4{x}^{2}+4x}[/latex]
Answer: The denominator factors are [latex]x{\left(x - 2\right)}^{2}[/latex]. To allow for the repeated factor of [latex]\left(x - 2\right)[/latex], the decomposition will include three denominators: [latex]x,\left(x - 2\right)[/latex], and [latex]{\left(x - 2\right)}^{2}[/latex]. Thus, [latex-display]\frac{-{x}^{2}+2x+4}{{x}^{3}-4{x}^{2}+4x}=\frac{A}{x}+\frac{B}{\left(x - 2\right)}+\frac{C}{{\left(x - 2\right)}^{2}}[/latex-display] Next, we multiply both sides by the common denominator.
[latex]\begin{array}{l}x{\left(x - 2\right)}^{2}\left[\frac{-{x}^{2}+2x+4}{x{\left(x - 2\right)}^{2}}\right]=\left[\frac{A}{x}+\frac{B}{\left(x - 2\right)}+\frac{C}{{\left(x - 2\right)}^{2}}\right]x{\left(x - 2\right)}^{2}\hfill \\ \text{ }-{x}^{2}+2x+4=A{\left(x - 2\right)}^{2}+Bx\left(x - 2\right)+Cx\hfill \end{array}[/latex]
On the right side of the equation, we expand and collect like terms.[latex]\begin{array}{l}-{x}^{2}+2x+4=A\left({x}^{2}-4x+4\right)+B\left({x}^{2}-2x\right)+Cx\hfill \\ \text{ }=A{x}^{2}-4Ax+4A+B{x}^{2}-2Bx+Cx\hfill \\ \text{ }=\left(A+B\right){x}^{2}+\left(-4A - 2B+C\right)x+4A\hfill \end{array}[/latex]
Next, we compare the coefficients of both sides. This will give the system of equations in three variables:[latex]-{x}^{2}+2x+4=\left(A+B\right){x}^{2}+\left(-4A - 2B+C\right)x+4A[/latex]
[latex]\begin{array}{rr}\hfill A+B=-1& \hfill \text{(1)}\\ \hfill -4A - 2B+C=2& \hfill \text{(2)}\\ \hfill 4A=4& \hfill \text{(3)}\end{array}[/latex]
Solving for [latex]A[/latex] , we have[latex]\begin{array}{l}4A=4\hfill \\ \text{ }A=1\hfill \end{array}[/latex]
Substitute [latex]A=1[/latex] into equation (1).[latex]\begin{array}{l}\text{ }A+B=-1\hfill \\ \left(1\right)+B=-1\hfill \\ \text{ }B=-2\hfill \end{array}[/latex]
Then, to solve for [latex]C[/latex], substitute the values for [latex]A[/latex] and [latex]B[/latex] into equation (2).[latex]\begin{array}{r}\hfill \text{ }-4A - 2B+C=2\\ \hfill -4\left(1\right)-2\left(-2\right)+C=2\\ \hfill \text{ }-4+4+C=2\\ \hfill \text{ }C=2\end{array}[/latex]
Thus,[latex]\frac{-{x}^{2}+2x+4}{{x}^{3}-4{x}^{2}+4x}=\frac{1}{x}-\frac{2}{\left(x - 2\right)}+\frac{2}{{\left(x - 2\right)}^{2}}[/latex]
Try It
Find the partial fraction decomposition of the expression with repeated linear factors.[latex]\frac{6x - 11}{{\left(x - 1\right)}^{2}}[/latex]
Answer: [latex]\frac{6}{x - 1}-\frac{5}{{\left(x - 1\right)}^{2}}[/latex]
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- College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].
- Ex 1: Partial Fraction Decomposition (Linear Factors) . Authored by: Sousa, James (Mathispower4u.com). License: CC BY: Attribution.
- Ex 3: Partial Fraction Decomposition (Repeated Linear Factors) . Authored by: Sousa, James (Mathispower4u.com). License: CC BY: Attribution.
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- Precalculus. Provided by: OpenStax Authored by: OpenStax College. Located at: https://cnx.org/contents/[email protected]:1/Preface. License: CC BY: Attribution.
