Quadratic Factors
Learning Outcomes
- Decompose a rational expression with non-repeated irreducible quadratic factors.
- Decompose a rational expression that has repeated irreducible quadratic factors.
tip for success
As with decomposing fractions with linear factors in the previous section, the techniques to decompose fractions with quadratic factors are challenging and will require some practice to become familiar to you. Do work through the given examples on paper first, slowly, step by step before trying to make sense of the General Note and How To boxes. It will take time and effort, so don’t be discouraged if it takes multiple attempts for each example.A General Note: Decomposition of [latex]\frac{P\left(x\right)}{Q\left(x\right)}:Q\left(x\right)[/latex] Has a Nonrepeated Irreducible Quadratic Factor
The partial fraction decomposition of [latex]\dfrac{P\left(x\right)}{Q\left(x\right)}[/latex] such that [latex]Q\left(x\right)[/latex] has a nonrepeated irreducible quadratic factor and the degree of [latex]P\left(x\right)[/latex] is less than the degree of [latex]Q\left(x\right)[/latex] is written as[latex]\dfrac{P\left(x\right)}{Q\left(x\right)}=\dfrac{{A}_{1}x+{B}_{1}}{\left({a}_{1}{x}^{2}+{b}_{1}x+{c}_{1}\right)}+\dfrac{{A}_{2}x+{B}_{2}}{\left({a}_{2}{x}^{2}+{b}_{2}x+{c}_{2}\right)}+\cdot \cdot \cdot +\dfrac{{A}_{n}x+{B}_{n}}{\left({a}_{n}{x}^{2}+{b}_{n}x+{c}_{n}\right)}[/latex]
The decomposition may contain more rational expressions if there are linear factors. Each linear factor will have a different constant numerator: [latex]A,B,C[/latex], and so on.How To: Given a rational expression where the factors of the denominator are distinct, irreducible quadratic factors, decompose it.
- Use variables such as [latex]A,B[/latex], or [latex]C[/latex] for the constant numerators over linear factors, and linear expressions such as [latex]{A}_{1}x+{B}_{1},{A}_{2}x+{B}_{2}[/latex], etc., for the numerators of each quadratic factor in the denominator.
[latex]\dfrac{P\left(x\right)}{Q\left(x\right)}=\dfrac{A}{ax+b}+\dfrac{{A}_{1}x+{B}_{1}}{\left({a}_{1}{x}^{2}+{b}_{1}x+{c}_{1}\right)}+\dfrac{{A}_{2}x+{B}_{2}}{\left({a}_{2}{x}^{2}+{b}_{2}x+{c}_{2}\right)}+\cdot \cdot \cdot +\dfrac{{A}_{n}x+{B}_{n}}{\left({a}_{n}{x}^{2}+{b}_{n}x+{c}_{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 [latex]\frac{P\left(x\right)}{Q\left(x\right)}[/latex] When Q(x) Contains a Nonrepeated Irreducible Quadratic Factor
Find a partial fraction decomposition of the given expression.[latex]\dfrac{8{x}^{2}+12x - 20}{\left(x+3\right)\left({x}^{2}+x+2\right)}[/latex]
Answer: We have one linear factor and one irreducible quadratic factor in the denominator, so one numerator will be a constant and the other numerator will be a linear expression. Thus,
[latex]\dfrac{8{x}^{2}+12x - 20}{\left(x+3\right)\left({x}^{2}+x+2\right)}=\dfrac{A}{\left(x+3\right)}+\dfrac{Bx+C}{\left({x}^{2}+x+2\right)}[/latex]
We follow the same steps as in previous problems. First, clear the fractions by multiplying both sides of the equation by the common denominator.[latex]\begin{align}\left(x+3\right)\left({x}^{2}+x+2\right)\left[\frac{8{x}^{2}+12x - 20}{\left(x+3\right)\left({x}^{2}+x+2\right)}\right]&=\left[\frac{A}{\left(x+3\right)}+\frac{Bx+C}{\left({x}^{2}+x+2\right)}\right]\left(x+3\right)\left({x}^{2}+x+2\right) \\[2mm] 8{x}^{2}+12x - 20&=A\left({x}^{2}+x+2\right)+\left(Bx+C\right)\left(x+3\right) \end{align}[/latex]
Notice we could easily solve for [latex]A[/latex] by choosing a value for [latex]x[/latex] that will make the [latex]Bx+C[/latex] term equal 0. Let [latex]x=-3[/latex] and substitute it into the equation.[latex]\begin{align}8{x}^{2}+12x - 20&=A\left({x}^{2}+x+2\right)+\left(Bx+C\right)\left(x+3\right) \\ 8{\left(-3\right)}^{2}+12\left(-3\right)-20&=A\left({\left(-3\right)}^{2}+\left(-3\right)+2\right)+\left(B\left(-3\right)+C\right)\left(\left(-3\right)+3\right) \\ 16&=8A \\ A&=2 \end{align}[/latex]
Now that we know the value of [latex]A[/latex], substitute it back into the equation. Then expand the right side and collect like terms.[latex]\begin{align} &8{x}^{2}+12x - 20=2\left({x}^{2}+x+2\right)+\left(Bx+C\right)\left(x+3\right) \\ &8{x}^{2}+12x - 20=2{x}^{2}+2x+4+B{x}^{2}+3B+Cx+3C \\ &8{x}^{2}+12x - 20=\left(2+B\right){x}^{2}+\left(2+3B+C\right)x+\left(4+3C\right) \end{align}[/latex]
Setting the coefficients of terms on the right side equal to the coefficients of terms on the left side gives the system of equations.[latex]\begin{align}2+B=8 && \text{(1)} \\ 2+3B+C=12 && \text{(2)} \\ 4+3C=-20 && \text{(3)} \end{align}[/latex]
Solve for [latex]B[/latex] using equation (1) and solve for [latex]C[/latex] using equation (3).[latex]\begin{align}2+B=8 && \text{(1)} \\ B=6 \\ \\ 4+3C=-20 && \text{(3)} \\ 3C=-24 \\ C=-8 \end{align}[/latex]
Thus, the partial fraction decomposition of the expression is[latex]\dfrac{8{x}^{2}+12x - 20}{\left(x+3\right)\left({x}^{2}+x+2\right)}=\dfrac{2}{\left(x+3\right)}+\dfrac{6x - 8}{\left({x}^{2}+x+2\right)}[/latex]
Q & A
Could we have just set up a system of equations to solve Example 3?
Yes, we could have solved it by setting up a system of equations without solving for [latex]A[/latex] first. The expansion on the right would be:[latex]\begin{align} 8{x}^{2}+12x - 20&=A{x}^{2}+Ax+2A+B{x}^{2}+3B+Cx+3C \\ 8{x}^{2}+12x - 20&=\left(A+B\right){x}^{2}+\left(A+3B+C\right)x+\left(2A+3C\right) \end{align}[/latex]
So the system of equations would be:[latex]\begin{align}A+B=8 \\ A+3B+C=12 \\ 2A+3C=-20 \end{align}[/latex]
tip for success
Sometimes, a combination of methods is helpful to obtain the decomposition. In the example above, you could have set the system up first, then used [latex]x=-3[/latex] to obtain the value for [latex]A[/latex], and it would be quick work to obtain [latex]B[/latex] and [latex]C[/latex] from the system. Remember that creativity is a key component of doing mathematics and that there is often more than one good way to reach a conclusion.Try It
Find the partial fraction decomposition of the expression with a nonrepeating irreducible quadratic factor.[latex]\dfrac{5{x}^{2}-6x+7}{\left(x - 1\right)\left({x}^{2}+1\right)}[/latex]
Answer: [latex-display]\dfrac{3}{x - 1}+\dfrac{2x - 4}{{x}^{2}+1}[/latex-display]
[embed]Decomposing P(x) / Q(x), When Q(x) Has a Repeated Irreducible Quadratic Factor
tip for success
Do work through the example below carefully and more than once. The problem may appear to be complicated, but it relies on the same techniques you've already practiced. Keep your work well-organized to avoid making mistakes.A General Note: Decomposition of [latex]\frac{P\left(x\right)}{Q\left(x\right)}[/latex] When Q(x) Has a Repeated Irreducible Quadratic Factor
The partial fraction decomposition of [latex]\dfrac{P\left(x\right)}{Q\left(x\right)}[/latex], when [latex]Q\left(x\right)[/latex] has a repeated irreducible quadratic factor and the degree of [latex]P\left(x\right)[/latex] is less than the degree of [latex]Q\left(x\right)[/latex], is[latex]\dfrac{P\left(x\right)}{{\left(a{x}^{2}+bx+c\right)}^{n}}=\dfrac{{A}_{1}x+{B}_{1}}{\left(a{x}^{2}+bx+c\right)}+\dfrac{{A}_{2}x+{B}_{2}}{{\left(a{x}^{2}+bx+c\right)}^{2}}+\dfrac{{A}_{3}x+{B}_{3}}{{\left(a{x}^{2}+bx+c\right)}^{3}}+\cdot \cdot \cdot +\dfrac{{A}_{n}x+{B}_{n}}{{\left(a{x}^{2}+bx+c\right)}^{n}}[/latex]
Write the denominators in increasing powers.How To: Given a rational expression that has a repeated irreducible factor, decompose it.
- Use variables like [latex]A,B[/latex], or [latex]C[/latex] for the constant numerators over linear factors, and linear expressions such as [latex]{A}_{1}x+{B}_{1},{A}_{2}x+{B}_{2}[/latex], etc., for the numerators of each quadratic factor in the denominator written in increasing powers, such as
[latex]\dfrac{P\left(x\right)}{Q\left(x\right)}=\dfrac{A}{ax+b}+\dfrac{{A}_{1}x+{B}_{1}}{\left(a{x}^{2}+bx+c\right)}+\dfrac{{A}_{2}x+{B}_{2}}{{\left(a{x}^{2}+bx+c\right)}^{2}}+\cdots +\text{ }\dfrac{{A}_{n}+{B}_{n}}{{\left(a{x}^{2}+bx+c\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 a Rational Function with a Repeated Irreducible Quadratic Factor in the Denominator
Decompose the given expression that has a repeated irreducible factor in the denominator.[latex]\dfrac{{x}^{4}+{x}^{3}+{x}^{2}-x+1}{x{\left({x}^{2}+1\right)}^{2}}[/latex]
Answer: The factors of the denominator are [latex]x,\left({x}^{2}+1\right)[/latex], and [latex]{\left({x}^{2}+1\right)}^{2}[/latex]. Recall that, when a factor in the denominator is a quadratic that includes at least two terms, the numerator must be of the linear form [latex]Ax+B[/latex]. So, let’s begin the decomposition.
[latex]\dfrac{{x}^{4}+{x}^{3}+{x}^{2}-x+1}{x{\left({x}^{2}+1\right)}^{2}}=\dfrac{A}{x}+\dfrac{Bx+C}{\left({x}^{2}+1\right)}+\dfrac{Dx+E}{{\left({x}^{2}+1\right)}^{2}}[/latex]
We eliminate the denominators by multiplying each term by [latex]x{\left({x}^{2}+1\right)}^{2}[/latex]. Thus,[latex]{x}^{4}+{x}^{3}+{x}^{2}-x+1=A{\left({x}^{2}+1\right)}^{2}+\left(Bx+C\right)\left(x\right)\left({x}^{2}+1\right)+\left(Dx+E\right)\left(x\right)[/latex]
Expand the right side.[latex]\begin{align}&{x}^{4}+{x}^{3}+{x}^{2}-x+1=A\left({x}^{4}+2{x}^{2}+1\right)+B{x}^{4}+B{x}^{2}+C{x}^{3}+Cx+D{x}^{2}+Ex \\ &=A{x}^{4}+2A{x}^{2}+A+B{x}^{4}+B{x}^{2}+C{x}^{3}+Cx+D{x}^{2}+Ex \end{align}[/latex]
Now we will collect like terms.[latex]{x}^{4}+{x}^{3}+{x}^{2}-x+1=\left(A+B\right){x}^{4}+\left(C\right){x}^{3}+\left(2A+B+D\right){x}^{2}+\left(C+E\right)x+A[/latex]
Set up the system of equations matching corresponding coefficients on each side of the equal sign.[latex]\begin{align}A+B=1 \\ C=1 \\ \\ 2A+B+D=1 \\ C+E=-1 \\ A=1 \end{align}[/latex]
We can use substitution from this point. Substitute [latex]A=1[/latex] into the first equation.[latex]\begin{align}1+B=1 \\ B=0 \end{align}[/latex]
Substitute [latex]A=1[/latex] and [latex]B=0[/latex] into the third equation.[latex]\begin{align}2\left(1\right)+0+D=1 \\ D=-1 \end{align}[/latex]
Substitute [latex]C=1[/latex] into the fourth equation.[latex]\begin{align} 1+E=-1\\ E=-2\end{align}[/latex]
Now we have solved for all of the unknowns on the right side of the equal sign. We have [latex]A=1[/latex], [latex]B=0[/latex], [latex]C=1[/latex], [latex]D=-1[/latex], and [latex]E=-2[/latex]. We can write the decomposition as follows:[latex]\dfrac{{x}^{4}+{x}^{3}+{x}^{2}-x+1}{x{\left({x}^{2}+1\right)}^{2}}=\dfrac{1}{x}+\dfrac{1}{\left({x}^{2}+1\right)}-\dfrac{x+2}{{\left({x}^{2}+1\right)}^{2}}[/latex]
Try It
Find the partial fraction decomposition of the expression with a repeated irreducible quadratic factor.[latex]\dfrac{{x}^{3}-4{x}^{2}+9x - 5}{{\left({x}^{2}-2x+3\right)}^{2}}[/latex]
Answer: [latex-display]\dfrac{x - 2}{{x}^{2}-2x+3}+\dfrac{2x+1}{{\left({x}^{2}-2x+3\right)}^{2}}[/latex-display]
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