# 9.4 Partial fractions  (Page 4/7)

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## Decomposing $\text{\hspace{0.17em}}\frac{P\left(x\right)}{Q\left(x\right)}\text{\hspace{0.17em}}$ When Q(x) Contains a nonrepeated irreducible quadratic factor

Find a partial fraction decomposition of the given expression.

$\frac{8{x}^{2}+12x-20}{\left(x+3\right)\left({x}^{2}+x+2\right)}$

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,

$\frac{8{x}^{2}+12x-20}{\left(x+3\right)\left({x}^{2}+x+2\right)}=\frac{A}{\left(x+3\right)}+\frac{Bx+C}{\left({x}^{2}+x+2\right)}$

We follow the same steps as in previous problems. First, clear the fractions by multiplying both sides of the equation by the common denominator.

Notice we could easily solve for $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ by choosing a value for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ that will make the $\text{\hspace{0.17em}}Bx+C\text{\hspace{0.17em}}$ term equal 0. Let $x=-3\text{\hspace{0.17em}}$ and substitute it into the equation.

Now that we know the value of $\text{\hspace{0.17em}}A,\text{\hspace{0.17em}}$ substitute it back into the equation. Then expand the right side and collect like terms.

$\begin{array}{l}\hfill \\ 8{x}^{2}+12x-20=2\left({x}^{2}+x+2\right)+\left(Bx+C\right)\left(x+3\right)\hfill \\ 8{x}^{2}+12x-20=2{x}^{2}+2x+4+B{x}^{2}+3B+Cx+3C\hfill \\ 8{x}^{2}+12x-20=\left(2+B\right){x}^{2}+\left(2+3B+C\right)x+\left(4+3C\right)\hfill \end{array}$

Setting the coefficients of terms on the right side equal to the coefficients of terms on the left side gives the system of equations.

Solve for $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ using equation (1) and solve for $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ using equation (3).

Thus, the partial fraction decomposition of the expression is

$\frac{8{x}^{2}+12x-20}{\left(x+3\right)\left({x}^{2}+x+2\right)}=\frac{2}{\left(x+3\right)}+\frac{6x-8}{\left({x}^{2}+x+2\right)}$

Could we have just set up a system of equations to solve [link] ?

Yes, we could have solved it by setting up a system of equations without solving for $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ first. The expansion on the right would be:

$\begin{array}{l}\begin{array}{l}\\ 8{x}^{2}+12x-20=A{x}^{2}+Ax+2A+B{x}^{2}+3B+Cx+3C\end{array}\hfill \\ 8{x}^{2}+12x-20=\left(A+B\right){x}^{2}+\left(A+3B+C\right)x+\left(2A+3C\right)\hfill \end{array}$

So the system of equations would be:

Find the partial fraction decomposition of the expression with a nonrepeating irreducible quadratic factor.

$\frac{5{x}^{2}-6x+7}{\left(x-1\right)\left({x}^{2}+1\right)}$

$\frac{3}{x-1}+\frac{2x-4}{{x}^{2}+1}$

## Decomposing $\text{\hspace{0.17em}}\frac{P\left(x\right)}{Q\left(x\right)}\text{\hspace{0.17em}}$ When Q(x) Has a repeated irreducible quadratic factor

Now that we can decompose a simplified rational expression with an irreducible quadratic factor, we will learn how to do partial fraction decomposition when the simplified rational expression has repeated irreducible quadratic factors. The decomposition will consist of partial fractions with linear numerators over each irreducible quadratic factor represented in increasing powers.

## Decomposition of $\text{\hspace{0.17em}}\frac{P\left(x\right)}{Q\left(x\right)}\text{\hspace{0.17em}}$ When Q(x) Has a repeated irreducible quadratic factor

The partial fraction decomposition of $\text{\hspace{0.17em}}\frac{P\left(x\right)}{Q\left(x\right)},\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}Q\left(x\right)\text{\hspace{0.17em}}$ has a repeated irreducible quadratic factor and the degree of $\text{\hspace{0.17em}}P\left(x\right)\text{\hspace{0.17em}}$ is less than the degree of $\text{\hspace{0.17em}}Q\left(x\right),\text{\hspace{0.17em}}$ is

$\frac{P\left(x\right)}{{\left(a{x}^{2}+bx+c\right)}^{n}}=\frac{{A}_{1}x+{B}_{1}}{\left(a{x}^{2}+bx+c\right)}+\frac{{A}_{2}x+{B}_{2}}{{\left(a{x}^{2}+bx+c\right)}^{2}}+\frac{{A}_{3}x+{B}_{3}}{{\left(a{x}^{2}+bx+c\right)}^{3}}+\cdot \cdot \cdot +\frac{{A}_{n}x+{B}_{n}}{{\left(a{x}^{2}+bx+c\right)}^{n}}$

Write the denominators in increasing powers.

Given a rational expression that has a repeated irreducible factor, decompose it.

1. Use variables like $\text{\hspace{0.17em}}A,B,\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ for the constant numerators over linear factors, and linear expressions such as $\text{\hspace{0.17em}}{A}_{1}x+{B}_{1},{A}_{2}x+{B}_{2},\text{\hspace{0.17em}}$ etc., for the numerators of each quadratic factor in the denominator written in increasing powers, such as
2. Multiply both sides of the equation by the common denominator to eliminate fractions.
3. Expand the right side of the equation and collect like terms.
4. 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.

A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
i want to sure my answer of the exercise
what is the diameter of(x-2)²+(y-3)²=25
how to solve the Identity ?
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
what is a complex number used for?
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
Is there any rule we can use to get the nth term ?