Partial fraction decomposition of the rational expression

[mistalopez]

Homework Statement

Write the partial fraction decomposition of the rational expression. Check your result algebraically.

(x² – 7x + 16)/[(x + 2)(x2 – 4x + 5)]

The Attempt at a Solution

[A/(x+2)] + [(Bx+C)/(x²-4x+5)]

x²-7x+16= A(X²-4x+5)+(Bx+C)(x+2)
X²-7x+16=[A+B]x²+[-4A+B+C]x+[5A+C]

A+B=1 => B=1-A
-4A+B+C=-7 => Will need to plugin later
5A+C=16 => C=16-5A

-4A+B+C=-7 => 4A+1-A+16-5A= -10A=17 A=(-17/10)

Now I plug A back into the others

B=1-(-17/10) => B=(27/10)

C=16-5(-17/10) => C= (49/2)


Result: [(-17/10)/(x+2)] + [((27x/10)+(49/2))/(x²-4x+5)]

However, I am being told the answer is wrong. What is it that I am doing wrong?

 

[Mark44]

Homework Statement

Write the partial fraction decomposition of the rational expression. Check your result algebraically.

(x² – 7x + 16)/[(x + 2)(x2 – 4x + 5)]

The Attempt at a Solution

[A/(x+2)] + [(Bx+C)/(x²-4x+5)]

x²-7x+16= A(X²-4x+5)+(Bx+C)(x+2)
X²-7x+16=[A+B]x²+[-4A+B+C]x+[5A+C]

I see a couple of mistakes on the line above, in the coefficients of x and the constant term.

A+B=1 => B=1-A
-4A+B+C=-7 => Will need to plugin later
5A+C=16 => C=16-5A

-4A+B+C=-7 => 4A+1-A+16-5A= -10A=17 A=(-17/10)

Now I plug A back into the others

B=1-(-17/10) => B=(27/10)

C=16-5(-17/10) => C= (49/2)


Result: [(-17/10)/(x+2)] + [((27x/10)+(49/2))/(x²-4x+5)]

However, I am being told the answer is wrong. What is it that I am doing wrong?

 

[mistalopez]

I realize there is a mistake, but you could please elaborate upon my mistake without giving me the answer? Your response did not make it obvious or clear for me. Maybe a more indepth response would help me better understand my own mistake. Thank you in advance for taking the time to help me with this problem!

Is it suppose to be X²-7x+16=[A+B]x²+[-4A+2B+C]x+[5A+2C]

Is that correct?

 

[Mark44]

That's what I get

 

[mistalopez]

That's what I get

But now I am stuck because I can only get one value.

A+B=1 => A=1+B

-4A+2B+C=-7 =>

5A+2C=16 =>

 

[Mark44]

But now I am stuck because I can only get one value.

A+B=1 => A=1+B

Mistake above. A = 1 - B

-4A+2B+C=-7 =>

5A+2C=16 =>

You have three equations in three unknowns, so you should be able to solve for them.

 

[mistalopez]

Mistake above. A = 1 - BYou have three equations in three unknowns, so you should be able to solve for them.

That was my question. How am I suppose to solve for 3 unknowns when I cannot cancel anything out. I need atleast 2 known variables to solve any of the equations. Do I subtract equations or some sort of method?

 

[rs1n]

Use one of the three equations to get a substitution formula. For example, you have A=1-B so far. By substituting into the other two equations, you should be able to reduce your system down to 2 equations and 2 unknowns. Go from there...

 

[GreatEscapist]

Homework Statement

Write the partial fraction decomposition of the rational expression. Check your result algebraically.

(x² – 7x + 16)/[(x + 2)(x2 – 4x + 5)]

The Attempt at a Solution

[A/(x+2)] + [(Bx+C)/(x²-4x+5)]

You need to change the Bx+C. That polynomial can be reduced. So change it to a B and a C.

 

[Mark44]

Homework Statement

Write the partial fraction decomposition of the rational expression. Check your result algebraically.

(x² – 7x + 16)/[(x + 2)(x2 – 4x + 5)]

The Attempt at a Solution

[A/(x+2)] + [(Bx+C)/(x²-4x+5)]

You need to change the Bx+C. That polynomial can be reduced. So change it to a B and a C.

NO! mistalopez has the correct decomposition.

 

[GreatEscapist]

NO! mistalopez has the correct decomposition.

Are you sure? We learned to break down polynomials to factors first, before making it a Bx+c. We learned that the only time you don't use that is when it does not come out evenly in a factor.

 

[Mark44]

Yes, I'm sure. x² -4x + 5 can't be factored into linear factors with real coefficients.

 

[GreatEscapist]

Oh my god, how stupid of me. I read that wrong.
I'm so sorry. >.<

 

[Mark44]

Oh my god, how stupid of me. I read that wrong.
I'm so sorry. >.<

We all screw up from time to time...