Partial Fraction Decomposition

Partial Fractions - How it Works - Video

Guidedlines

Guideline 1 is important here.

In the first example, the exponent in the denominator is higher so we don't need long division.

In the second example, the exponents in the numerator and the denominator are the same so we need long division.

In the third example, the exponent in the numerator is higher so we need long division.

Guideline 2 tells us to factor when necessary.

Guideline 3A tells us that when we separate there are different numbers in the numerator.

Guideline 3B tells us the same except we have a variable and a constant in the numerator.

Guideline 4 tells us to find the numbers. We can do that using matrices, substitution, elimination, or ...

Example 1

Example 1:

Since the exponent in the numerator is lower, we do not need long division. That is good news.

But we do need to factor the trinomial in the denominator so x2 + 9x + 20 => (x + 4)(x +5). Now that we factored, each parenthesis will be a denominator. Order does not matter here, but people usually put the smaller number first.

Now we have (3x + 10) / (x2 + 9x + 20) = A / (x + 4) + B / (x + 5).

We need to get rid of the denominators in order to do that we need to by our original denominator, x2 + 9x + 20. Let's use (x + 4)(x + 5) since it is easier to cancel different terms.

Now we have (3x + 10)= A * (x + 5) + B * (x + 4) after multiplying (x + 4)(x + 5) to each fraction.

3x + 10 = A * (x + 5) + B * (x + 4)

3x + 10 = Ax + 5A + Bx + 4B

3x + 10 = Ax + Bx + 5A + 4B

3x + 10 = (A + B)x + [5A + 4B]


Distributed the number out in front.

Rearranged the terms.

Factored out any shared terms.

Now we now that 3 = (A + B) and 10 = [5A + 4B]. We can set up of a system of equations so we can solve for A and B. We can use Gauss-Jordan to find our answers, but we are going to show how to get the answers in a different way.

5A + 4B = 10

A + B = 3

=>

=>

5A + 4B = 10

4A + 4B = 12


Multiplied each term by 4.

Now we are going to subtract the two equations so we get 1A - 0B = -2 => A = -2. We need to substitute that number, A = -2, in for either equation to solve for B, (-2) + B = 5 => B = 5.

Our last step is to substitute those numbers into the partial fraction decomposition, (3x + 10) / (x2 + 9x + 20) = -2 / (x + 4) + 5 / (x + 5).

Example 2

Example 2:

Here we the same question. The first part is the same, but all the factors in our denominator are linear, and they don't repeat. So, we can solve in different way.

We have 3x + 10= A * (x + 5) + B * (x + 4). We can substitute values that make one variable disappear. What number(s) could make our variable disappear? They are -5 and -4.

3x + 10 = A * (x + 5) + B * (x + 4)

3 * [-5] + 10 = A * ([-5] + 5) + B * ([-5] + 4)

-15 + 10 = A * 0 + B * (-1)

-5 = -1 * B

B = 5


Substituted -5 to eliminate the variable, A.

Multiplied.

Simplified by adding.

Divided each side by -1 and put B on the left side.

3x + 10 = A * (x + 5) + B * (x + 4)

3 * [-4] + 10 = A * ([-4] + 5) + B * ([-4] + 4)

-12 + 10 = A * 1 + B * (0)

-2 = 1 * A

A = -2


Substituted -4 to eliminate the variable, B.

Multiplied.

Simplified by adding.

Put A on the left side.

Our last step is to substitute those numbers into the partial fraction decomposition, (3x + 10) / (x2 + 9x + 20) = -2 / (x + 4) + 5 / (x + 5).

Example 3

Example 3:

Since the exponent in the numerator is not lower, we need long division to move forward.

After completing long division, we get 2, our quotient, and 16x +4, our remainder. We can set aside our quotient and now just focus on the remainder. Now we have 2 + (16x + 4) / (x2 - 6x + 5).

We need to factor x2 - 6x + 5 => (x - 1)(x - 5).

Now that we factored, each parenthesis will be a denominator. Order does not matter here, but people usually put the smaller number first.

Now we have (16x + 4) / (x2 - 6x + 5) = A / (x - 1) + B / (x - 5).

We need to get rid of the denominators in order to do that we need to by our original denominator,x2 - 6x + 5. Let's use (x - 1)(x - 5) since it is easier to cancel different terms.

Now we have (16x + 4)= A * (x - 5) + B * (x - 1) after multiplying (x - 1)(x - 5) to each fraction.

16x + 4 = A * (x - 5) + B * (x - 1)

16x + 4 = Ax - 5A + Bx - 1B

16x + 4 = Ax + Bx - 5A - 1B

16x + 4 = (A + B)x + [-5A - 1B]


Distributed the number out in front.

Rearranged the terms.

Factored out any shared terms.

Now we now that 16 = (A + B) and 4 = [-5A - 1B]. We can set up of a system of equations so we can solve for A and B. We can use Gauss-Jordan to find our answers, but we are going to show how to get the answers in a different way.

-5A - B = 4

A + B = 16

Now we are going to add the two equations so we get -4A - 0B = 20 => -4A = 20 => A = -5. We need to substitute that number, A = -5, in for either equation to solve for B, (-5) + B = 16 => B = 21.

Our last step is to substitute those numbers into the partial fraction decomposition, (16x + 4) / (x2 - 6x + 5) = -5 / (x - 1) + 21 / (x - 5). We can't forget our quotient, 2. So our final answer is 2 + (16x + 4) / (x2 - 6x + 5) = 2 + -5 / (x - 1) + 21 / (x - 5).

Example 4

Example 4:

Here we the same question. The first part is the same, but all the factors in our denominator are linear, and they don't repeat. So, we can solve in different way.

We have 16x + 4= A * (x - 5) + B * (x - 1). We can substitute values that make one variable disappear. What number(s) could make our variable disappear? They are 1 and 5.

16x + 4 = A * (x - 5) + B * (x - 1)

16 * [5] + 4 = A * ([5] - 5) + B * ([5] - 1)

80 + 4 = A * (0) + B * (4)

84 = 4B

B = 21


Substituted 5 to eliminate the variable, A.

Multiplied.

Simplified by adding.

Divided each side by 4 and put A on the left side.

16x + 4 = A * (x - 5) + B * (x - 1)

16 * [1] + 4 = A * ([1] - 5) + B * ([1] - 1)

16 + 4 = A * (-4) + B * (0)

20 = -4A

A = -5


Substituted 1 to eliminate the variable, B.

Multiplied.

Simplified by adding.

Divided each side by -4 and put A on the left side.

Our last step is to substitute those numbers into the partial fraction decomposition, So our final answer is 2 + (16x + 4) / (x2 - 6x + 5) = 2 + -5 / (x - 1) + 21 / (x - 5).

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