Partial Fraction Decomposition

Example:  Find the partial fraction decomposition for the rational expression.

11/{x(x^2+11)}

Solution:

The rational expression
 11/{x(x^2+11)}
Factor the denominator completely
11/{x(x^2+11)}
Write the expression with each factor as a separate fraction.
11/{x(x^2+11)}=?/x+?/{x^2+11}
Decide what type of expression to put in the numerator

x is linear so we put a constant in the numerator

x^2+11 is a quadratic so we put a linear expression in the numerator

11/{x(x^2+11)}=?/x+?/{x^2+11}
11/{x(x^2+11)}=A/x+?/{x^2+11}
11/{x(x^2+11)}=A/x+{Bx+C}/{x^2+11}
Now that the set up is done we need to solve for the unknowns A, B and C
11/{x(x^2+11)}=A/x+{Bx+C}/{x^2+11}
Multiply both sides of the equation by the common denominator {x(x^2+11)} and simplify by canceling out common factors
11/{x(x^2+11)}=A/x+{Bx+C}/{x^2+11}
{x(x^2+11)}11/{x(x^2+11)}={x(x^2+11)}(A/x+{Bx+C}/{x^2+11})
11={x(x^2+11)}{A/x}+{x(x^2+11)}{Bx+C}/{x^2+11}
11={(x^2+11)}{A}+x(Bx+C)
Use the distributive property and collect the like terms with the x’s
11={(x^2+11)}{A}+x(Bx+C)
11=Ax^2+11A+Bx^2+Cx
11=Ax^2+Bx^2+Cx+11A
11=(A+B)x^2+Cx+11A
Equate coefficients to create a system of equations to solve
11=(A+B)x^2+Cx+11A
0x^2+0x+11=(A+B)x^2+Cx+11A
On the left 11 is the constant term.  There is no linear term so the coefficient is zero and there is no quadratic term so the coefficient is zero.

On the right the constant (no x’s) is 11A, the linear coefficient is C and the quadratic coefficient is A+B.

System of equations from equating the coefficients
matrix{3}{1}{{11=11A} {0=C} {0=A+B}}

Now solve the system of equations.

The first equation is  11=11A.  Since it only has one variable, I can solve for A by hand.

11=11A
11/11={11A}/11
1=A

The second equation is 0=C .  It is already solved.

The third equation is 0=A+B.  Since I know A=1, I can substitute and solve for B.

0=A+B
0=1+B
0-1=1-1+B
-1=0+B
-1=B

Using the original set up of 11/{x(x^2+11)}=A/x+{Bx+C}/{x^2+11}, substitute the values of A, B, and C.

11/{x(x^2+11)}=A/x+{Bx+C}/{x^2+11}
11/{x(x^2+11)}=1/x+{-1x+0}/{x^2+11}
11/{x(x^2+11)}=1/x+{-x}/{x^2+11}

The partial fraction decomposition of 11/{x(x^2+11)} is 1/x+{-x}/{x^2+11}

Application of Exponential Functions: Finding the Interest Rate

Example:

What is the interest rate necessary for an investment to quadruple after 7 year of continuous compound interest?

Solution:

Since this question involve continuous compound interest, we will use the associated formula.

A=Pe^{rt}

We are given that the invest quadruples in 7 years.  This tells me that when t=7 that A will be 4 times P.  I can write that in symbols A=4P.

Substitute these values into the continuous compound formula and solve for the interest rate.

Continuous compound formula
A=Pe^{rt}
Substitute the values of t and A into the formula
A=Pe^{rt}
4P=Pe^{r*7}
4P=Pe^{7r}
Solve for r by dividing both sides by P and simplifying
4P=Pe^{7r}
{4P}/P={Pe^{7r}}/P
4=e^{7r}
Solve for r by taking the log of both sides.
4=e^{7r}
ln 4=ln e^{7r}
Solve for r by using the power rule and simplifying
ln 4=ln e^{7r}
ln 4=7r ln e
ln 4=7r (1)
ln 4=7r
Solve for r by dividing both sides by 7 and simplifying
ln 4=7r
{ln 4}/7={7r}/7
{ln 4}/7=r
Find the value in the calculator
{ln 4}/7=r
0.1980420516=r
Write the answer as a percentage rounded to two decimal places
r=0.1980420516
r=19.80420516%
r=19.80%

 

Solve an exponential equation: Take the log of both sides

Example:

(1.41)^x = (sqrt{2})^{1-4x}

Solution:

 

The exponential equation
 (1.41)^x = (sqrt{2})^{1-4x}
Since the bases cannot be easily written the same use the method of taking the log of both sides
(1.41)^x = (sqrt{2})^{1-4x}
ln (1.41)^x = ln (sqrt{2})^{1-4x}
Use the power rule for logarithms
ln (1.41)^x = ln (sqrt{2})^{1-4x}
x ln (1.41) = (1-4x) ln sqrt{2}
Use the distributive law
x ln (1.41) = (1-4x) ln sqrt{2}
x ln (1.41) = ln sqrt{2}-4x ln sqrt{2}
Collect the terms with x to one side and collect the terms without x on the other side
x ln (1.41) = ln sqrt{2}-4x ln sqrt{2}
x ln (1.41) +4x ln sqrt{2}= ln sqrt{2}-4x ln sqrt{2} +4x ln sqrt{2}
x ln (1.41) +4x ln sqrt{2}= ln sqrt{2}
Factor the common x
x ln (1.41) +4x ln sqrt{2}= ln sqrt{2}
x( ln (1.41) +4 ln sqrt{2})= ln sqrt{2}
Solve for x by dividing both sides by the factor in the parenthesis and simplify
x( ln (1.41) +4 ln sqrt{2})= ln sqrt{2}
{x( ln (1.41) +4 ln sqrt{2})}/{ ln (1.41) +4 ln sqrt{2}}= {ln sqrt{2}}/{ ln (1.41) +4 ln sqrt{2}}
x = {ln sqrt{2}}/{ ln (1.41) +4 ln sqrt{2}}
The solution
x = {ln sqrt{2}}/{ ln (1.41) +4 ln sqrt{2}}
x =0.2003

When you type this into a calculator be sure to use parenthesis around the numerator and around the denominator.  Here is an example of how you might enter it.

(ln (sqrt{2}))/(ln(1.41)+4 ln(sqrt{2}))

Solve the Logarithmic Equation by the one to one property

Example:

2 log_3(7-x)-log_3 2=log_3 18

Solution:

 

The logarithmic equation
2 log_3(7-x)-log_3 2=log_3 18
Use the power rule and the quotient rule to condense to a single logarithm
2 log_3(7-x)-log_3 2=log_3 18
log_3(7-x)^2-log_3 2=log_3 18
log_3((7-x)^2/ 2)=log_3 18
Since both sides of the equation have the same log base the expressions inside the logarithms must be equal
log_3((7-x)^2/ 2)=log_3 18
(7-x)^2/ 2= 18
Clear the denominator by multiplying by 2 on both sides and simplifying
(7-x)^2/ 2= 18
2*(7-x)^2/ 2= 2*18
(7-x)^2= 36
Get rid of the square by square rooting both sides and simplifying
(7-x)^2= 36
sqrt{(7-x)^2}= sqrt{36}
7-x= pm 6
Get x by itself by subtracting 7 on both sides
7-x= pm 6
7-7-x=-7 pm 6
-x=-7 pm 6
Get x by itself by dividing both sides by negative 1
-x=-7 pm 6
-x/-1={-7 pm 6}/-1
x=7 pm 6
x=7 + 6 or x=7 - 6
x=13 or x=1
Check x=13
2 log_3(7-13)-log_3 2=log_3 18
2 log_3(-6)-log_3 2=log_3 18
Log of a negative is undefined.  Exclude this solution.
Check x=1
2 log_3(7-1)-log_3 2=log_3 18
2 log_3(6)-log_3 2=log_3 18
log_3(6)^2-log_3 2=log_3 18
log_3 36-log_3 2=log_3 18
log_3 36/2=log_3 18
log_3 18=log_3 18
Keep this solution.

The solution to the equation is x=1.

Solve an Exponential Equation: Take the log of both sides

Problem:  Solve the exponential equation.

16^{3x-3}=3^{x-3}

Solution:

 

The exponential equation
16^{3x-3}=3^{x-3}
Since the bases cannot be easily written the same, use the method of taking the log of both sides
ln (16^{3x-3})=ln (3^{x-3})
Use the power rule for logarithms.
(3x-3)ln16=(x-3)ln3
Use the distributive law
3xln16-3ln16=xln3-3ln3
Collect the terms with x to one side and collect the terms without x on the other side
3xln16-3ln16=xln3-3ln3
3xln16-3ln16+3ln16=xln3-3ln3+3ln16
3xln16-xln3=xln3-xln3-3ln3+3ln16
3xln16-xln3=-3ln3+3ln16
3xln16-xln3=3ln16-3ln3
Factor the common x
3xln16-xln3=3ln16-3ln3
x(3ln16-ln3)=3ln16-3ln3
Solve for x by dividing both sides by the factor in the parenthesis and simplify
x(3ln16-ln3)=3ln16-3ln3
{x(3ln16-ln3)}/{3ln16-ln3}={3ln16-3ln3}/{3ln16-ln3}
x={3ln16-3ln3}/{3ln16-ln3}
The solution
x={3ln16-3ln3}/{3ln16-ln3}
x = 0.6956

When you type this into a calculator be sure to use parenthesis around the numerator and around the denominator.  Here is an example of how you might enter it.

(3ln(16)-3ln(3))/(3ln(16)-ln(3))

 

Solving an Exponential Equation: Relating the Bases

Example:  Solve the exponential equation.

4^{x-9}=1/1024

Solution:

The exponential equation
4^{x-9}=1/1024
Try to write both sides of the equation with the same base.  Try 4 since there is a base of 4 on the left 4^{x-9}=1/4^5
Using a property of negative exponents move the base to the numerator  4^{x-9}=4^{-5}
Now that that the bases are the same the exponents must be equal  4^{x-9}=4^{-5}

x-9=-5

Solve for x
 x-9=-5

x-9+9=-5+9

x=4

The solution the the exponential equation is 4.

 

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