Partial Fractions

Question 1

Partial Fractions -

Answer 1

The smaller fractions that when manipulated will form the large rational fraction.

Question 2

How can partial fractions be found?

Answer 2

By using long algebraic division

Question 3

Set up of long algebraic devision -

Answer 3

1. Draw out the division:__|—————
2. Put the denominator of the rational fraction on the left side on top of the line.
3. Put the numerator on right hand side below the line leaving space between components, should there be a step missing (x^2 +7) add in a 0x to the equation where it should be.

Question 4

Process of long algebraic division -

Answer 4

1. Find what the denominators first component can be multiplied by to be equal to first component of the numerator - then place this value at top of division.
2. Multiply entire denominator by this value and place below the numerator.
3. Take the manipulated denominator away from the numerator - first value should equal 0.
4. Repeat this process with the remainder adding the value to top of the line until the denominator cannot be multiplied to equal first component.

Question 5

How to show answer for long algebraic division -

Answer 5

Have rational fraction = top line of division plus (final remainder over normal denominator)

Question 6

Three or more distinct linear factors:
N/(x-2)((x+3)(x-6)

Answer 6

A/(x-2)+B/(x+3)+C/(x-6)

Question 7

Repeated linear factor:
N/(x+3)(x-4)^2

Answer 7

A/(x+3)+B/((x-4)+C/(x-4)^2

Question 8

Irreducible quadratic factor:
N/(2x+1)(x^2+x+2)

Answer 8

A/(2x+1)+(Bx+C)/(x^2+x+2)

Question 9

What is an improper rational fraction?

Answer 9

Degree of numerator = degree of denominator

Question 10

How to work through improper rational fraction.

Answer 10

Use algebraic long division to simplify fraction. Use partial fractions on remainder/denominator. Solve this then substitute in to solved algebraic long division.