How do you set up the partial fractions for $$\frac{P(x)}{Q(x)}$$ where $$Q(x)$$ factors into linear factors?

Set up the partial fractions as $$\frac{A}{(x-a)} + \frac{B}{(x-b)} + ...$$ for each linear factor $$(x-a)$$ , where $$A, B, ...$$ are constants to be determined.

What if $$Q(x)$$ has repeated linear factors?

For repeated linear factors like $$(x-a)^n$$ , include terms like $$\frac{A_1}{(x-a)} + \frac{A_2}{(x-a)^2} + ... + \frac{A_n}{(x-a)^n}$$ .

How do you handle irreducible quadratic factors in $$Q(x)$$ ?

For irreducible quadratic factors like $$ax^2 + bx + c$$ , use the form $$\frac{Ax + B}{ax^2 + bx + c}$$ .

Example: Decompose $$\frac{2x+3}{(x-1)(x+2)}$$ .

$$\frac{2x+3}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}$$ , where you solve for $$A$$ and $$B$$ .

What is the result of decomposing $$\frac{-x+34}{x^2-3x-4}$$ ?

The decomposition is $$\frac{6}{x-4} - \frac{7}{x+1}$$ .

What is the significance of the degree of the numerator in Partial Fraction Decomposition?

The degree of the numerator must be less than the degree of the denominator to apply Partial Fraction Decomposition directly.

How can you verify your Partial Fraction Decomposition?

You can verify by combining the fractions back into a single fraction and checking if it equals the original rational function.

What is a common mistake when performing Partial Fraction Decomposition?

A common mistake is forgetting to account for repeated factors or incorrectly setting up the equations for the constants.

What is the role of polynomial long division in Partial Fraction Decomposition?

Polynomial long division is used when the degree of the numerator is greater than or equal to the degree of the denominator, simplifying the expression first.

How do you approach a problem with multiple variables in Partial Fraction Decomposition?

Treat each variable separately and apply the same principles of decomposition for each variable's rational function.

What is the final step after finding the constants in Partial Fraction Decomposition?

The final step is to write the original rational function as the sum of the decomposed fractions, ready for integration or further analysis.

Resource Library

Math

Algebra

Partial Fraction Decomposition

Flashcard

•

Mathematics

•

9th - 12th Grade

•

Practice Problem

•

Hard

•

CCSS

HSA.APR.D.6, HSA.APR.D.7

Standards-aligned

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FLASHCARD QUESTION

Front

What is Partial Fraction Decomposition?

Back

Partial Fraction Decomposition is a method used to express a rational function as the sum of simpler fractions, making it easier to integrate or analyze.

FLASHCARD QUESTION

Front

What is the first step in Partial Fraction Decomposition?

Back

The first step is to ensure that the degree of the numerator is less than the degree of the denominator. If not, perform polynomial long division.

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