Question 1

Which of the following is the correct first step when performing partial fraction decomposition on the rational function $$\frac{2x^3$ + 3x^2$ + 2x + 1}{3x + 1}$$?

Accepted Answer

Perform long division since the degree of the numerator is greater than the denominator.

Question 2

Given $$\frac{5x}{x^3$ + 2x}$$, which of the following is the correct form for its partial fraction decomposition?

Accepted Answer

$$\frac{A}{x} + \frac{B}{x^2$} + \frac{C}{x + 2}$$

Question 3

Suppose you have $$\frac{4x^2$}{x^2$ + 5x}. $$After factoring the denominator, which of the following is the correct setup for partial fractions?

Accepted Answer

$$\frac{A}{x} + \frac{B}{x + 5}$$

Question 4

For the rational function $$\frac{1}{x(x^2$ + 1)}$$, what is the correct partial fraction decomposition form?

Accepted Answer

$$\frac{A}{x} + \frac{Bx + C}{x^2$ + 1}$$

Question 5

After performing long division on $$\frac{2x^3$ + 3x^2$ + 2x + 1}{3x + 1}$$, the result is a polynomial plus a remainder. What is the next step to fully decompose the remainder?

Accepted Answer

Factor the remainder's denominator and set up partial fractions.

Question 6

Which of the following statements about partial fraction decomposition is TRUE?

Accepted Answer

Partial fraction decomposition can only be applied if the degree of the numerator is less than the degree of the denominator.

Question 7

Given $$\frac{4x}{x^2$ + 5x}$$, which real-world scenario could this represent before decomposition?

Accepted Answer

The rate of change of a population where growth depends on both the current population and an external factor.

Question 8

If the denominator of a rational function is $$x^2 + 1$, which is irreducible over the reals, what form should the partial fraction decomposition take?

Accepted Answer

$$\frac{Ax + B}{x^2$ + 1}$$

Question 9

Which of the following is the correct partial fraction decomposition for $$\frac{3x^2$}{(x - 1)^2$ (x + 2)}$$?

Accepted Answer

$$\frac{A}{x - 1} + \frac{B}{(x - 1)^2$} + \frac{C}{x + 2}$$

Question 10

A chemical reaction rate is modeled by $$\frac{1}{x(x + 1)^2$}. $$What is the correct setup for partial fraction decomposition?

Accepted Answer

$$\frac{A}{x} + \frac{B}{x + 1} + \frac{C}{(x + 1)^2$}$$

Question 11

When decomposing $$\frac{4x^2$}{x^2$ + 5x}$$, what is the value of $$A in$$ $$\frac{A}{x} + \frac{B}{x + 5}$$ after equating coefficients?

Accepted Answer

$$A = 4$$

Question 12

Which of the following is NOT a valid step in partial fraction decomposition?

Accepted Answer

Leaving the numerator unfactored and not solving for unknowns.

Question 13

If $$\frac{1}{x^2$ + 1} is $$decomposed, which integral technique is most appropriate for integrating the resulting terms?

Accepted Answer

Trigonometric substitution

Question 14

A medical dosage model uses $$\frac{1}{x(x^2$ + 1)}. $$After decomposition, which term would represent the effect of the dosage at $$x = 0$$?

Accepted Answer

$$\frac{A}{x}$$

Question 15

Which of the following is the correct decomposition for $$\frac{4x}{x^2$ + 5x}$$?

Accepted Answer

$$\frac{4}{x} - \frac{4}{x + 5}$$

Question 16

If a rational function has a repeated quadratic factor in the denominator, such as $$(x^2$$ + $$1)^2$$, what is the correct form for partial fraction decomposition?

Accepted Answer

$$\frac{Ax + B}{x^2$ + 1} + \frac{Cx + D}{(x^2$ + 1)^2$}$$

Question 17

Which of the following is a necessary condition for partial fraction decomposition?

Accepted Answer

The denominator must be factorable into irreducible linear or quadratic terms.

Question 18

In a real-world scenario, why might partial fraction decomposition be useful for integrating rational functions in physics or engineering?

Accepted Answer

It simplifies complex expressions into manageable terms for integration.

Question 19

Given $$\frac{3x^2$}{(x - 1)^2$ (x + 2)}$$, what is the value of $$B in$$ $$\frac{A}{x - 1} + \frac{B}{(x - 1)^2$} + \frac{C}{x + 2} if$$ $$B is $$the coefficient for the repeated linear factor?

Accepted Answer

$$B = 3$$

Partial Fraction Decomposition in Precalculus

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