11.4 Partial Fractions

Find the least common denominator of the following rationals:

$\frac{2}{3} , \frac{5}{12} , a n d \frac{1}{18}$

Find the least common denominator of the following rational expressions:

$\frac{x + 1}{x + 3}$ and $\frac{2}{x^{2} - 9}$

Solve the system of equations by elimination.

$\left\{\begin{matrix} 3 x + y = 5 \\ 2 x - 3 y = 7 \end{matrix}\right$

Decompose the given rational expression with distinct linear factors.

Find the partial fraction decomposition of the following expression.

Decompose the given rational expression with repeated linear factors.

Find the partial fraction decomposition of the expression with repeated linear factors.

Find a partial fraction decomposition of the given expression.

Find the partial fraction decomposition of the expression with a nonrepeating irreducible quadratic factor.

Decompose the given expression that has a repeated irreducible factor in the denominator.

Find the partial fraction decomposition of the expression with a repeated irreducible quadratic factor.

Can any quotient of polynomials be decomposed into at least two partial fractions? If so, explain why, and if not, give an example of such a fraction

Can you explain how to verify a partial fraction decomposition graphically?

Once you have a system of equations generated by the partial fraction decomposition, can you explain another method to solve it? For example if you had $\frac{7 x + 13}{3 x^{2} + 8 x + 15} = \frac{A}{x + 1} + \frac{B}{3 x + 5}$, we eventually simplify to $7 x + 13 = A \left(3 x + 5\right) + B \left(x + 1\right) .$ Explain how you could intelligently choose an $x$ -value that will eliminate either $A$ or $B$ and solve for $A$ and $B .$

Find the decomposition of the partial fraction for the nonrepeating linear factors.

$\frac{3 x −79}{x^{2} −5 x −24}$

Find the decomposition of the partial fraction for the nonrepeating linear factors.

$\frac{10 x + 47}{x^{2} + 7 x + 10}$

Find the decomposition of the partial fraction for the nonrepeating linear factors.

$\frac{32 x −11}{20 x^{2} −13 x + 2}$

Find the decomposition of the partial fraction for the nonrepeating linear factors.

$\frac{5 x}{x^{2} −9}$

Find the decomposition of the partial fraction for the nonrepeating linear factors.

$\frac{6 x}{x^{2} −4}$

Find the decomposition of the partial fraction for the nonrepeating linear factors.

$\frac{4 x −1}{x^{2} - x −6}$

Find the decomposition of the partial fraction for the nonrepeating linear factors.

$\frac{3 x −1}{x^{2} −5 x + 6}$

Find the decomposition of the partial fraction for the repeating linear factors.

$\frac{x}{\left(x −2\right)^{2}}$

Find the decomposition of the partial fraction for the repeating linear factors.

$\frac{−24 x −27}{\left(4 x + 5\right)^{2}}$

Find the decomposition of the partial fraction for the repeating linear factors.

$\frac{5 - x}{\left(x −7\right)^{2}}$

Find the decomposition of the partial fraction for the repeating linear factors.

$\frac{5 x^{2} + 20 x + 8}{2 x \left(x + 1\right)^{2}}$

Find the decomposition of the partial fraction for the repeating linear factors.

$\frac{54 x^{3} + 127 x^{2} + 80 x + 16}{2 x^{2} \left(3 x + 2\right)^{2}}$

Find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor.

$\frac{4 x^{2} + 6 x + 11}{\left(x + 2\right) \left(x^{2} + x + 3\right)}$

Find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor.

$\frac{−2 x^{2} + 10 x + 4}{\left(x −1\right) \left(x^{2} + 3 x + 8\right)}$

Find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor.

$\frac{4 x^{2} + 17 x −1}{\left(x + 3\right) \left(x^{2} + 6 x + 1\right)}$

Find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor.

$\frac{4 x^{2} + 5 x + 3}{x^{3} −1}$

Find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor.

$\frac{3 x^{2} −7 x + 33}{x^{3} + 27}$

Find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor.

$\frac{4 x^{2} + 4 x + 12}{8 x^{3} −27}$

Find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor.

$\frac{−2 x^{3} −30 x^{2} + 36 x + 216}{x^{4} + 216 x}$

Find the decomposition of the partial fraction for the irreducible repeating quadratic factor.

$\frac{x^{3} + 6 x^{2} + 5 x + 9}{\left(x^{2} + 1\right)^{2}}$

Find the decomposition of the partial fraction for the irreducible repeating quadratic factor.

$\frac{x^{2} + 5 x + 5}{\left(x + 2\right)^{2}}$

Find the decomposition of the partial fraction for the irreducible repeating quadratic factor.

$\frac{x^{2} + 25}{\left(x^{2} + 3 x + 25\right)^{2}}$

Find the decomposition of the partial fraction for the irreducible repeating quadratic factor.

$\frac{5 x + 2}{x \left(x^{2} + 4\right)^{2}}$

Find the decomposition of the partial fraction for the irreducible repeating quadratic factor.

$\frac{2 x −9}{\left(x^{2} - x\right)^{2}}$

Find the partial fraction expansion.

$\frac{x^{2} + 4}{\left(x + 1\right)^{3}}$

Perform the operation and then find the partial fraction decomposition.

$\frac{7}{x + 8} + \frac{5}{x −2} - \frac{x −1}{x^{2} −6 x −16}$

Perform the operation and then find the partial fraction decomposition.

$\frac{2 x}{x^{2} −16} - \frac{1 −2 x}{x^{2} + 6 x + 8} - \frac{x −5}{x^{2} −4 x}$