About Partial Fraction Decomposition Linear Factors:
We already know that we may combine two or more rational expressions into one single rational expression. In some cases, we will need to reverse this process and break our single rational expression up into the sum of two or more rational expressions. This process is known as partial fraction decomposition.
Test Objectives
- Demonstrate the ability to perform long division with polynomials
- Demonstrate the ability to factor a polynomial
- Demonstrate the ability to find the partial fraction decomposition with distinct linear factors
- Demonstrate the ability to find the partial fraction decomposition with repeated linear factors
#1:
Instructions: Find the partial fraction decomposition of each.
$$a)\hspace{.2em}\frac{-8x^2 - 16x + 36}{x^3 + 5x^2 - 4x - 20}$$
$$b)\hspace{.2em}\frac{-2x + 1}{x^2 - x}$$
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#2:
Instructions: Find the partial fraction decomposition of each.
$$a)\hspace{.2em}\frac{-10x + 10}{x^2 - 2x}$$
$$b)\hspace{.2em}\frac{-3x - 16}{x^2 + 2x}$$
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#3:
Instructions: Find the partial fraction decomposition of each.
$$a)\hspace{.2em}\frac{-x - 21}{x^2 - 9}$$
$$b)\hspace{.2em}\frac{2x^3 - 14x^2 + 17x + 20}{x^3 - 9x^2 + 20x}$$
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#4:
Instructions: Find the partial fraction decomposition of each.
$$a)\hspace{.2em}\frac{2x^2 + 9x - 10}{x^2 + 5x}$$
$$b)\hspace{.2em}\frac{-4x^2 - 4x + 4}{x^3 + 4x^2 + 4x}$$
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#5:
Instructions: Find the partial fraction decomposition of each.
$$a)\hspace{.2em}\frac{4x - 13}{x^2 - 6x + 9}$$
- Pull out the GCF:
- $$x(x^3 + 9x^2 + 27x + 27)$$
- Rearrange the inside:
- $$x(x^3 + 27 + 9x^2 + 27x)$$
- The first two terms inside the parentheses can be factored using the pattern:
- $$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$
- Pull out the GCF from the final two terms inside the parentheses and look for a common binomial factor
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}{-}\frac{1}{x-2}- \frac{4}{x + 5}- \frac{3}{x + 2}$$
$$b)\hspace{.2em}{-}\frac{1}{x}- \frac{1}{x - 1}$$
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#2:
Solutions:
$$a)\hspace{.2em}{-}\frac{5}{x}- \frac{5}{x - 2}$$
$$b)\hspace{.2em}{-}\frac{8}{x}+ \frac{5}{x + 2}$$
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#3:
Solutions:
$$a)\hspace{.2em}{-}\frac{4}{x-3}+ \frac{3}{x + 3}$$
$$b)\hspace{.2em}2 + \frac{1}{x}+ \frac{2}{x - 4}+ \frac{1}{x - 5}$$
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#4:
Solutions:
$$a)\hspace{.2em}2 - \frac{2}{x}+ \frac{1}{x + 5}$$
$$b)\hspace{.2em}\frac{1}{x}- \frac{5}{x + 2}+ \frac{2}{(x + 2)^2}$$
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#5:
Solutions:
$$a)\hspace{.2em}\frac{4}{x - 3}- \frac{1}{(x - 3)^2}$$
$$b)\hspace{.2em}\frac{1}{x}+ \frac{5}{x + 3}+ \frac{2}{(x + 3)^2}- \frac{1}{(x + 3)^3}$$
