Question: Decompose into separate partial fractions 
| Foundations
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| 1) How many fractions will this decompose into? What are the denominators?
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| 2) How do you solve for the numerators?
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| Answer:
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| 1) Since each of the factors are linear, and one has multipliclity 2, there will be three denominators. The linear term, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x-1}
, will appear once in the denominator of the decomposition. The other two denominators will be Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x+3{\text{, and }}(x+3)^{2}}
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| 2) After writing the equality, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {3x^{2}+6x+7}{(x+3)^{2}(x-1)}}={\frac {A}{x-1}}+{\frac {B}{x+3}}+{\frac {C}{(x+3)^{2}}}}
, clear the denominators, and evaluate both sides at x = 1, -3, and any third value. Each evaluation will yield the value of one of the three unknowns.
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Solution:
| Step 1:
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From the factored form of the denominator we can observe that there will be three denominators:  , and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (x+3)^{2}}
. So the final answer will be of the form: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {A}{x-1}}+{\frac {B}{x+3}}+{\frac {C}{(x+3)^{2}}}}
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| Step 2:
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| Now we have the equality Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {3x^{2}+6x+7}{(x+3)^{2}(x-1)}}={\frac {A}{x-1}}+{\frac {B}{x+3}}+{\frac {C}{(x+3)^{2}}}}
. Now clearing the denominators we end up with Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A(x+3)^{2}+B(x-1)(x+3)+C(x-1)=3x^{2}+6x+7}
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| Step 3:
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To proceed we start by evaluating both sides at different x-values. We start with x = 1, since this will zero out the B and C. This leads to  , Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 16A=3+6+7}
, and finally A = 1.
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| Step 4:
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| Now evaluate at -3 to zero out both A and B. This yields the following equations Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle C((-3)-1)=3(-3)^{2}+6(-3)+7}
, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle -4C=3(9)-18+7}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -4C = 27 - 11}
, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C = -4}
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| Step 5:
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| To obtain the value of B we can evaluate x at any value except 1, and -3. We do not want to evaluate at 1 and -3 since both of these will zero out the B. Evaluating at x = 0 will make the arithmetic easier, and gives us Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(0 + 3)^2 + B(0 - 1)(0 + 3) + C(0 - 1) = 9A -3B -C = 7}
. However, we know the values of both A and C, which are 1 and -4, respectively. So Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 9 -3B - (-4) = 7}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -3B + 13 = 7}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -3B = -6}
, and finally Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B = 2}
. This means the final answer is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{x - 1} + \frac{2}{x + 3} - \frac{4}{(x + 3)^2}}
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| Final Answer:
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| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{x - 1} + \frac{2}{x + 3} - \frac{4}{(x + 3)^2}}
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