|
|
| Line 46: |
Line 46: |
| | |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 <math>A(0 + 3)^2 + B(0 - 1)(0 + 3) + C(0 - 1) = 9A -3B -C = 7</math>. However, we know the values of both A and C, which are 1 and -4, respectively. So <math>9 -3B - (-4) = 7</math>, <math>-3B + 13 = 7</math>, <math>-3B = -6</math>, and finally <math>B = 2</math>. This means the final answer is <math>\frac{1}{x - 1} + \frac{2}{x + 3} - \frac{4}{(x + 3)^2}</math> | | |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 <math>A(0 + 3)^2 + B(0 - 1)(0 + 3) + C(0 - 1) = 9A -3B -C = 7</math>. However, we know the values of both A and C, which are 1 and -4, respectively. So <math>9 -3B - (-4) = 7</math>, <math>-3B + 13 = 7</math>, <math>-3B = -6</math>, and finally <math>B = 2</math>. This means the final answer is <math>\frac{1}{x - 1} + \frac{2}{x + 3} - \frac{4}{(x + 3)^2}</math> |
| | |} | | |} |
| | + | |
| | + | [[8AF11Final|<u>'''Return to Sample Exam</u>''']] |
Revision as of 14:36, 6 April 2015
Question: Decompose into separate partial fractions 
| Foundations
|
| 1) How many fractions will this decompose into? What are the denominators?
|
| 2) How do you solve for the numerators?
|
| Answer:
|
| 1) Since each of the factors are linear, and one has multipliclity 2, there will be three denominators. The linear term, x -1, will appear once in the denominator of the decomposition. The other two denominators will be x + 3, 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 (x + 3)^2}
.
|
| 2) After writing the equality, 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{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 use the cover up method to solve for A, B, and C. After you clear the denominators, the cover up method is to evaluate both sides at x = 1, -3, and any third value. Each evaluation will yield the value of one of the three unknowns.
|
Solution:
| Step 1:
|
| From the factored form of the denominator we can observe that there will be three denominators: 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 x - 1, x + 3}
, 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 (x + 3)^2}
. So the final answer will be of the form: 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{A}{x - 1} + \frac{B}{x + 3} + \frac{C}{(x + 3)^2}}
|
| Step 2:
|
| Now we have the equality 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{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 (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(x + 3)^2 + B(x - 1)(x + 3) + C(x - 1) = 3x^2 + 6x + 7}
.
|
| Step 3:
|
| 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 (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(1 + 3)^2 = 3(1)^2 + 6(1) + 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 16A = 3 + 6 + 7}
, and finally A = 1.
|
| Step 4:
|
| Now evaluate at -3 to zero out both A and B. This yields the following equations 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((-3) - 1) = 3(-3)^2 + 6(-3) + 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 = 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}
|
| Step 5:
|
| 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}}
|
Return to Sample Exam