Mister Exam
Expression simplification/
Fraction Decomposition into the simple/
y/(b*y-2*b^2)-2/(y^2+y-2*b*y-2*b)*(1+(3*y+y^2)/(3+y))
How do you y/(b*y-2*b^2)-2/(y^2+y-2*b*y-2*b)*(1+(3*y+y^2)/(3+y)) in partial fractions?
The solution
You have entered
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/ 2\
y 2 | 3*y + y |
---------- - --------------------*|1 + --------|
2 2 \ 3 + y /
b*y - 2*b y + y - 2*b*y - 2*b
$$\frac{y}{- 2 b^{2} + b y} - \left(1 + \frac{y^{2} + 3 y}{y + 3}\right) \frac{2}{- 2 b + \left(- 2 b y + \left(y^{2} + y\right)\right)}$$
y/(b*y - 2*b^2) - 2/(y^2 + y - 2*b*y - 2*b)*(1 + (3*y + y^2)/(3 + y))
Numerical answer
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y/(-2.0*b^2 + b*y) - 2.0*(1.0 + (y^2 + 3.0*y)/(3.0 + y))/(y + y^2 - 2.0*b - 2.0*b*y)
y/(-2.0*b^2 + b*y) - 2.0*(1.0 + (y^2 + 3.0*y)/(3.0 + y))/(y + y^2 - 2.0*b - 2.0*b*y)
Rational denominator
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/ 2 \ / 2\ / 2 \
\- 2*b + b*y/*\-6 - 8*y - 2*y / + y*(3 + y)*\y + y - 2*b - 2*b*y/
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/ 2 \ / 2 \
(3 + y)*\- 2*b + b*y/*\y + y - 2*b - 2*b*y/
$$\frac{y \left(y + 3\right) \left(- 2 b y - 2 b + y^{2} + y\right) + \left(- 2 b^{2} + b y\right) \left(- 2 y^{2} - 8 y - 6\right)}{\left(- 2 b^{2} + b y\right) \left(y + 3\right) \left(- 2 b y - 2 b + y^{2} + y\right)}$$
((-2*b^2 + b*y)*(-6 - 8*y - 2*y^2) + y*(3 + y)*(y + y^2 - 2*b - 2*b*y))/((3 + y)*(-2*b^2 + b*y)*(y + y^2 - 2*b - 2*b*y))
Powers
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/ 2 \
2*\y + 3*y/
-2 - ------------
y 3 + y
------------ + --------------------
2 2
- 2*b + b*y y + y - 2*b - 2*b*y
$$\frac{y}{- 2 b^{2} + b y} + \frac{-2 - \frac{2 \left(y^{2} + 3 y\right)}{y + 3}}{- 2 b y - 2 b + y^{2} + y}$$
/ 2 \
| y + 3*y|
2*|1 + --------|
y \ 3 + y /
------------ - --------------------
2 2
- 2*b + b*y y + y - 2*b - 2*b*y
$$\frac{y}{- 2 b^{2} + b y} - \frac{2 \left(1 + \frac{y^{2} + 3 y}{y + 3}\right)}{- 2 b y - 2 b + y^{2} + y}$$
y/(-2*b^2 + b*y) - 2*(1 + (y^2 + 3*y)/(3 + y))/(y + y^2 - 2*b - 2*b*y)
Assemble expression
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/ 2 \
| y + 3*y|
2*|1 + --------|
y \ 3 + y /
------------ - ----------------------
2 2
- 2*b + b*y y - 2*b + y*(1 - 2*b)
$$\frac{y}{- 2 b^{2} + b y} - \frac{2 \left(1 + \frac{y^{2} + 3 y}{y + 3}\right)}{- 2 b + y^{2} + y \left(1 - 2 b\right)}$$
/ 2 \
| y + 3*y|
2*|1 + --------|
y \ 3 + y /
------------ - --------------------
2 2
- 2*b + b*y y + y - 2*b - 2*b*y
$$\frac{y}{- 2 b^{2} + b y} - \frac{2 \left(1 + \frac{y^{2} + 3 y}{y + 3}\right)}{- 2 b y - 2 b + y^{2} + y}$$
/ 2 \
| y + 3*y|
2*|1 + --------|
y \ 3 + y /
------------ - ---------------------
2 2
- 2*b + b*y y + y + b*(-2 - 2*y)
$$\frac{y}{- 2 b^{2} + b y} - \frac{2 \left(1 + \frac{y^{2} + 3 y}{y + 3}\right)}{b \left(- 2 y - 2\right) + y^{2} + y}$$
y/(-2*b^2 + b*y) - 2*(1 + (y^2 + 3*y)/(3 + y))/(y + y^2 + b*(-2 - 2*y))
Trigonometric part
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/ 2 \
| y + 3*y|
2*|1 + --------|
y \ 3 + y /
------------ - --------------------
2 2
- 2*b + b*y y + y - 2*b - 2*b*y
$$\frac{y}{- 2 b^{2} + b y} - \frac{2 \left(1 + \frac{y^{2} + 3 y}{y + 3}\right)}{- 2 b y - 2 b + y^{2} + y}$$
y/(-2*b^2 + b*y) - 2*(1 + (y^2 + 3*y)/(3 + y))/(y + y^2 - 2*b - 2*b*y)
Combining rational expressions
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y*(-2*b + y*(1 + y - 2*b)) - 2*b*(1 + y)*(y - 2*b)
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b*(y - 2*b)*(-2*b + y*(1 + y - 2*b))
$$\frac{- 2 b \left(- 2 b + y\right) \left(y + 1\right) + y \left(- 2 b + y \left(- 2 b + y + 1\right)\right)}{b \left(- 2 b + y\right) \left(- 2 b + y \left(- 2 b + y + 1\right)\right)}$$
(y*(-2*b + y*(1 + y - 2*b)) - 2*b*(1 + y)*(y - 2*b))/(b*(y - 2*b)*(-2*b + y*(1 + y - 2*b)))