# How do you combine \frac { 1} { x - 3} + \frac { 1} { ( x - 3) ^ { 2} } - \frac { 1} { ( x - 3) ^ { 3} } into one term?

Dec 5, 2017

$\frac{{x}^{2} - 5 x + 5}{{x}^{3} + 9 {x}^{2} + 27 x - 27} = \frac{{x}^{2} - 5 x + 5}{x - 3} ^ 3$

#### Explanation:

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

Take $\left(x - 3\right) = a$, So,

${\left(x - 3\right)}^{2} = {x}^{2} - 6 x + 9$ ------------ =${a}^{2}$ and
${\left(x - 3\right)}^{3} = {x}^{3} - 27 - 3 \times {x}^{2} \left(3\right) + 3 \times \times x \left(9\right)$

${\left(x - 3\right)}^{3} = {x}^{3} - 27 - 9 \times {x}^{2} + 27 x$

${\left(x - 3\right)}^{3} = {x}^{3} + 9 {x}^{2} + 27 x - 27$------${a}^{3}$

And, given expression can be written as:

$\implies \setminus \frac{1}{a} + \setminus \frac{1}{{a}^{2}} - \setminus \frac{1}{{a}^{3}}$

Now solve by equating the denominators:

$\implies \setminus \frac{1}{a} + \setminus \frac{1}{{a}^{2}} - \setminus \frac{1}{{a}^{3}}$

$\implies \setminus \frac{1}{a} \left({a}^{2} / {a}^{2}\right) + \setminus \frac{1}{{a}^{2}} \left(\frac{a}{a}\right) - \setminus \frac{1}{{a}^{3}}$

$\implies \setminus \frac{{a}^{2}}{{a}^{3}} + \setminus \frac{a}{{a}^{3}} - \setminus \frac{1}{{a}^{3}}$

$\implies \setminus \frac{{a}^{2} + a - 1}{{a}^{3}}$

Substitute values:

$\implies \frac{{x}^{2} - 6 x + 9 + x - 3 - 1}{{x}^{3} + 9 {x}^{2} + 27 x - 27}$

$\implies \frac{{x}^{2} - 5 x + 5}{{x}^{3} + 9 {x}^{2} + 27 x - 27}$

$\implies \frac{{x}^{2} - 5 x + 5}{x - 3} ^ 3$

Dec 5, 2017

$\frac{{x}^{2} - 5 x + 5}{x - 3} ^ 3$

#### Explanation:

$\text{we require the fractions to have a "color(blue)"common denominator}$

$\text{the common denominator of }$

$\left(x - 3\right) , {\left(x - 3\right)}^{2} \text{ and "(x-3)^3" is } {\left(x - 3\right)}^{3}$

$\text{multiply numerator/denominator of }$

$\frac{1}{x - 3} \text{ by } {\left(x - 3\right)}^{2}$

$\Rightarrow \frac{1}{x - 3} \times {\left(x - 3\right)}^{2} / {\left(x - 3\right)}^{2} = {\left(x - 3\right)}^{2} / {\left(x - 3\right)}^{3}$

$\text{multiply numerator/denominator of}$

$\frac{1}{x - 3} ^ 2 \text{ by } \left(x - 3\right)$

$\Rightarrow \frac{1}{x - 3} ^ 2 \times \frac{x - 3}{x - 3} = \frac{x - 3}{x - 3} ^ 3$

$\text{putting this together gives}$

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

$\text{the fractions have a common denominator so add}$
$\text{the numerators leaving the common denominator}$

$= \frac{{x}^{2} - 6 x + 9 + x - 3 - 1}{x - 3} ^ 3$

$= \frac{{x}^{2} - 5 x + 5}{x - 3} ^ 3 \to \left(x \ne 3\right)$