# Further Maths - Partial Fractions

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/partial-fractions/ · Updated: 2021-01-27 · Tags: further-maths, algebra

##### What is the simple case of partial fractions??
Where the denominator is $(ax + b)(cx + d)$.

##### What is the harder case of partial fractions??
Where the denominator is $(ax + b)(cx + d)^2$

##### $$\frac{5x}{(x + 2)(x - 3)}$$ What's the first step to finding the partial fractions??
Rewriting as

$$
\frac{A}{x + 2} + \frac{B}{x - 3}
$$

##### $$\frac{A}{x + 2} + \frac{B}{x - 3}$$ How could you add these two fractions together??
$$
\frac{A(x - 3) + B(x + 2)}{(x+2)(x-3)}
$$

##### $$\frac{5x}{(x + 2)(x - 3)} = \frac{A(x - 3) + B(x + 2)}{(x+2)(x-3)}$$ How could you simplify this??
$$
5x = A(x - 3) + B(x + 2)
$$

##### $$5x = A(x - 3) + B(x + 2)$$ If you're solving this, what could you set $x$ equal to in order to make one of the unknowns dissapear??
* $3$
* $-2$

##### $$5x = A(x - 3) + B(x + 2)$$ If $x = 3$, what is $B$ equal to??
$$
B = 3
$$

##### If the denominator is $$(ax + b)(cx + d)^2$$, how many partial fractions would there be??
$$
3
$$

##### If the denominator is $$(ax + b)(cx + d)^2$$, what would the partial fractions look like??
$$
\frac{A}{(ax + b)} + \frac{B}{(cx + d)^2} + \frac{C}{(cx + d)}
$$

##### If the denominator is $$(ax + b)(cx + d)^2$$, in partial fractions would would be in the denominator of the $A$ term??
$$
(ax + b)
$$

##### If the denominator is $$(ax + b)(cx + d)^2$$, in partial fractions would would be in the denominator of the $B$ term??
$$
(cx + d)^2
$$

##### If the denominator is $$(ax + b)(cx + d)^2$$, in partial fractions would would be in the denominator of the $C$ term??
$$
(cx + d)
$$
##### $$\frac{A}{(ax + b)} + \frac{B}{(cx + d)^2} + \frac{C}{(cx + d)}$$ In partial fractions, why is the denominator of the $B$ term $(cx + d)^2$ rather than just $(cx + d)$??
Otherwise when you add the fractions together they don't reduce properly.

##### $$\frac{A}{(ax + b)} + \frac{B}{(cx + d)^2} + \frac{C}{(cx + d)}$$ What does this look like as one fraction??
$$
\frac{A(cx + d)^2 + B(ax + b) + C(ax + b)(cx + d)}{(ax + b)(cx + d)^2}
$$

##### $$\frac{A}{(ax + b)} + \frac{B}{(cx + d)^2} + \frac{C}{(cx + d)}$$ What does $A$ multiply in the numerator when this is written as one fraction??
$$
A(cx + d)^2
$$

##### $$\frac{A}{(ax + b)} + \frac{B}{(cx + d)^2} + \frac{C}{(cx + d)}$$ What does $B$ multiply in the numerator when this is written as one fraction??
$$
B(ax + b)
$$

##### $$\frac{A}{(ax + b)} + \frac{B}{(cx + d)^2} + \frac{C}{(cx + d)}$$ What does $C$ multiply in the numerator when this is written as one fraction??
$$
C(ax + b)(cx + d)
$$

##### $$\frac{A}{x + 2} + \frac{B}{(x-1)^2} + \frac{C}{(x-1)}$$ What is the numerator of this fraction when all the terms are added together??
$$
A(x - 1)^2 + B(x + 2) + C(x + 2)(x - 1)
$$

##### $$\frac{x^2 + 8x + 30}{(x + 2)(x - 3)^2}$$ In partial fractions, what is $x^2 + 8x + 30$ equivalent to in terms of $A$, $B$ and $C$??
$$
x^2 - 8x + 30 \equiv A(x - 3)^2 + B(x + 2) + C(x + 2)(x - 1)
$$

##### $$A(x - 3)^2 + B(x + 2) + C(x + 2)(x - 1)$$ What would be the 'gotcha' for substituting in $x = -2$??
You have to square $(x - 3)^2$.

##### $$x^2 - 8x + 30 \equiv A(x - 3)^2 + B(x + 2) + C(x + 2)(x - 3)$$ If you know $A = 2$ and $B = 3$, what two different techniques could you use here in order to find the value of $C$??
* Equating coefficients
* Substituting in a value of $x$ and seeing what value of $C$ makes it true.

### 2021-05-05
##### $$\frac{5x^2 + 5x + 8}{(x + 2)(x^2 + 5)}$$ How would you write this for a partial fractions question??
$$
\frac{A}{x+2} + \frac{Bx + C}{x^2 + 5}
$$

##### When do you use $Bx + C$ for a partial fractions question??
When there is a quadratic $x^2$ term under the fraction, like $(x^2 + 5)$ or $(x^2 - 6)$.

### 2022-01-20
##### What's the quick way of getting to the numerator equivalence in partial fractions??
Multiplying both sides by the denominator.

### 2022-05-12
##### $$\frac{8x^2 - 12}{(2x^2 + 3)(x+1)}$$ What would you write this equivalent to for a partial fractions question??
$$
\frac{Ax + B}{2x^2 + 3} + \frac{C}{x + 1}
$$

##### $$\frac{8x^2 - 12}{(x+2)(x+3)^2}$$ What would you write this equivalent to for a partial fractions question??
$$
\frac{A}{x + 2} + \frac{B}{x + 3} + \frac{C}{(x + 3)^2}
$$

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