Further Maths - Partial Fractions
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 $(ax + b)(cx + d)^2$
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\[\frac{5x}{(x + 2)(x - 3)}\]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??
- $3$
- $-2$
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\[5x = A(x - 3) + B(x + 2)\]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.
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\[\frac{A}{(ax + b)} + \frac{B}{(cx + d)^2} + \frac{C}{(cx + d)}\]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 $(x - 3)^2$.
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\[x^2 - 8x + 30 \equiv A(x - 3)^2 + B(x + 2) + C(x + 2)(x - 3)\]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
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\[\frac{5x^2 + 5x + 8}{(x + 2)(x^2 + 5)}\]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
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\[\frac{8x^2 - 12}{(2x^2 + 3)(x+1)}\]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??
\[\frac{A}{x + 2} + \frac{B}{x + 3} + \frac{C}{(x + 3)^2}\]