Unit 1 - Number and Algebra
Required Prior Knowledge
Questions
Solve
a) $$\begin{matrix}x&+&2y&=&11\\2x&+&3y&=&18\end{matrix}$$
b) $$\begin{matrix}2x&-&y&=&8\\3x&+&y&=&17\end{matrix}$$
Solutions
Get Ready
Questions
Calculate
a) $$\frac{2}{x+1}+\frac{1}{x-2}$$
b) $$\frac{5}{x-3}+\frac{3}{2x+1}$$
c) $$\frac{4}{x+2}+\frac{1}{\left(x+2\right)^{2}}$$
d) $$\frac{1}{x-1}+\frac{3}{x+4}+\frac{2}{\left(x+4\right)^{2}}$$
Solutions
Notes
In each of the questions in the Get Ready above you started with a sum of partial fractions to find a single algebraic fraction.
Sometimes it is useful to be able to go the other way.
This is called splitting (or decomposing) a fraction in to partial fractions.
The first step is to factorise the denominator.
Then we find the possible denominators of the partial fractions, leaving the numerators as constants we need to find.
Finally we equate the numerators after expansion to find the values of the unknown constants.
Examples and Your Turns
Example
Write the following as a sum of partial fractions$$\frac{7x+1}{x^{2}+x-6}$$
Your Turn
Write the following as a sum of partial fractions$$\frac{3x+4}{x^{2}+3x+2}$$
Your Turn
Write the following as a sum of partial fractions$$\frac{2x-7}{x^{2}-3x+2}$$
Your Turn
Write the following as a sum of partial fractions$$\frac{2x+9}{x^{2}-x-6}$$
Example
Write the following as a sum of partial fractions in the given form$$\frac{33-12x}{\left(x+1\right)\left(x-2\right)^{2}}=\frac{A}{x+1}+\frac{B}{x-2}+\frac{C}{\left(x-2\right)^{2}}$$
Your Turn
Write the following as a sum of partial fractions in the given form$$\frac{3x^{2}+16}{\left(1-3x\right)\left(2+x\right)^{2}}=\frac{A}{1-3x}+\frac{B}{2+x}+\frac{C}{\left(2+x\right)^{2}}$$
Your Turn
Write the following as a sum of partial fractions in the given form$$\frac{3x-1}{\left(1-2x\right)^{2}}=\frac{A}{1-2x}+\frac{B}{\left(1-2x\right)^{2}}$$
Your Turn
Write the following as a sum of partial fractions in the given form$$\frac{9x-32}{\left(x-3\right)^{2}}=\frac{A}{x-3}+\frac{B}{\left(x-3\right)^{2}}$$
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.