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LearnAQA7367First teaching 2017Paper 2 · Compulsory contentFurther algebra and functions

Paper 2 · Module

Further algebra and functions

4 subtopics≈ 155 min in total

Start the module: Roots of polynomial equations Open graphing calculator

Roots of polynomial equations
For $ax^2+bx+c=0$ : $\alpha+\beta=-\frac ba$ , $\alpha\beta=\frac ca$ . Cubics and quartics have similar sums of roots taken one, two, three at a time. These let you evaluate expressions in the roots, and build new equations, without solving.
≈ 40 min
Summation of series
$\sum r=\frac12n(n+1)$ , $\sum r^2=\frac16n(n+1)(2n+1)$ , $\sum r^3=\frac14n^2(n+1)^2$ . Combine them for polynomial sums. When each term is a difference $f(r)-f(r+1)$ , almost everything cancels: the method of differences.
≈ 40 min
Rational functions and asymptotes
A rational function $y=\frac{P(x)}{Q(x)}$ has vertical asymptotes where $Q(x)=0$ , and its behaviour for large $|x|$ (horizontal or oblique asymptote) comes from dividing. Writing $y=f(x)$ as a quadratic in $x$ and using the discriminant gives the range and turning points.
≈ 40 min
Rational inequalities
Never multiply an inequality by an expression that might be negative. Multiply by the square of the denominator instead (always positive), collect on one side, factorise, and use a sketch or sign table. Exclude values that make a denominator zero.
≈ 35 min