A-Level Maths & Further Maths/Polynomials

Polynomials are expressions involving powers of a term, often x, where the power is two or more. A power of 1 would be a linear expression, and zero would make it a constant (just 1). They should become easier to understand as you work through them.

Quadratics should be a familiar term from GCSE learning. Quadratic equations are algebraic equations which involve a squared term, and often a linear and constant term.

The first skill you need to have is to factorise quadratics. An example of the method to do so is below.

Example

Factorise ${\displaystyle x^{2}+6x+8}$ . If you can recall from GCSE, you may look for two numbers that add together to make 6, and multiply to make 8. If you thought of 4 and 2, you would be correct. This method works for all quadratics where the coefficient of x2 is one, such as ${\displaystyle x^{2}+42x+377}$ . You just need to find the two factors, then split the x term, and factorise the first and last two terms, as below.

{\displaystyle {\begin{aligned}x^{2}+42x+377&=x^{2}+13x+29x+377\\&=x\cdot (x+13)+29\cdot (x+13)\\&=(x+13)\cdot (x+29)\end{aligned}}}

What about when the coefficient isn't one? You may be able to take the coefficient out as a factor of the whole expression. If not, split the middle term and try to factorise; while this gets harder with bigger coefficients, it will always work, and you'll get better with more practise.

 To do:- finish example; - quadratics - completing square, formula; - higher order polynomials; - sketching and properties of graphs;