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OCR A-level Maths Factorising techniques

Factorising techniques

This page covers the following topics:

1. Factorising linear equations
2. Factorising quadratic equations
3. The quadratic equation
4. Factor Theorem

When solving linear equations we generally want to rearrange it, and make some variable the 'subject' of the formula. Something is the subject of the formula if it appears by itsel on one side of the equation. For example, if y = 2x + 3, then y is the subject. To solve an equation for y, we want to isolate it like that on one side, and then evaluate the terms on the other side with numbers we know.

Factorising linear equations

When given a quadratic such as (x + 1)(x + 2), we can multiply out the brackets using FOIL (meaning first, outside, inside, last), giving xΒ² + 2x + x + 2, which is just xΒ² + 3x + 2. However, for solving a quadratic it's more useful to do this process in reverse. Given some quadratic like xΒ² + 3x + 2, we want to find 2 values to fit in (x + _)(x + _). We know these values need to equal 2 when multiplied together, and when summed they need to equal 3. Once we spot the values 1 and 2 for this, we have (x + 1)(x + 2), and it's very easy to solve an equation like (x + 1)(x + 2) = 0, because this just gives our solutions x = –1 and –2.

Factorising quadratic equations

We have a quadratic equation of the form axΒ² + bx + c = 0, where a, b, and c are just coefficients. The solutions to the quadratic are given by the quadratic formula: x = (–b +– √(bΒ² – 4ac))/2a.

The quadratic equation

For a polynomial f(x), if f(a) = 0 then we know (x – a) is a factor of the polynomial. Likewise, is we know (x – a) is a factor then f(a) = 0. While not particularly useful for quadratics, this can prove helpful in factorising more difficult polynomials of order 3 or above.

Factor Theorem

1

Identify a factor of f(x) = x⁡ + 1.

Identify a factor of f(x) = x⁡ + 1.

2

Factorise xΒ² + 5x + 6.

Factorise xΒ² + 5x + 6.

3

Solve 22x + xΒ² = –21.

Solve 22x + xΒ² = –21.

4

Identify a factor of f(x)= xΒ³ + xΒ² + x – 3.

Identify a factor of f(x)= xΒ³ + xΒ² + x – 3.

5

Solve s – 4 = b + 3 for b when s = 4.

Solve s – 4 = b + 3 for b when s = 4.

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