 # Factorising techniques for SQA Higher Maths 3. Factor Theorem

When given a quadratic such as (x + 1)(x + 2), we can multiply out the brackets using FOIL (meaning first, outside, inside, last), giving x² + 2x + x + 2, which is just x² + 3x + 2. However, for solving a quadratic it's more useful to do this process in reverse. Given some quadratic like x² + 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. We have a quadratic equation of the form ax² + bx + c = 0, where a, b, and c are just coefficients. The solutions to the quadratic are given by the quadratic formula: x = (–b +– √(b² – 4ac))/2a. For a polynomial f(x), if f(a) = 0 then we know that (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. # 1

Identify a factor of f(x) = x⁵ + 1.

Notice that f(–1) = 0, hence by the factor theorem we know (x + 1) is a factor of the polynomial. # 2

Factorise x² + 5x + 6.

We want something of the form (x + _)(x + _) and we know when multiplied together, the values in the gaps should equal 6. They could either be 1 and 6, or 2 and 3 (or also –1 and –6, or –2 and –3). However the only pair of values here that add up to 5 is 2 and 3, so we have (x + 2)(x + 3). # 3

Solve 22x + x² = –21.

One needs to rearrange and be aware of the ordering of terms. We have a = 1, b = 22 and c = 21, which in the quadratic formula gives our solutions x = –1 and x = –21. # 4

Identify a factor of f(x)= x³ + x² + x – 3.

Notice that f(1) = 0, hence by the factor theorem (x – 1) is a factor of the polynomial. # 5

Factorise x² + x – 2.

One needs to notice that –1 and 2 multiplied together give –2, but when added together give 1 (the coefficient of x in our equation). The factorised equation is therefore (x – 1)(x + 2). End of page