Algebraic division

Degree – the highest power of x in the polynomial (e.g. the degree of 4x5 + 6x2 – 3x – 1 is 5)

Divisor – this is the thing you're dividing by (e.g. if you divide x2 + 4x – 3 by x + 2, the divisor is x + 2)

Quotient – the stuff you get when you divide by the divisor (not including the remainder)

Remainder – the part that's left over at the end (for A-level maths this will be constant)

Factor theorem links roots and factors. If f(x) is a polynomial and f(a) = 0, then (x – a) is a factor of f(x)

If f(b/a) = 0, then (ax – b) is a factor of f(x)

To show (2x + 1) is a factor of f(x) = 2x3 – 3x2 + 4x + 3, show that f(-1/2) = 0

The quartic polynomial g(x) = x4 – 8x3 – 87x2 + 342 + 1512 has roots at x = 12, 6, -3, and -7

If A is a root of g(x) then g(a) = 0, so (x – a) is a factor of g(x)

To divide by subtracting multiples of the divisor, always get rid of the highest power of x

Method 2 of algebraic division uses long division, similar to numerical division

The formula for algebraic division is f(x) = q(x)d(x) + r(x) where q(x) is the quotient, d(x) is the divisor, and r(x) is the remainder

The degree of the quotient is deg f(x) – deg d(x), and the degree of the remainder must be less than the degree of the divisor

To work out constants in algebraic division, substitute values for x and equate coefficients

In the example, when dividing x4 – 3x3 – 3x2 + 10x + 5 by x – 2, the degree of the remainder is 0

Putting the values into the original identity gives the final answer for the division

So the final answer for dividing x4 – 3x3 – 3x2 + 10x + 5 by x – 2 is (x3 – x2 – 5x) remainder 5