A-level Mathematics/OCR/C2/Dividing and Factoring Polynomials

Remainder TheoremEdit

The remainder theorem states that: If you have a polynomial f(x) divided by x + c, the remainder is equal to f(-c). Here is an example.

What will the remainder be if x^3 + 8x^2 - 4x^2 + 17x - 40 is divided by x - 3?

f(3)= 3^3 + 8 \left ( 3 \right )^2 - 4\left ( 3 \right )^2 + 17\left ( 3 \right ) - 40 = 74

The remainder is 74.

FactorisingEdit

When you factor an equation you try to "unmultiply" the equation. The N-Roots Theorem states that if f(x) is a polynomial of degree greater than or equal to 1, then f(x) has exactly n roots, providing that a root of multiplcity k is counted k times. The last part means that if an equation has 2 roots that are both 6, then we count 6 as 2 roots.


The Factor TheoremEdit

The factor theorem allows us to check whether a number is a factor. It states:

A polynomial f(x) has a factor x - c if and only if f(c) = 0.

For example:

Determine if x + 2 is a factor of 2x^2 + 3x -2.

Since c is positive instead of negative we need to use this basic identity:

x + 2 = x - \left ( - 2 \right )

Now we can use the factor theorem.

2 \left (-2 \right )^2 + 3 \left (-2 \right ) -2 = 8 - 6 - 2 = 0.

Since the resultant is 0, (x+2) is a factor of 2x^2 + 3x -2.

This means it is possible to re-state the polynomial in the form (x+2)( some linear expression of x).

So 2x^2 + 3x -2 = (x+2)(ax+b)

Expanding the right hand side we get :

2x^2 + 3x -2 = ax^2 + x( 2a+b) +2b

Equating like terms we get :

2= a

2a+b = 3 and

2b = -2

Giving a= 2, b= -1 from the first and third equations and this works in the second, so

2x^2 + 3x -2 = (x+2)(2x-1)


This is part of the C2 (Core Mathematics 2) module of the A-level Mathematics text.


Dividing and Factoring Polynomials / Sequences and Series / Logarithms and Exponentials / Circles and Angles / Integration

Appendix A: Formulae
Last modified on 13 November 2010, at 14:42