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

< A-level Mathematics‎ | OCR‎ | C2

## Remainder Theorem

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.

## Factorising

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 Theorem

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