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

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:

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)