A-level Mathematics/OCR/C2/Dividing and Factoring Polynomials
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 is divided by x - 3?
The remainder is 74.
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:
Determine if x + 2 is a factor of .
Since c is positive instead of negative we need to use this basic identity:
Now we can use the factor theorem.
Since the resultant is 0, (x+2) is a factor of .
This means it is possible to re-state the polynomial in the form (x+2)( some linear expression of x).
So = (x+2)(ax+b)
Expanding the right hand side we get :
Equating like terms we get :
2a+b = 3 and
2b = -2
Giving a= 2, b= -1 from the first and third equations and this works in the second, so