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 is divided by x - 3? 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 has a factor x - c if and only if . For example: 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 : 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 = (x+2)(2x-1)