Need Maths help - Factor and Remainder Theorems

[Malachi]

Soleron said:

I have a A-level Maths Core 1 mock paper tomorrow, and while I am confident in most areas I am struggling with using the factor and remainder theorems in real questions (and recogninsing where to use them).Could someone explain how and when to apply them (not from scratch, I do understand the terminology). I also understand polynomial divsion, of which these thorems are shortcuts to.

Factor Theorem: If (x-a) is a factor of polynomial f(x) then f(a)=0 and x=a is a root of the equation f(x)=0

Remainder Theorem: f(a) is the remainder when polynomial f(x) is divided by (x-a).

 

Some example questions I have trouble working through - I can't see what to do (but these are only examples; I need to understand the general method rather than these specifically):

1.

a) A polynomial equation f(x) can be written in the form -2(x+3)(x-2)(x-2). Find f(0).

Change the Xs by whatever number in parenthesis, in this case 0 then make the calculation

b) The expression can also be written as f(x) = -2x^3 + bx^2 + cx + d. Find b, c and d.

Just do the previous multiplication(-2(x+3)(x-2)(x-2)), it would give you an answer of this form, ax^3 + bx^2 + cx + d, the number that are in the place of b, c and d is your answer

2.

a) f(x) = x^3 - 5x + 2. x=2 is a root of f(x)=0. Show that f(x-3) = x^3 - 9x^2 + 22x -10.

Again, put x-3 in the place of the Xs then make the calculation, it will give you x^3 - 9x^2 + 22x -10

b) Write down the roots of f(x-3)=0

This one a little tricky, see, they told you that if X=2 in f(x) it would give you zero(So f(2)=0), so that mean you need to find what number you need to put in place of X in f(x-3) for it to give you f(2)

3. f(x) = 2x^3 - x^2 - 11x - 12. The only real root of f(x) is (x-3). Show that x=2 is a root of f(x) = -22.

Again, make the calculation by putting 2 in place of the X and it will give you -22

 

Hope that help