Thread: Remainder and Factor Theorem HELP

well i missed a week off school, and i am completely in the dark on my homework, if someone could please tell me what to do it would be GREATLY appreciated
1. The function f is given by f(x) = 2x 3 + x 2 – 13x + 6.
(a) Find f(–2), f(½) and f(2).
(i) Write down the remainder when f(x) is divided by x + 2.
(ii) Explain why 2x – 1 is a factor of f(x).
(b) Factorise f(x) and hence solve f(x) = 0.
(c) Sketch the curve y = f(x).

2. Factorise p(x) = 13x - 6 - 4x 3. HINT you may have to divide for this one!
Solve p(x) = 0.
Sketch the graph of y = 13x - 6 - 4x 3.

3. (a) Use the remainder theorem to find the remainder when x 3 – 2x 2 + 8x – 3 is divided by x – 5.
(b) When x 3 – 4x 2 + ax + 6 is divided by x – 3 the remainder is 18.
Find the value of the constant a.

4. When x 3 + px 2 + qx + 1 is divided by x – 2 the remainder is 9 and when divided by x + 3
the remainder is 19. Find the values of the constants p and q.

5. Find the value of the constant k given that x + 4 is a factor of f(x), where
f(x) = 3x 3 + 10x 2 – kx - 4.
Hence factorise f(x) and solve f(x) = 0.


in general i have no idea what f(x) means, is it something times what x is, please please help me.

f(x) is the given function. I'll try to help you with the first one, and hopefully you'll understand how to do it from there.
a.)

Plug in -2 whenever you see x

, might be off on the arithmetic
Do the same for the other values, just plug in the given value whenever you see a x and simplify.
i.)
Polynomial Long Division

Here is a website that shows you how to do polynomial long division.
ii.)
If there is no remainder, than the divisor is a factor of the entire polynomial. (e.g. 24/3 = 8, no remainder, so 3 is a factor)
b.)
To factor the polynomial, you divide it by its factors, and you are already given one. Divide first by , and you will get a quadratic equation of the form , and that should be simply to factor.

Once you have it all factored, say it turns out to be something like (2x-1)(x+3)(x-2). Set this equal to 0, and solve for x
This is only true when any one of the factors is 0,so you have




Solving these for x, we see that