c2 maths - 4 damn hard questions!!

hello, i am having reaall trouble on these 4 qquestions, id greatly appreciate help from even one of these, ive been stuck on these qquestions for embarrisingly long amounts of time.
pls can u explain ur steps too please, cause im a lil stupid.lol thanks in advance

q)
given (x+1) is a factor of f(x)= x^3 + x^2 - 9x - 9

solve the equation f(x) = 0

q2)a sequence of positive numbers is u1,u2,u3, u4,....un (numbers are small and below u as in C02) , where un = 2^n

prove that un+1 - un = un for all positive integers n.

b) fina an expression for un+m - un in terms of n and m, where n and m are positive integers.

q3)when x^4 -75 is divided by (x-B) where B is a positive integer , the remainder is 6. Find B.

b) find the remainder when x^4 -75 is divided by (x-2B)

q4) determine the remainder when x^4 -4 is divided by (x-2)
b)determine the remainder when x^4 -4 is divided by (x - root 2)
c) determine two factors of x^4 - 4

Reply 1

Geminus

Q1.

You're given that (x+1) is a factor, so you can divide f(x) by (x+1). You can either use long division method/equating coefficients method - depends which way you've been taught/prefer. By taking out this factor you should be left with a quadratic equation, which you can then go on to solve. The equation that you're left with is x^2 - 9; which you can then solve. Post again if you don't understand where that's come from.

Reply 2

Geminus

Q3.

You can use the remainder theorem here. It states that when a function is divided by a factor (e.g. x-B), then the remainder will equal f(B). If you substitute in, then you should get a value for B (as B^4 - 75 = 6). I got that B was 3.

For the second part, just use the value of B that you've found in part a, and use the remainder theorem again. Again, if you want full working, just shout!

Q4 is based upon the same sort of thing really, so you might be able to have a go at them now :smile: