firstly I divide f(x) by one of the factors (x + 1). Then I take the quotient and divide that by the other factor (x - 1). I am now left with the quadratic 3x^2 + 7x + 4 which I factorise into (3x + 4) and (x + 1).

Thus the factors are:

(3x + 4)(x - 1)(x + 1)(x + 1) == (3x + 4)(x - 1)(x + 1)^2

As always, thanks for the help ]]>

Those two terms are both included in f(x), so dividing cancels them out.

f(x)/(x² - 1) = (4+3x)(x+1)

]]>f(x)= (x^2-1)(4+3x)(1+x)= (x-1)(4+3x)(1+x)^2,

which is divisible by x^2-1

Well done ricky and fgarb!]]>

I cannot seem to make this work

x^2 - 1 / 3x^4 + 7x^3 (just to start)

so I divide 3x^4 by x^2 to give me 3x^2.

I then multiply 3x^2 by x^2 and then by -1 to give me 3x^4 - 3x^2.

I place this underneath 3x^4 + 7x^3 and subtract it, 3x^4 - 3x^4 = 0 as I would expect but I cannot subtract -3x^2 from 7x^3...

]]>Do you know what is the polynomial division?

Nope, no idea

I just looked at what I could do to (x - 1) to make (x^2 - 1). I only got (x - 1) in the first place beause I misread the book! So that was luck, then fgarb pointed out that it was one of the factors anyway.

f(1) = 3(1)^4 + a(1)^3 + b(1)^2 - 7(1) - 4 = 0

a + b = 8

f(-1) = 3(-1)^4 + a(-1)^3 + b(-1)^2 - 7(-1) - 4 = 0

-a + b = -6

Thus: a = 7 and b = 1.

How do I go about factorising f(x) now? Do I use the same long division method I have been using on cubic polynomials?

Well done!!

"Given that (x² - 1) is a factor of the polynomial f(x), where f(x) = 3x^4 + ax^3 + bx^2 - 7x - 4, find the values of a and b and hence factorise f(x) completely."

Do you know what is the polynomial division?

It may help you to reduce the power of f(x).

Just divide f(x) by (x^2-1) and then the remainder which you will get must be divisible to (x^2-1).

Maybe I must simply guess the other factors and put them into f(x) and test different values of a and b from f(1) till I find another that makes f(x) = 0?

(x^2 - 1) / (x - 1) = ???

]]>Then, you know that f(x) has to be divisible by both x-1 and the other factor that you found. See if you can use both of those conditions to find a and b. Good luck!

]]>Try breaking (x^2-1) up into its constituent factors and applying them part by part

How do I do this? I've never seen such a factor before.

]]>