# Factor the following over the set of rational numbers

## Homework Statement

Factor the following over the set of rational numbers. Simplify if possible.

cos³ x-1

I do not know how to deal with the cubic cosine. Help is greatly appreciated.

Dick
Homework Helper
If you put cos(x)=1 then that expression is zero. That tells you that (cos(x)-1) is a factor. Divide (cos(x))^3-1 by cos(x)-1. More generally any expression of the form a^3-b^3 can be factored in the same way.

so would this be fully factored over the set of rational numbers?

cos (x-1)(x^2+x+1)

Dick
Homework Helper
Nooo. That's all mishmashed. I thought the expression you gave was a^3-1 where a=cos(x). That factors into (a-1)(a^2+a+1) all right. Substitute a=cos(x) into that. There shouldn't be any cos(x-1) or bare powers of x floating around.

I'm quite confused right now. On the assignment page, it is written as: cos³ x-1

Dick
Homework Helper
By the usual rules of precedence, that is interpreted as (cos(x))^3-1. Not cos^3(x-1). They are two different things.

(cosx - 1)(cos2x + cosx + 1) ???

Dick
Homework Helper
Write carefully. Does cos2x mean cos^2(x) or cos(2x)?

Acutally, would it not be cos(x)^2 ??

HallsofIvy
Homework Helper
cos2(x) is a standard notation for (cos(x))2.

so what you are saying is: Cos³ x-1 = cos(x-1)³ = (cosx-cos1)(cosx-cos1)(cosx-cos1)

Dick
Homework Helper
so what you are saying is: Cos³ x-1 = cos(x-1)³ = (cosx-cos1)(cosx-cos1)(cosx-cos1)

No!!! Worse, and worse. You basically had it when you wrote "(cosx - 1)(cos2x + cosx + 1)". I was just suggesting it would be clearer to write cos^2(x) rather than cos2x because I assumed that's what you meant. The more you write, the more I worry about you. cos(x-1) IS NOT equal to cos(x)-cos(1).

Sorry, I have never dealt with cosine to any degree like this before.

Let me double check that I have it correct now. Is cos³x-1 the same as cosx³-1 ?

Dick