Factor the following over the set of rational numbers

[dranseth]

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]

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.

 

[dranseth]

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

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

 

[Dick]

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.

 

[dranseth]

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

 

[Dick]

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

 

[dranseth]

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

 

[Dick]

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

 

[dranseth]

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

 

[HallsofIvy]

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

 

[dranseth]

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

 

[Dick]

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).

 

[dranseth]

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]

I just knew you would do that next. Use parentheses to clarify your meaning. cos(x^3)-1, cos(x)^3-1, cos(x^3-1), cos((x-1)^3), (cos(x-1))^3, they are ALL completely different. Get a calculator and pick a number for x, say x=0.5 and evaluate them all and collect the answers. Parentheses indicate that the thing inside of the parentheses is evaluated before the operations outside are done. You will get different numbers for each one. They are not equal or equivalent. You've switched, I think now, ALL of them. If you don't understand which of these the question concerns, please ask your instructor. It's pretty clearly, to me, (cos(x))^3-1. If you don't understand what that means, please ask your instructor. That factors to (cos(x)-1)*(cos(x)^2+cos(x)+1). If you don't believe me try verifying it with a calculator for your selected value of x. If you get the wrong answer, please ask your instructor.