Factor the following over the set of rational numbers

In summary, the conversation is about factoring the expression cos³ x-1 over the set of rational numbers and simplifying if possible. The expression is interpreted as (cos(x))^3-1 and factors to (cos(x)-1)*(cos(x)^2+cos(x)+1). The conversation also touches on the use of parentheses to clarify meaning and the importance of asking for clarification when unsure.
  • #1
dranseth
87
0

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.
 
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  • #2
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.
 
  • #3
so would this be fully factored over the set of rational numbers?

cos (x-1)(x^2+x+1)
 
  • #4
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.
 
  • #5
I'm quite confused right now. On the assignment page, it is written as: cos³ x-1
 
  • #6
By the usual rules of precedence, that is interpreted as (cos(x))^3-1. Not cos^3(x-1). They are two different things.
 
  • #7
(cosx - 1)(cos2x + cosx + 1) ?
 
  • #8
Write carefully. Does cos2x mean cos^2(x) or cos(2x)?
 
  • #9
Acutally, would it not be cos(x)^2 ??
 
  • #10
cos2(x) is a standard notation for (cos(x))2.
 
  • #11
so what you are saying is: Cos³ x-1 = cos(x-1)³ = (cosx-cos1)(cosx-cos1)(cosx-cos1)
 
  • #12
dranseth said:
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).
 
  • #13
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 ?
 
  • #14
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.
 

Related to Factor the following over the set of rational numbers

1. What does it mean to "factor over the set of rational numbers"?

Factoring over the set of rational numbers means finding the rational number factors of an expression. This involves breaking down the expression into smaller parts that can be multiplied together to get the original expression.

2. Why is it important to factor over the set of rational numbers?

Factoring over the set of rational numbers is important because it helps simplify complicated expressions and equations. This can make solving problems and finding solutions much easier and more efficient.

3. How do you factor an expression over the set of rational numbers?

To factor an expression over the set of rational numbers, you can use various techniques such as finding common factors, using the distributive property, or using the quadratic formula. It also helps to have a good understanding of the rules of fractions and rational numbers.

4. Can all expressions be factored over the set of rational numbers?

No, not all expressions can be factored over the set of rational numbers. Some expressions may have irrational or imaginary solutions that cannot be represented as rational numbers. In these cases, other methods must be used to solve the expression.

5. Are there any strategies or tips for factoring over the set of rational numbers?

One strategy for factoring over the set of rational numbers is to start by finding common factors and then using the distributive property to break down the expression further. It can also be helpful to practice and familiarize yourself with different factoring techniques and to check your work by multiplying the factors back together to ensure you have factored correctly.

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