Whats the best way to factor this

[shreddinglicks]

Homework Statement

2cos^2(θ) + cos(θ) - 1

Homework Equations

2cos^2(θ) + cos(θ) - 1

The Attempt at a Solution

the answer is (2cos(θ)-1)(cos(θ)+1)

How do I factor this to give me that answer?

 

[Student100]

Do a substitution, x = cos(theta)

Now you have 2x^2 + x -1.

You know the signs of the factors must be a plus and a minus, so ( + ) ( - ) the two on your x squared let's you know that you need a two x. And the 1 tells you both end digits must be 1. (2x-1) (x +1) is the only combination that works to leave you with one quantity of positive x.

 

[HakimPhilo]

Here's an outline of the solution: As pointed out by Student100 the substitution $\cos\theta\leadsto x$ produces a quadratic, and if you know the roots of a quadratic -- say $ax^2+bx+c$ with roots $r_1,r_2$ if any -- then you can factor it as $a(x-r_1)(x-r_2)$. The only step that remains is to “undo” the substitution by replacing $x$ with the original $\cos\theta$. I hope you can achieve all of those steps to find your desired answer. ;-)