Quadratic formula: 2sin^3x+3sin^2x-3sinx-2

[messa]

Hello, for the following equation, is it okay to take a sinx out of each of the terms, then use the quadratic formula on (2sin^2x+3sinx-5)?

2sin^3x+3sin^2x-3sinx-2

 

[arthur ohlsten]

2 sin^3x +3sin^2x-3sinx-2

let z=sinx
2z^3+3z^2-3z-2 but at z=1 f[z]=0, divide by z-1
[z-1][2z^2+5z+2] factor second term
[z-1][2z+1][z+2]
substitute

[sinx -1][2sinx +1][ sinx+2] answer

Arthur

 

[messa]

how do you know that at z=1 f(z)=0?

 

[arthur ohlsten]

Usually when the instructor gives you a problem, the point is to teach something you are studying. I doubt if you are studying how to find roots.
I will assume the roots,or at least one, is a integer. try simple integers, such as 0,1,-1,2,-2
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2z^3+3z^2-3z-2
let z=0
2[0]+3[0]-3[0]-2
remainder -2 not a root

let z=-1
2[-1]^3+3[-1]^2-3[-1] -2
-2+3+3-2
remainder 2 not a root

let z=1
2[1]^3 +3[1]^2-3[1]-2
2+3-3-2
remainder 0; 1 is a root
z=1 or [z-1] is a factor

then factor the equation where one of the factors is [z-1]
[z-1][whatever we got, I don't remember]

Arthur

 

[messa]

Thank you, I understand that.

 

[arthur ohlsten]

There are other ways to determine how many real positive roots, or real negative roots. a equation has.
I assume you haven't studied these methods yet, so you can determine them by other means,.
Just keep in mind that all problems given to you by an instructor have solutions, and probably have simple integer solutions.
hope this helps
Arthur