How do I factor a polynomial with a coefficient in front of the highest power?

[Matt1234]

Homework Statement

8x^3 + 4X^2 - 2x -1

Homework Equations

remainder theorem
factor theorem

The Attempt at a Solution

None yet, not sure how to go about it when there is a coefficient in front of the highest power of X.

 

[rock.freak667]

so if 8x³ + 4x² - 2x -1=0 and we divide by 8, the last term is -1/8. What product of fractions will give you -1/8?

-1/8 and 1 would give -1/8 do either of these satisfy the equation?

What are the other options to choose from?

 

[Matt1234]

is that the way we do it? Divide all by 8. I thought about that but thought it was wrong.

Just noticed you made them equal to zero. Why did you do that? The original question does not make them equal to zero.

I divided all by 8 and noticed neither -1/8 nor 1 satisfy the remainder theorem

Also, i can't seem to find a number that makes them zero when using the remainder theorem. I need some more help, sorry.

 

[rock.freak667]

is that the way we do it? Divide all by 8. I thought about that but thought it was wrong.

Just noticed you made them equal to zero. Why did you do that? The original question does not make them equal to zero.

I divided all by 8 and noticed neither -1/8 nor 1 satisfy the remainder theorem

Also, i can't seem to find a number that makes them zero when using the remainder theorem. I need some more help, sorry.

Well we don't need to divide by 8, but we are looking at solution of the form ax+b for when the coefficient of the highest degree in the polynomial is not 1.

to make -1/8...you can multiply -1/2 and 1/4 or -1/4 and 1/2. Do any of these make the equation zero?

 

[Matt1234]

f(1/2) works

So one factor is (x - 1/2)

I will divide to find the second part:
Second part = 8x^2 +8x +2

And i forgot how to factor that, lol. using quadratic formula cause i can't do it in my head right now.

 

[Matt1234]

Here is what i got:

(x- 1/2) (4x+2) (2x +1) I don't have an answer key is this correct?

Sorry for asking so much of this forum, i promise one day if i learn i will repay by helping others.

 

[rock.freak667]

Here is what i got:

(x- 1/2) (4x+2) (2x +1) I don't have an answer key is this correct?

Sorry for asking so much of this forum, i promise one day if i learn i will repay by helping others.

yes but that can be further simplified as 4x+2 =2(x+1), that 2 can be expanded into the x-1/2 to give integer coefficients.

 

[Dick]

Here is what i got:

(x- 1/2) (4x+2) (2x +1) I don't have an answer key is this correct?

Sorry for asking so much of this forum, i promise one day if i learn i will repay by helping others.

That's right. You could also write that as (2x-1)*(2x+1)^2. Factor a 2 out of the second factor and move it into the first. Looking forward to your helping help!

 

[lurflurf]

ley y=2x
8x^3 + 4X^2 - 2x -1
(2x)^3+(2x)^2-(2x)-1
y^3+y^2-y-1
factor by grouping
gcd(y^3,y^2)=y^2
gcd(-y,-1)=-1
(y^2)(y+1)+(-1)(y+1)
and so on

 

[Matt1234]

Thanks guys i got it now. :)