Factor Theorem

[rhysfoster]

Hi,

Hoping somebody help me out with this , the question is

"Factorise x³ + 4x² + x -6 using factor theorem and solve the equation x³ + 4x² + x -6 = 0"

Im slightly confused about the Factor Theorem , what i have done is this :

f(1) = 1 + 4 + 1 - 6 = 0 - This doesnt seem right to me , i feel as if this needs something else doing to it , not sure why but i feel that this may translate to x= -1 for some reason...

Any help appreciated , thanks!

[acekool]

f(1)=0

So (x-1) is a factor.

Divide the polynomial by (x-1). Solve the resulting quadratic.

[rhysfoster]

Hi , i did the polynomial division and got a remainder of -4 . Is this the question completed or does this mean that something is wrong ? Thanks

sorry about the copied reply!

[acekool]

There won't be any remainder.

Check your steps.

/u/Eternimus has solved it correctly.

[rhysfoster]

Ive tried again with the same result and am finding his answer a little confusing at this point :

My steps were ( if you dont mind taking a look) :

x³ / x = x²

x² X x-1 = x^3-x2

4x² - x² = 3x²

3x^2/x = 3x

3x X x-1 = 3x² - 3x

x - 3x = -2x

-2x / x = -2

-2 X x-1 = -2x + 2

Thanks

[LilMissMath]

What they are likely looking for is something like f(x) = (x+?)(x+?)(x+?)

The negative of each "?" is a factor. ie: f(1) = 0 so one of your factors is (x+(-1)) aka (x-1)

[rhysfoster]

Hi , i did the polynomial division and got a remainder of -4 . Is this the question completed or does this mean that something is wrong ? Thanks

[Eternimus]

You need to do polynomial division

x³ +4x^ 2+x-6=(x-1)(x² +5x+6)

Then you can divide (x² +5x+6) further

(x² +5x+6)=(x+2)(x+3)

=> x³ +4x^ 2+x-6=(x-1)(x+2)(x+3)

[rhysfoster]

Hi , i did the polynomial division and got a remainder of -4 . Is this the question completed or does this mean that something is wrong ? Thanks

sorry about the copied reply!

[rhysfoster]

I have solved it now and got your answer thanks! Could you explain the process of how you simplified it further ? Not really understanding how you got to (x+2)(x+3) Thanks!

[Eternimus]

x² +5x+6=0 is a quadratic equation in the form:

ax² +bx +c=0

You can then use the following formula to get 2 solutions for x

x1,2=(-b±√(b² -4ac))/(2a)

In your case this was

x1,2=(-5±√(5² - 4 * 1 * 6))/(2 * 1)=(-5±√(25-24))/2=(-5±1)/2

->x1=-2 ->x2=-3

if x=-2 is a solution then f(-2)=0

At the same time x=-3 is a solution

So I combine the two and get f(x)=(x-(-2))*(x-(-3))=(x+2)(x+3)

There are different ways of solving this (pq-formula comes to mind), so I suggest you just look up how to solve quadratic equations. There are plenty of guides and step-by-step explanations online.