Polynomials Question HELP (1 Viewer)

[kayven]

Hello,

How would i approach this question and how would i solve it? Please show all working out and reasoning.

P(x) is a monic polynomial of degree 4 with integer coefficents and constant term 3. One zero is 'i' , another zero is rational and the sum of the zeros is negative. Factorise P(x) fully over R.

Thank you so much.

 

[SpiralFlex]

By conjugate root theorem,



So,






The zeros are

But the constant term is 3.

Let us discuss, since the sum is negative, the zeroes must be and

So,

(Over the real field.)

 

[bleakarcher]

spiralflex, if you dont mind me asking can you tell me what the conjugate root theorem is?

 

[SpiralFlex]

spiralflex, if you dont mind me asking can you tell me what the conjugate root theorem is?

Conjugate root theorem in teenage language terms is if, you have a real polynomial (ie. a polynomial with all real or integer coefficients.)

Eg.

And you know something is a root, eg.

Then the conjugate, is also a root.

 

[kayven]

Spiral Flex,

How can you just say that the zeroes are +- 1 and +-3? Thats the only part i dont understand.

Thanks again.

 

[SpiralFlex]

Remember,

If I have something like

If I factorised quickly in my head.



Notice how the factors multiple to get a constant term of 6?

Similarly,





So it must equal to 3, since constant term is three.

 

[kayven]

oh i see thank you so much.

How do you do it so quickly in your head, do you use factor/remainder theorem and then use long division? :O

 

[SpiralFlex]

oh i see thank you so much.

How do you do it so quickly in your head, do you use factor/remainder theorem and then use long division? :O

Inspection method:






We look at as the coefficient. Our polynomial,

We already have

To get rid of it, we must