Polynomials (1 Viewer)

[cutemouse]

A monic polynomial P(x) of degree 4 has exactly two zeros at x=3 and x=-3 and it is even.

(i) Show that there are many such polynomials and find their general form.

(ii) If this polynomial has a value of -45 when x=2 find the unique polynomial P(x)

Thanks =D

 

[Timothy.Siu]

A monic polynomial P(x) of degree 4 has exactly two zeros at x=3 and x=-3 and it is even.

(i) Show that there are many such polynomials and find their general form.

(ii) If this polynomial has a value of -45 when x=2 find the unique polynomial P(x)

I just need help with (i) really... WTH is the general form?

i) P(x)=k(x-3)^2(x+3)^2

ii)P(2)=25k=-45
k=-9/5

 

[cutemouse]

Eh? How does that work out? :S Where does it say that they are double roots?

EDIT: Also P(x) is monic, so what's the 'k' all about?

 

[Timothy.Siu]

my bad i'll do it again

 

[gurmies]

Let P(x) = (x+3)(x-3)(x^2+bx+c) ===> (x^2-9)(x^2+bx+c)

P(x) = x^4 + bx^3 + cx^2 - 9x^2 - 9bx - 9c

Since even polynomials can only have even powers, b = 0

P(x) = (x+3)(x-3)(x^2+c)

(ii) P(2) = -45

(2+3)(2-3)(4+c) = -45

-5(4+c) = -45

4+c = 9

c = 5

Therefore, P(x) = (x-3)(x+3)(x^2+5)

Sorry Tim if you were mid-solution or something

 

[kurt.physics]

The polynomial will be in the form



with













edit: beaten

 

[gurmies]

The polynomial will be in the form



with













edit: beaten

Kurt, where did you get that d>0?

EDIT: Just read the question and noticed ONLY two roots =]

 

[Timothy.Siu]

lol, nah it was too hard for me.