Polynomials questions (1 Viewer)

[enigma_1]

a) Prove that every odd polynomial function is zero at x=0

b) Prove that every odd polynomial p(x) is divisible by x.

c) Find the polynomial p(X) that is known to be monic of degree 3 and an odd function and has one zero at x=2.

 

[braintic]

(a) If P(x) is odd, then P(-x) = -P(x).
Specifically, P(-0) = -P(0)
ie. P(0) = -P(0)
2P(0) = 0
P(0) = 0

(b) Just follows straight on from part (a) - if a polynomial has a zero at x=0, then x is a factor

(c) Since P(x) is odd, it has only 2 terms - an x^3 term and an x term
Given also that it is monic, then P(x) = x^3 + ax
Substituting P(2) = 0 gives 0 = 8 + 2a
a=-4

So P(x) = x^3 - 4x