easy polynomials (1 Viewer)

[AFGHAN22]

A polynomial P(x) is odd, i.e. P(-x) = - P(x)
1. prove that P(0) = 0 and hence show that p(x) is divisible by x.
2. if b is a zero of P(x), show that –b is also a zero of P(x).
3. a polynomial P(x) is known to be odd, to be monic and have a zero -2. Show that p(x) is of at least degree 3.
State the paticukar polynomial Q(x) of degree 3 with the above properties. Is Q(x) unique?
4. A polynomial s(x) is odd, monic and has a zero -2. State the most general form of s(x) with degree 4</ d </ 6
note: </ denotes is less than or equal to

answers
3. Q(x) = 1x(x+2) (x-2) unique
4. s(x) = 1x (x-2)(x+2) (x-b)(x+b) where b is not equal to 2 or -2 , degree must be 5 since if b is zero of s(x) then so is –b

 

[haboozin]

A polynomial P(x) is odd, i.e. P(-x) = - P(x)
1. prove that P(0) = 0 and hence show that p(x) is divisible by x.
2. if b is a zero of P(x), show that –b is also a zero of P(x).

1. P(x) = axⁿ + .... + a_n-1x + a_n
P(0) = a_n = 0
ie. P(x) = axⁿ + .... + a_n-1x
P(x) = x[ax^n-1 + .... + a_n-1]

ie divisible by x

2. P(-b) = - P(b) by definition
but P(b) = 0
so P(-b) = - 0
= 0
so -b is also a root.