Question

A polynomial p(x) of degree n is such that p(a) = 0 and p(-b) = 0. Which of the following is the factored form of the polynomial?
a) (x − a)(x + b)g(x); where g(x) is a polynomial of degree n − 2
d) (x − a)(x + b)g(x); where g(x) is a polynomial of degree n
c) (x + a)(x + b)g(x); where g(x) is a polynomial of degree n − 2
d) (x + a)(x + b)g(x); where g(x) is a polynomial of degree n​

Answer 1

Answer:

f(x) → degree(n)

f(1)=f

2

f(3)=π

f(x)=ax

n

+bx

n−1

+cx

n−2

If degree 0 f(x)=a=constant f(1)=f(3) but it is not true

If degree1 f(x)=ax+b f(1)=a+b 2 f(3)=3a+b=π

Two variables & two unknown

a & b can be found uniquly

∴ one polynomial used only

For n > 1

Let n = 2 ax

2

+ bx + c

We have 3 variable & only 2 equations can be formed from given condition

Hence infinite such polynomial can be formed.