Question

The number of polynomials P(x) satisfying the equation P(x²) + 2x² + 10x = 2x. P(x + 1) + 3 is

• A. 2
• B. 1
• C. 3
• D. Infinite

Answer 1

The correct option is B

1

Let P(x) be a polynomial of degree n. then the left hand side is of degree 2n in x while the right hand side is of degree ( n + 1). This means that n = 1. Let us take P(x) = ax + b. then

ax² + b + 2x² + 10x ≡ 2x [a (x + 1) + b] + 3

i.e., ( a + 2)x² + 10x + b ≡ 2ax² + 2 (a+b) x + 3

this implies a = 2 and b = 3. Hence P(x) is the polynomial 2x + 3 and is of the first degree. P(x) = 2x + 3 is the unique polynomial satisfying the given identity.