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