What's the polynomial?

Algebra Level 2

x 2 − 2 x + 2 ≤ p ( x ) ≤ 2 x 2 − 4 x + 3 x^2-2x+2 \leq p(x) \leq 2x^2-4x+3

Given that p ( x ) p(x) is a polynomial and satisfies the above for x ∈ R x\in \mathbb{R} and given that p ( 11 ) = 141 p(11)=141 , find the value of p ( 21 ) p(21) .


The answer is 561.

Vilakshan Gupta by Vilakshan Gupta , 19, India

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1 solution

Atomsky Jahid
May 24, 2020

Red, black and blue curves represent f(x), p(x) and g(x) respectively. Red, black and blue curves represent f(x), p(x) and g(x) respectively.

f ( x ) : = x 2 − 2 x + 2 = ( x − 1 ) 2 + 1 f(x) := x^2-2x+2 \\ = (x-1)^2+1 g ( x ) : = 2 x 2 − 4 x + 3 = 2 ( x − 1 ) 2 + 1 g(x) := 2x^2-4x+3 \\ = 2(x-1)^2+1 We then conclude, p ( x ) p(x) is of the following form. p ( x ) : = k ( x − 1 ) 2 + 1 w h e r e , k ∈ [ 1 , 2 ] p(x) := k(x-1)^2+1 \\ where, k \in [1,2] So, p ( 11 ) = k ( 11 − 1 ) 2 + 1 = 141 ⟹ k = 7 5 p(11) = k(11-1)^2 + 1=141 \\ \implies k = \frac{7}{5} Therefore, p ( 21 ) = 7 5 ( 21 − 1 ) 2 + 1 = 561 p(21) = \frac{7}{5} (21-1)^2 +1 = \boxed{561}

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