Polynomial

Given $P$ is a polynomial that is not constant, satisfy $P\left( P(x) \right) = (x^2 + x +1) \times P(x)$ for every real $x$ . Find $P(10)$ .

#Algebra

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Hint: Let $d$ denote the degree of polynomial $P(x)$ , then the degree of LHS is $2d$ , and the degree of RHS is $d+2$ . So can you solve for $d$ ? And can you determine the function $P(x)$ ?

Pi Han Goh - 3 years, 11 months ago

I got $P(x) = x^2 + x$ but I dont know, maybe there's another possible formula for $P(x)$

Fidel Simanjuntak - 3 years, 11 months ago

No, there isn't. If you try for (constant of quadratic polynomial not equal 0), then you get a system of equations that has no solution.

I know that this isn't the best/fastest way to get the answer, but it works!

Pi Han Goh - 3 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$