# 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 $$P(10)$$.

Note by Fidel Simanjuntak
12 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears 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} $$\boxed{123}$$

Sort by:

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 $$P(x)$$?

- 12 months ago

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

- 12 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!

- 12 months ago