Anything's possible, am I right?

The polynomials $x^2 + 2$ and $x^2+x-1$ are written on a board. If the polynomials $f(x)$ and $g(x)$ are already on the board ( $f(x)$ may equal to $g(x)$ ) we are allowed to add any of $f(x) g(x), f(x) -g(x)$ or $f(x) + g(x)$ to the board.

(a) Can the polynomial $x$ ever be made to appear on the board?

(b) Can the polynomial $4x-1$ ever be made to appear on the board?

Give proof.

$f(x)$ and $g(x)$ can represent any two polynomials on the board

#Algebra #Sharky

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

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

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}`	$\boxed{123}$

Comments

(a) Suppose $a, b$ are integers. If $f(a), g(a)$ are divisible by $b$ , then $f(a)+g(a), f(a)-g(a), f(a)g(a)$ are all divisible by $b$ as well. Thus the property "divisible by $b$ at $x = a$ " is an invariant; they will hold for all polynomials generated.* At $x = 3$ , $f(x) = g(x) = 11$ and thus both of them are divisible by $11$ , so all polynomials generated will be divisible by $11$ on $x=3$ . But $x$ doesn't satisfy this (since it gives $3$ on $x=3$ ). So it cannot appear.

(*) We can use induction to formalize this, inducting on the number of operations required to generate a polynomial. The base case is $f, g$ that takes zero steps; the induction step lies on the fact that if a polynomial $p$ can be generated (in the shortest way) from either of $f+g, f-g, fg$ , then $f,g$ must be generated before $p$ , so we can use induction there.

(b) It can appear. Let $F_1(x) = x^2 + 2, F_2(x) = x^2 + x - 1$ . Then,

$F_3(x) = F_1(x) - F_2(x) = x - 3$
$F_4(x) = F_3(x) F_3(x) = x^2 - 6x + 9$
$F_5(x) = F_1(x) - F_4(x) = 7x - 10$
$F_6(x) = ((F_5(x) - F_3(x)) - F_3(x)) - F_3(x) = 4x - 1$

Ivan Koswara - 5 years, 7 months ago

What is the reason for choosing $x = 3$ ? In particular, if we were given 2 other (quadratic) polynomials, how do we know what to use?

Can we classify the set of polynomials which can be reached through these operations?

Hint: Bezout's Identity.

Calvin Lin Staff - 5 years, 7 months ago

Nice question :)

Hint: Find an Invariant.

Hint: Evaluate the polynomial functions at a particular point.

Calvin Lin Staff - 5 years, 7 months ago

I don't really get your problem. What are g(x) and f(x)? Are g(x) and f(x) equal to $x^{2}+1$ and/or $x^{2} +x-1$

Julian Poon - 5 years, 7 months ago

I think f(x) and g(x) some other polynomial function

Department 8 - 5 years, 7 months ago

It's any $f(x), g(x)$ that is written on the board.

Jake Lai - 5 years, 7 months ago

$f(x)$ and $g(x)$ are any two polynomials on the board.

Sharky Kesa - 5 years, 7 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}`	$\boxed{123}$