Problem 4: Symmetrical Properties of Roots

Given that $\alpha$ is a root of the equation $x^2-x+3=0$ , show that

(a) $\alpha^3+2\alpha+3=0$

(b) $\alpha^4+5\alpha^2+9=0$

#Algebra

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

You simply need to show that a^2 - a + 3 is a root for both these equations(used a instead of alpha)

Tanya Gupta - 6 years, 10 months ago

Is there a more efficient way of doing so? (Hint: Manipulate the given equation)

Victor Loh - 6 years, 10 months ago

multiply by a. plug in value of a^3.

hemang sarkar - 6 years, 10 months ago

Hint: (a) Multiply the first equation by $x$ . Clearly $\alpha$ is also a root to this equation. Now we substitute $x^2$ from the original equation.

Exercise: Solve (b)

A Brilliant Member - 6 years, 10 months ago

Since a (using instead of alpha) is a root of x^2 - x + 3 =0 so a satisfies it.hence a^2 - a + 3 =0. Now dividing a^3 + 2a +3 by a^2 - a + 3 we get quotient=a+1& remainder=0. So a^3 +2a+3=(a+1)(a^2 -a+3) + 0 (divident = quotient*divisor + remainder). Since a^2 -a+3=0, so a^3 +2a+3=0.hence the ans.... Similarly we divide a^4 +5a^2 +9 by a^2 -a+3 getting quotient=a^2 +a+3 & remainder =0. Since a^2 -a+3=0 so we get, a^4+ 5a^2 +9 =0...

Ankush Gogoi - 6 years, 10 months ago

a^3+2a+3=a(a^2+2)+3=a(a^2-a+3+a-3+2)+3=a(0+a-1)+3=a^2-a+3=0 similarly second one can be done

A P Rao - 6 years, 6 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}`	$\boxed{123}$