Think like a Mathematician!

Prove the following algebraic identities:

(1) a3+b3+c3+d3−3abc−3abd−3acd−3bcd=(a+b+c+d)(a2+b2+c2+d2−ab−ac−ad−bc−bd−cd)a^3+b^3+c^3+d^3-3abc-3abd-3acd-3bcd = (a+b+c+d)(a^2+b^2+c^2+d^2-ab-ac-ad-bc-bd-cd)

(2) (a+b+c)3=a3+b3+c3+3(a+b)(b+c)(c+a)=a3+b3+c3−3abc+3(a+b+c)(ab+bc+ca)(a+b+c)^3 = a^3+b^3+c^3+3(a+b)(b+c)(c+a) = a^3+b^3+c^3-3abc+3(a+b+c)(ab+bc+ca)

(3) (a+b+c+d)4=a4+b4+c4+d4+4ab(a2+b2)+4ac(a2+c2)+4ad(a2+d2)+4bc(b2+c2)+4bd(b2+d2)+4cd(c2+d2)+6a2b2+6a2c2+6a2d2+6b2c2+6b2d2+6c2d2+12abc(a+b+c)+12abd(a+b+d)+12acd(a+c+d)+12bcd(b+c+d)+24abcd(a+b+c+d)^4 = a^4+b^4+c^4+d^4+4ab(a^2+b^2)+4ac(a^2+c^2)+4ad(a^2+d^2)+4bc(b^2+c^2)+4bd(b^2+d^2)+4cd(c^2+d^2)+6a^2b^2+6a^2c^2+6a^2d^2+6b^2c^2+6b^2d^2+6c^2d^2+12abc(a+b+c)+12abd(a+b+d)+12acd(a+c+d)+12bcd(b+c+d)+24abcd

Note: You aren't allowed to use binomial or multinomial Theorem to prove the 3rd identity!

#Algebra

Note by Vaibhav Priyadarshi
1 year, 8 months ago

No vote yet
1 vote

  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.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list
  • bulleted
  • list

1. numbered
2. 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 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"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai−1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 ∑i=13 \sum_{i=1}^3
\sin \theta sin⁡θ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Shouldn't you have used the identity sign instead of the equal sign, @Vaibhav Priyadarshi?

A Former Brilliant Member - 1 year, 1 month ago

Log in to reply

It doesn't matter.

Vaibhav Priyadarshi - 9 months ago
×

Problem Loading...

Note Loading...

Set Loading...