Olympiad Question | Nice Algebra Problem | Math Olympiad Preparation

Can you solve the given algebraic problem? Find the value of the expression x^5 +x^4 + 1 if x^2 + x+1= 0

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Olympiad Question | Nice Algebra Problem | Math Olympiad Preparation

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[PriyanshuShil_]

Sir your videos are the best,
It helps me to learn new things eveytime

Love from india☺☺

[johnbrennan]

Since x^2+x+1=0 this implies x^3=1. Sox^5+ x^4 +1=(x)^3 by x^2 plus x^4 +1. That gives x^2 +x^4 +1 which equals -x-1+x+1= 0

[mykiits]

Solved it in a bit another way. I just added and subtracted x^3 to expression we need to find: we obtain x^5+x^4+x^3-x^3+1. First three terms = 0, because we can factor out of them x^3 and obtain x^3*(x^2+x+1), and x^2+x+1 = 0 by condition. Next step - factorizing 1-x^3 which gives us (1-x)(1+x+x^2). And once again we obtain 0

[bentels]

I did it differently. I used long division to divide x⁵+x⁴+1 by x²+x+1 and concluded that

x⁵+x⁴+1 = (x²+x+1)(x³-x+1)

Since x⁵+x⁴+1 is a multiple of x²+x+1 and x²+x+1 = 0, the value of x⁵+x⁴+1 must be 0.

[fevengr]

So I started by solving for X. Using the quadratic formula, X = (-1 +/- SQRT(-3))/2. After calculating values for X^2, X^4 and X^5, I noticed that X=X^4 and X^2=X^5, THUS, X^2 + X + 1 = X^5 + X^4 + 1 = 0. So, once again I have proven that doing it the hard way works too. Thanks!

[j.r.]

I used a completely different approach, which I like because of its symmetry: Start by rearranging the original equation to get x^2 = -x -1. Then x^4 = x^2 + 2x +1, which is the same as (x^2 + x +1) + x, or 0 + x. This means that x^4 = x. Then we get x^5 = (x)x^4 = x^2. So x^5 + x^4 + 1 can be rewritten as x^2 + x +1, which is the original equation! Thus, the answer remains 0.

[SuperYoonHo]

you rock PreMath nobody can possibley beat you

[geetavansiya]

Add x^3 on both side
And take 1 on onther side
X^5+x^4+x^3+1=x^3
X^3(x^2+x+1)+1=x^3
1=x^3
X^3-1=0
(X-1)(x^2+x+1)=0
=0

[johngreen]

I did this without using fractions. Since x^2+x+1 =o then x^2=-(x +1)
so square both sides you get x^4= x^2 +2x +1 = x^2+x+1 +x = 0 + x = x
Now x^5+x^4+1 =x^4(x + 1) +1 =x(x + 1) +1 = x^2 +x +1 =0.
This seemed easier than the other commenters work, as well.

[guyhoghton]

x² + x + 1 = 0
Multiply through by x and add 1 to both sides:
⇒ x³ + (x² + x + 1) = 1
⇒ x³ = 1.
∴ x⁵ + x⁴ + 1 = x³(x² + x) + 1 = x² + x + 1 = 0.

[HappyFamilyOnline]

Great explanation👍
Thanks for sharing😊

[hsuan-hungkuo]

When you wrote down x^2+x+1=0, you should know that x^3=1 and hence the question is reduced from
y=x^5+x^4+1
To
y= x^2+x+1

And the answer shall be zero! Any senior high should be able to get it within 3 seconds!

[rcnayak_]

An alternate approach. x² + x + 1 = 0 (given). Multiply both sides by (x - 1). Which becomes x³ - 1 = 0 so that x³ = 1. Now x^5 = x².x³ = x². Similarly x ^4 = x³ . x = x. So that x^5 + x^4 + 1 reduces to x² + x +1 which is equal to 0 (as given).

[VIKASCHOUDHARY-idez]

We can solve it by using complex number property Omega w²+w+1=0 and w³=1

[pdean]

Very nice ! You are polishing the the knowledge of your viewers as no one can claim that (s)he knows everything. Only sky is the limit. Keep making such videos, please.

[math]

your solution is very well explained, thanks for sharing this algebra problem

[SladeMacGregor]

AMAZING!!! Love it!!! Thank you Professor!

[nathanricard]

By making an Euclidean division of x^5+x^4+1 by x^2+x+1. The result is :
x^5+x^4+1 = (x^2+x+1)(x^3-x+1) = 0, based on the assumed.

[АлександрИванов-фэя]

Как всегда великолепно. Большое спасибо

[ovalteen]

It looks like there are a lot of different solutions here.  Mine was to multiply the first equation by (x^3+1) which results in x^5+x^4+x^3+x^2+x+1=0.  Move the unwanted terms to the right and factor out x to get -x(x^2+x+1) and substitute 0 for the polynomial part since we know its result, and 0 times -x is 0.