Find the sum of real solutions

Algebra Level 4

Find the sum of all real solutions of the equation:

$7{ x }^{ 7 } + { 6x }^{ 6 } + 5{ x }^{ 5 } + 4{ x }^{ 4 } + 3{ x }^{ 3 } + 2{ x }^{ 2 } + x - 76543210 = 0$

The answer is 10.

4 solutions

Thành Trần
Nov 30, 2014

We have 7.x^7+6.x^6+5.x^5+4.x^4+3.x^3+2.x^2+x = 76,543,210 You can quickly realize that the total above is made of 70,000,000+6,000,000+500,000+40,000+3,000+200+10 Or 7x10,000,000+6x1,000,000+5x100,000+4x10,000+3x1,000+2x100+10 Then we have the only solution is x=10

But how can you show that that is the only real solution?

Aloysius Ng - 6 years, 6 months ago

I also used the same method

Parv Maurya - 6 years, 6 months ago

I know, it's just a detailed explaination.

Thành Trần - 6 years, 6 months ago

By the Descartes Rule of Sign, There are 7 Real roots of the given equation and its sum is given ax^7 +bx^6+...........+h=0, Sum of Roots Taken one at a time is -b/a.

Surya Pratap Singh - 6 years, 6 months ago

be careful , you are right about what vieta's tells but how are you sure that all the roots are real

U Z - 6 years, 5 months ago

Thats what I do, but shouldn't the result be -6/7 instead of 10?

Lorraine Xie - 6 years, 5 months ago

Josep Mauri
Nov 3, 2016

We have 7.x^7+6.x^6+5.x^5+4.x^4+3.x^3+2.x^2+x+0=7.10^7+6.10^6+5.10^5+4.10^4+3.10^3+2.10^2+1.10+0 Then we have the only solution x=10

Labib El Khawarizmy
Nov 30, 2014

from the equation we know that one of roots is 10. then you can just know that the equation has just one real root, that is 10

Then how do you know this is its only real root?

Kenny Lau - 6 years, 6 months ago

I think there are 7 Real Roots.....

Surya Pratap Singh - 6 years, 6 months ago

Ibraheem Akram
Nov 30, 2014

Put x=10 which satisfy the equation.

Then how do you know there are no other real roots?

Kenny Lau - 6 years, 6 months ago

Find the sum of real solutions

The answer is 10.

4 solutions

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