# TGIF Problem of the Week 2

Good job to those who have sent their answers in our last week’s problem. You may check the names and the solution to that problem here.

Here comes this week’s problem. It’s easier this time.

Sum of the Cubes – August 22-29, 2014.

Find the sum of the cubes of the roots of cubic $3x^3-2x+4=0$

Hint: Vieta’s Formula and Algebraic Identity. Use the custom search in the upper right of the page 🙂

Send your answer using the form below. Make sure to include your school or company. Solution will be presented on this page same day next week. Only answers using the button below will be accepted.  Goodluck!

[contact-form-7 id=”2341″ title=”Submitter”]

1. John Lester Tan. DEE HWA LIONG ACADEMY

2. John Mark Guimba. University of the Philippines – Diliman

3. Highryll Tan. Statefields School Inc.

4. Chin Yi Xiang

Solution:

Let a,b,c are the roots of equation. Using Vieta’s formula we have the following relations

$a+b+c=0$

$ab+bc+ac=\displaystyle\frac{2}{3}$

$abc=\displaystyle\frac{-4}{3}$

Using one of the basic algebraic identities we have

$a^3+b^3+c^3=(a+b+c)(a^2+b^2+c^2-ab-bc-ac)+3abc$

But  $a+b+c=0$

That reduces our equation to

$a^3+b^3+c^3=3abc$

But  $abc=\displaystyle\frac{-4}{3}$

$a^3+b^3+c^3=3(\displaystyle\frac{-4}{3})$

$\boxed{a^3+b^3+c^3=-4}$

### Dan

Blogger and a Math enthusiast. Has no interest in Mathematics until MMC came. Aside from doing math, he also loves to travel and watch movies.