Can Vieta help?

Algebra Level 4

$\large { x }^{ 3 }-{ 453x }^{ 2 }+68402x-3442800$

If the expression above has 3 real roots ${ r }_{ 1 },{ r }_{ 2 },{ r }_{ 3 }$ in an arithmetic progression having common difference 1 then find the value of

${ { r }_{ 1 } }^{ 2 }+{ { r }_{ 2 } }^{ 3 }+{ { r }_{ 3 } }^{ 2 }.$

Note : The exponents in the expression are indeed $2, 3, 2$ .

The answer is 3488555.

2 solutions

Shivamani Patil
Sep 29, 2014

Sum of roots by vieta's formula is $453$ .

We know AP sum formula ${ S }_{ n }=\frac { n }{ 2 } \left( 2a+\left( n-1 \right) d \right)$ )

In this formula we have $n=3$ , $a={ r }_{ 1 } ,{ S }_{ 3 }={ r }_{ 1 }+{ r }_{ 2 }+{ r }_{ 3 }$

Therefore

${ S }_{ 3 }=\frac { 3 }{ 2 } \left( 2a+\left( 2 \right) \times 1 \right)$

${ S }_{ 3 }=3a+3$

We know sum of roots is $453$

Therefore

$453=3a+3\quad$

$\Rightarrow a=150={ r }_{ 1 }$

$\Rightarrow { r }_{ 2 }=151$

$\Rightarrow { r }_{ 3 }=152$

$\Rightarrow { { r }_{ 1 } }^{ 2 }+{ { r }_{ 2 } }^{ 3 }+{ { r }_{ 3 } }^{ 2 }={ 150 }^{ 2 }+{ 151 }^{ 3 }+{ 152 }^{ 2 }=22500+3442951+23104=3488555$

A more simplified solution Consider the roots be

$a-1$ , $a$ , $a+1$

$a-1 + a + a+1 = 453$

From here

$a = 151$

Hence

$150^{2} + 151^{3} + 152^{2} = \boxed{3488555}$

Krishna Sharma - 6 years, 8 months ago

Good solution.

shivamani patil - 6 years, 8 months ago

same as mine ! :)

Nihar Mahajan - 6 years, 4 months ago

Exactly same solution

Kushagra Sahni - 5 years, 11 months ago

I did the same way.

Niranjan Khanderia - 5 years, 7 months ago

The roots of your polynomial are NOT 150, 151, 152.

Assuming that the polynomial is correct, then $r_1 ^2 + r_2 ^2 + r_3 ^2 = ( r_1 + r_2 + r_3) ^2 - 2 (r_1 r_2 + r_2 r_3 + r_3 r_1 ) = 453^2 - 2 \times 68402 = 68405.$

Calvin Lin Staff - 6 years, 8 months ago

I have updated question. Now roots are 150,151,152

shivamani patil - 6 years, 8 months ago

I apologize for the error.

I did not notice that the term $r_2$ in the expression has a cube in it. I have updated the question to reflect that. 3488555 is the correct answer.

Sorry for the confusion that this caused.

Calvin Lin Staff - 6 years, 8 months ago

Calvin, the roots are 150,151 & 152. Why do you say they are not so?

ajit athle - 1 year, 10 months ago

Fox To-ong
Feb 13, 2015

since the sum of the roots = 453 and the first root = 150, then to become in AP, The roots are 150,151 and 152

Can Vieta help?

The answer is 3488555.

2 solutions

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