How will do do it? - Part 2

Algebra Level 4

x 4 − 64 x 3 + a x 2 − b x + 65536 = 0 \large x^4 - 64x^3 +ax^2 - bx + 65536 = 0

Find the value of real numbers a , b a,b such that the equation above has all real and positive roots.

Enter ⌊ b a ⌋ \left \lfloor \dfrac{b}{a} \right\rfloor as your answer.

Notation : ⌊ ⋅ ⌋ \lfloor \cdot \rfloor denotes the floor function .

Try part 1

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The answer is 10.

Neelesh Vij by neelesh vij , 21, India

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1 solution

Neelesh Vij
Mar 21, 2016

A very good method of doing the question is as follows:

Let the roots be x 1 , x 2 , x 3 , x 4 x_1 , x_2 , x_3 , x_4

Now from Vieta's

→ x 1 + x 2 + x 3 + x 4 = 64 \rightarrow x_1 + x_2 + x_3 + x_4 = 64

Also → x 1 x 2 x 3 x 4 = 65536 \rightarrow x_1x_2x_3x_4 = 65536

Now by CAREFUL observation:

A.M of roots = x 1 + x 2 + x 3 + x 4 4 = 64 4 = 16 \dfrac{x_1 + x_2 + x_3 + x_4}{4} = \dfrac{64}{4} = 16

G.M of roots = ( x 1 x 2 x 3 x 4 ) 1 / 4 = ( 65536 ) 1 / 4 = 16 (x_1 x_2 x_3 x_4)^{1/4} = (65536)^{1/4} = 16

So, x 1 = x 2 = x 3 = x 4 = 16 x_1 = x_2 = x_3 = x_4 = 16

Now, By Vieta's a = ∑ x 1 x 2 and − b = − ∑ x 1 x 2 x 3 \displaystyle a = \sum x_1x_2 \text{ and} -b=-\sum x_1x_2x_3

Solving we get a = 1536 , b = 15384 a =1536, b = 15384

So [ b a ] = 10 \left[ \dfrac ba \right] = \boxed{10}

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