Expansionless Solution

Algebra Level 4

If a a , b b , c c , and d d are roots of x 4 + 4 x 3 − 6 x 2 + 7 x − 9 = 0 x^4+4x^3 -6x^2+7x-9=0 , find the value of ( 1 + a 2 ) ( 1 + b 2 ) ( 1 + c 2 ) ( 1 + d 2 ) (1+a^2)(1+b^2)(1+c^2)(1+d^2) .

11 9 5 13

Benny Joseph by Benny Joseph , 20, India

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1 solution

Chew-Seong Cheong
Nov 30, 2017

Let f ( x ) = x 4 + 4 x 3 − 6 x 2 + 7 x − 9 = ( x − a ) ( x − b ) ( x − c ) ( x − d ) f(x) = x^4+4x^3-6x^2+7x-9=(x-a)(x-b)(x-c)(x-d) . Then f ( x ) = ( a − x ) ( b − x ) ( c − x ) ( d − x ) f(x)=(a-x)(b-x)(c-x)(d-x) and we note that:

f ( i ) f ( − i ) = ( 1 + a 2 ) ( 1 + b 2 ) ( 1 + c 2 ) ( 1 + d 2 ) = ( i 4 + 4 i 3 − 6 i 2 + 7 i − 9 ) ( ( − i ) 4 + 4 ( − i ) 3 − 6 ( − i ) 2 + 7 ( − i ) − 9 ) = ( − 2 + 3 i ) ( − 2 − 3 i ) = 13 \begin{aligned} f(i)f(-i)&=(1+a^2)(1+b^2)(1+c^2)(1+d^2) \\ &= (i^4+4i^3-6i^2+7i-9)((-i)^4+4(-i)^3-6(-i)^2+7(-i)-9) \\ &=(-2+3i)(-2-3i) \\ &= \boxed{13}\end{aligned}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Excellent idea!

Benny Joseph - 3 years, 6 months ago

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Thanks. Glad that you like it.

Chew-Seong Cheong - 3 years, 6 months ago

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