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Algebra Level 2

If ( x 2 − x = 1 { x }^{ 2 }\quad -\quad x\quad =\quad 1 ); Evaluate : ( x 4 − 2 x 3 + 3 x 2 − 2 x + 3 = ? ({ x }^{ 4 }\quad -\quad 2{ x }^{ 3 }\quad +\quad 3{ x }^{ 2 }\quad -\quad 2x\quad +\quad 3\quad =\quad ? ) Note: You should not find the roots x. Just simplify this expression.


The answer is 6.

John Doe by John Doe , 36, USA

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2 solutions

John Doe
Nov 11, 2014

x 4 − 2 x 3 + 3 x 2 − 2 x + 3 = x^{ 4 }-2x^{ 3 }+3x^{ 2 }-2x+3= x 4 − x 3 − x 3 + x 2 + 2 x 2 − 2 x + 3 = x^{ 4 }-x^{ 3 }-x^{ 3 }+x^{ 2 }+2x^{ 2 }-2x+3= x 2 ( x 2 − x ) − x 3 + x 2 + 2 x 2 − 2 x + 3 = x^2(x^2-x)-x^3+x^2+2x^2-2x+3= x 2 ( x 2 − x ) − x ( x 2 − x ) + 2 ( x 2 − x ) + 3 = x^{ 2 }(x^{ 2 }-x)-x(x^{ 2 }-x)+2(x^{ 2 }-x)+3= x 2 − x + 2 + 3 = 6 x^{ 2 }-x+2+3=6

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Parveen Soni
Nov 9, 2014

from given we have
x^-x-1=0 and required expression can be written as
x^4-2x^3+3x^2-2x+3=(x^2-x-1)(x^2-x+3)+6
answer=0+6=6


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