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Algebra Level 3

if A and B are roots of the equation x^2 - 2x + 3=0 , obtain the equation whose roots are A^3 - 3A^2 + 5A - 2 and B^3 - B^2 + B + 5

x^2 - 3x + 2 x^2 - 3x -6 x^2 + 3x + 2 x^2 - 4x + 5

Harsh Vazirani by Harsh Vazirani , 23, India

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1 solution

Aditya Sky
Apr 3, 2016

Since, A \color{#D61F06}{A} and B \color{#3D99F6}{B} are the roots to the polynomial x 2 − 2 x + 3 \color{#20A900}{x^{2}-2x+3} , therefore A 2 − 2 A + 3 = 0 \color{#D61F06}{A^{2}-2A+3=0} and B 2 − 2 B + 3 = 0 \color{#3D99F6}{B^{2}-2B+3=0} .

By synthetic division, it can be seen that :- ( i ) A 3 − 3 A 2 + 5 A − 2 = ( A 2 − 2 A + 3 ⏟ 0 ) ( A − 1 ) + 1 ( i i ) B 3 − B 2 + B + 5 = ( B 2 − 2 B + 3 ⏟ 0 ) ( B + 1 ) + 2 (i)\,\,\, \color{#624F41}{A^{3}-3A^{2}+5A-2}=(\color{#D61F06}{\underbrace{A^{2}-2A+3}_\text{0}})(A-1)+1 \,\,\,\,\,\,\,\,\,\,\,\ (ii)\,\,\, \color{#624F41}{B^{3}-B^{2}+B+5}=(\color{#3D99F6}{\underbrace{B^{2}-2B+3}_\text{0}})(B+1)+2 ⟹ A 3 − 3 A 2 + 5 A − 2 = 1 \implies \color{#624F41}{A^{3}-3A^{2}+5A-2}=1 and B 3 − B 2 + B + 5 = 2 \color{#624F41}{B^{3}-B^{2}+B+5}=2 .

So, the required polynomial is x 2 − ( 1 + 2 ) x + ( 1 ⋅ 2 ) = x 2 − 3 x + 2 \color{#BA33D6}{x^{2}}-(\color{#624F41}{1+2})\color{#BA33D6}{x}+(\color{#624F41}{1 \cdot 2})\,=\, x^{2}-3x+2 .

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