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  • Let  and  be the roots of the equation <img alt="x^{2}-x-1=0." src="https:/

Let \alpha and \beta be the roots of the equation x^{2}-x-1=0. If P_{k}=\left ( \alpha \right )^{k}+\left ( \beta \right )^{k},k\geq 1, then which one of the following statements is not true ?  
Option: 1 P_{1}+P_{2}+P_{3}+P_{_{4}}+P_{5} =26
Option: 2 P_{5}=11
Option: 3 P_{5}=P_{2}\cdot P_{3}
Option: 4 P_{3}=P_{5}- P_{4}
 

Answers (1)

best_answer

 

 

Polynomial Equation of Higher Degree, Remainder theorem -

\\\mathrm{An \;equation \;of \;the\; form\; a_0x^n+a_1x^{n-1}+...+a_{n-1}x+a_n=0, } \\\mathrm{where \; a_0, a_1,..., a_n \; are\; constant\; and \; a_0 \; \neq 0}

Is known as the polynomial equation of degree n which have n and only n roots.

 

\\\mathrm{sum \;of\; all\; roots=\sum \alpha_1 = \alpha_1 + \alpha_2 +...+\alpha_n-1+\alpha_n=(-1)\frac{a_1}{a_0}} \\\\\mathrm{sum\; of \; products\; taken \; two \; at \; a \; time\;} \\\mathrm{\;\;\;\sum \alpha_1\alpha_2=\alpha_1\alpha_2 + \alpha_1\alpha_3 + ... + \alpha_1\alpha_n + \alpha_2\alpha_3+...+\alpha_2\alpha_n+...+\alpha_{n-1}\alpha_n=(-1)^2\frac{a_2}{a_0}} \\ \\\mathrm{sum\; of \; products\; taken \; three \; at \; a \; time\;} \\\mathrm{\;\;\; \sum \alpha_1\alpha_2\alpha_3 = (-1)^3\frac{a_3}{a_0}} \\\\\mathrm{product\; of \; all \; roots = \alpha_1\alpha_2...\alpha_n = (-1)^n\frac{a_n}{a_0}}

For example, suppose n = 3 and ax3 + bx2 +cx + d = 0 is polynomial equation with a ≠ 0 and ?, ? and ? are the roots of the equation then :

\\\mathrm{\alpha+\beta+\gamma = -\frac{b}{a}} \\\mathrm{\sum \alpha \beta =\alpha\beta+\beta\gamma+\gamma\alpha= (-1)^2\frac{c}{a}=\frac{c}{a}} \\\mathrm{\alpha\beta \gamma=(-1)^3\frac{d}{a}=-\frac{d}{a}}

-

\alpha+\beta=1 \text{ and }\alpha\timex\beta=-1

\\P_1=\alpha+\beta=1\\P_2=\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta=1-2(-1)=3\\P_3=\alpha^3+\beta^3=(\alpha+\beta)(\alpha^2+\beta^2-\alpha\beta)=(1\times(3+1))=4\\P_4=\alpha^4+\beta^4=(\alpha^2+\beta^2)^2-2\alpha^2\beta^2=(P_2)^2-2(-1)^2=7

x^2=x+1

\\x^2=x+1\Rightarrow x^5=x^4+x^3\\ \alpha^5=\alpha^4+\alpha^3\text{ and }\beta^5=\alpha^4+\alpha^3

\alpha^5+\beta^5=P_4+P_3=11

\mathrm{P}_{5} \neq \mathrm{P}_{2} \times \mathrm{P}_{3}

Correct option (3)

Posted by

Ritika Jonwal

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