Question

find the quadratic polynomial whose zeroes are log 1000,log 0.01 0.1​

Answer 1

GiVeN :-

$\sf{log1000\ and\ log_{0.1} 0.01\ are\ the\ zeros\ of\ a\ quadratic }\\\sf{ polynomial.}$

To DeTeRmInE :-

$\sf{The\ quadratic\ polynomial\ whose\ zeros\ are\ given.}$

SoLuTiOn :-

$\sf{We\ know,}$

$\textsf{For a log with no base, the base is considered to be 10.}$

$\sf{Ex :-}$  $\sf{log\ x=log_{10}x}$

$\sf{log_1_0 1000=3}$

$\sf{(As\ 10^3=1000)}$

$\sf{log_0_._10.01=log_{\frac{1}{10} }\frac{1}{100} =2}$

$\sf{[As\ (\dfrac{1}{10})^2 =\dfrac{1}{100} ]}$

$\sf{Hence,\ the\ zeros\ are\ 3\ and\ 2.}$

$\rule{140}2$

$\bullet\ \sf{Sum\ of\ the\ zeros,\ S=3+2=5}\\\\\bullet\ \sf{Product\ of\ the\ zeros,\ P=3\times2=6}$

$\sf{A\ quadratic\ equation\ is\ of\ the\ form:-}\\\sf{x^2-(S)x+P}$

$\sf{The\ required\ polynomial\ is:-}$

$\large\boxed{\sf{x^2-5x+6}}$