Question

Pls answer this question

Answer 1

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$\underline{\blacksquare\:\:\footnotesize{Question}}$

$\footnotesize{\text{if (x-2) and}\:(x-\dfrac{1}{2})\:\text{are the factors of the polynomial}}$

$\footnotesize{px^2+5x+r\:\text{then prove that}\:\:\boxed{p=r} }$

$\underline{\blacksquare\:\:\footnotesize{AnsweR}}$

$\footnotesize{let\:,\: f(x)=px^2+5x+r}$

$\footnotesize{and\:\: g_1(x)=(x-2)}$

$\footnotesize{and \:\:g_2(x)=(x-\dfrac{1}{2})}$

$\footnotesize{g_1(x)\:\text{is a factor of f(x) , so the remainder is zero }}$

✏ $\footnotesize{x-2=0}$

✏ $\footnotesize{x=2}$

$\therefore\:\:\footnotesize{f(2)=p(2)^2+5(2)+r=0}$

✏ $\footnotesize{4p+10+r=0}$

✏ $\boxed{\footnotesize{4p+r=-10}} ....................(i)$

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$\footnotesize{g_2(x)\:\text{ is a factor of f(x) , so the remainder is zero }}$

✏ $\footnotesize{x-\dfrac{1}{2}=0}$

✏ $\footnotesize{x=\dfrac{1}{2}}$

$\therefore\:\:\footnotesize{f(2)=p(\dfrac{1}{2})^2+5(\dfrac{1}{2})+r=0}$

✏ $\footnotesize{\dfrac{p}{4}+\dfrac{5}{2}+r=0}$

✏ $\footnotesize{\dfrac{p+10+4r}{4}=0}$

✏ $\boxed{\footnotesize{p+4r=-10}}\:......................(ii)$

$\footnotesize{\text{comparing}\:equ^n\:(i)\:and\:(ii)\:,\:we\: get}$

✏ $\footnotesize{p+4r=4p+r}$

✏ $\footnotesize{4r-r=4p-p}$

✏ $\footnotesize{3r=3p}$

✏ $\boxed{\footnotesize{p=r}}$ (proved)

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