Ex.2.3 Q2 Polynomials Solution - NCERT Maths Class 9

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Find the remainder when \(\begin{align}x^{3}-a x^{2}+6 x-a\end{align}\) is divided by \(x - a.\)


 Video Solution
Ex 2.3 | Question 2

Text Solution


Let \(p(x)\) be any polynomial of degree greater than or equal to one and let \(a\) be any real number. If a polynomial \(p(x)\) is divided by \(\begin{align}x-a\end{align}\) then the remainder is \(p(a).\)


Let \(\begin{align}p(x)=x^{3}-a x^{2}+6 x-a\end{align}\)

The root of \(x-a = 0\) is \(a.\)

\[\begin{align} p(a) &=(a)^{3}-a(a)^{2}+6(a)-a \\ &=a^{3}-a^{3}+5 a \\ &=5 a \end{align}\]

Hence by remainder theorem, \(\begin{align}5 a\end{align}\) is the remainder when \(\begin{align}x^{3}-a x^{2}+6 x-a\end{align}\)  is divided by \(x - a.\)

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