(B) Let the remainder when $p(x)$ is divided by $x^2-a^2$ be $R(x) = mx + c$,where $x^2-a^2 = (x-a)(x+a)$.By the Remainder Theorem,$p(a) = -a$ and $p(

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(B) Let the remainder when $p(x)$ is divided by $x^2-a^2$ be $R(x) = mx + c$,where $x^2-a^2 = (x-a)(x+a)$.By the Remainder Theorem,$p(a) = -a$ and $p(-a) = a$.Since $p(x) = (x^2-a^2)q(x) + (mx+c)$,we have:$p(a) = m(a) + c = -a$ ... $(i)$$p(-a) = m(-a) + c = a$ ... (ii)Subtracting (ii) from $(i)$: $2ma = -2a \Rightarrow m = -1$.Substituting $m = -1$ into $(i)$: $-a + c = -a \Rightarrow c = 0$.Thus,the remainder $R(x) = -1(x) + 0 = -x$.

Class 11 Mathematics 4-2.Quadratic Equations and Inequations Solution of quadratic equations and Nature of roots

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Let $a \neq 0$ and $p(x)$ be a polynomial of degree greater than $2$. If $p(x)$ leaves remainders $a$ and $-a$ when divided respectively by $x+a$ and $x-a$,then the remainder when $p(x)$ is divided by $x^2-a^2$ is:

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$x$
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$-2x$
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$2x$

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Let $a \neq 0$ and $p(x)$ be a polynomial of degree greater than $2$. If $p(x)$ leaves remainders $a$ and $-a$ when divided respectively by $x+a$ and $x-a$,then the remainder when $p(x)$ is divided by $x^2-a^2$ is:

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