# Polynomials - Class 10 - Mathematics

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## Extra Questions - Polynomials | NCERT | Mathematics | Class 10

If $\alpha$, $\beta$, and $y$ are the roots of $x^{3} + 8 = 0$, then the equation whose roots are $\alpha^{2}$, $\beta^{2}$, and $y^{2}$ is

(A) $x^{3} - 8 = 0$ (B) $x^{3} - 16 = 0$ (C) $x^{3} + 64 = 0$ (D) $x^{3} - 64 = 0$.

The correct answer is **(D) $x^{3} - 64 = 0$**.

Given the equation: $$ x^{3} + 8 = 0 $$ where $\alpha$, $\beta$, and $y$ are roots, we want to find the equation with roots squared: $\alpha^{2}$, $\beta^{2}$, and $y^{2}$.

We start by substituting $x = \sqrt{y}$ into the original equation: $$ (\sqrt{y})^3 + 8 = 0 \implies y^{\frac{3}{2}} + 8 = 0 $$

Now, solving for $y^{\frac{3}{2}}$: $$ y^{\frac{3}{2}} = -8 \implies y^3 = (-8)^2 = 64 $$

Therefore, the equation $y^3 - 64 = 0$ or expressed in terms of $x$: $$ x^3 - 64 = 0 $$

This equation has roots $\alpha^2$, $\beta^2$, and $y^2$. Thus, the correct choice is **(D) $x^{3} - 64 = 0$**.

Which of the following is a binomial?

A) $x^{2} - x + 1$

B) $3x^{2} + 2x^{2} - x^{2}$

C) $\left(x^{2} - 2x\right) x^{-1}$

D) $2x^{2} + x^{2} - 2x + 3 - 2$

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If $x=0$ and $x=-1$ are the roots of the polynomial $f(x)=2x^{3}-3x^{2}+ax+b$, find the value of $a$ and $b.

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Let $f(x)$ be a polynomial of degree 5 such that $x = \pm 1$ are its critical points. If $\lim_{x \to 0} \left(2 + \frac{f(x)}{x^{3}}\right) = 4$, then which of the following is not true?

A $f(1) - 4 f(-1) = 4$.

B $x = 1$ is a point of maxima and $x = -1$ is a point of minimum of $f$.

C $f$ is an odd function.

D $x = 1$ is a point of minima and $x = -1$ is a point of maxima of $f$.