Q. Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(i) $4x^2-3x+7$
(ii) $y^2+\sqrt{2}$
(iii) $3\sqrt{t}+t\sqrt{2}$
(iv) $y+\frac{2}{y}$
(v) $x^{10}+y^3+t^{50}$

Q. Factorise:
(i) $x^3-2x^2-x+2$
(ii) $x^3-3x^2-9x-5$
(iii) $x^3+13x^2+32x+20$
(iv) $2y^3+y^2-2y-1$

Q. Find $p\left(0\right),p\left(1\right)$ and $p\left(2\right)$ for each of the following polynomials:
(i) $p\left(y\right)=y^2-y+1$
(ii) $p\left(t\right)=2+t+2t^2-t^3$
(iii) $p\left(x\right)=x^3$
(iv) $p\left(x\right)=(x-1)(x+1)$

Q. Use the Factor Theorem to determine whether $g\left(x\right)$ is a factor of $p\left(x\right)$ in each of the following cases:
(i) $p\left(x\right)=2x^3+x^2-2x-1,g\left(x\right)=x+1$
(ii) $p\left(x\right)=x^3+3x^2+3x+1,g\left(x\right)=x+2$
(iii) $p\left(x\right)=x^3-4x^2+x+6,g\left(x\right)=x-3$

Q. Evaluate the following products without multiplying directly:
(i) $103\times 107$
(ii) $95\times 96$
(iii) $104\times 96$

Q. Find the value of the polynomial $5x-4x^2+3$ at
(i) $x=0$
(ii) $x=-1$
(iii) $x=2$

Q. Verify whether the following are zeroes of the polynomial, indicated against them.
(i) $p\left(x\right)=3x+1,x=-\frac{1}{3}$
(ii) $p\left(x\right)=5x-\pi ,x=\frac{4}{5}$
(iii) $p\left(x\right)=x^2-1,x=1,-1$
(iv) $p\left(x\right)=(x+1)(x-2),x=-1,2$
(v) $p\left(x\right)=x^2,x=0$
(vi) $p\left(x\right)=lx+m,x=-\frac{m}{l}$
(vii) $p\left(x\right)=3x^2-1,x=-\frac{1}{\sqrt{3}},\frac{2}{\sqrt{3}}$
(viii) $p\left(x\right)=2x+1,x=\frac{1}{2}$

Q. Classify the following as linear, quadratic and cubic polynomials:
(i) $x^2+x$
(ii) $x-x^3$
(Iii) $y+y^2+4$
(iv) $1+x$
(v) $3t$
(vi) $r^2$
(vii) $7x^3$

Q. Find the zero of the polynomial in each of the following cases:
(i) $p\left(x\right)=x+5$
(ii) $p\left(x\right)=x-5$
(iii) $p\left(x\right)=2x+5$
(iv) $p\left(x\right)=3x-2$
(v) $p\left(x\right)=3x$
(vi) $p\left(x\right)=ax,a\ne 0$
(vii) $p\left(x\right)=cx+d,c\ne 0,c,d$ are real numbers.

Q. Determine which of the following polynomials has $(x+1)$ a factor:
(i) $x^3+x^2+x+1$
(ii) $x^4+x^3+x^2+x+1$
(iii) $x^4+3x^3+3x^2+x+1$
(iv) $x^3-x^2-(2+\sqrt{2})x+\sqrt{2}$

Q. Write the degree of each of the following polynomials:
(i) $5x^3+4x^2+7x$
(ii) $4-y^2$
(iii) $5t-\sqrt{7}$
(iv) $3$

Q. Check whether $7+3x$ is a factor of $3x^3+7x$.

Q. Give one example each of a binomial of degree $35$ and of a monomial of degree $100$.

Q. Write the coefficients of $x^2$ in each of the following:
(i) $2+x^2+x$ (ii) $2-x^2+x^3$ (iii) $\frac{\pi }{2}{x}^2+x$ (iv) $\sqrt{2}x-1$

Q. Use suitable identities to find the following products:
(i) $(x+4)(x+10)$
(ii) $(x+8)(x-10)$
(iii) $(3x+4)(3x-5)$
(iv) $(y^2+\frac{3}{2})(y^2-\frac{3}{2})$
(v) $(3-2x)(3+2x)$

Q. Factorise:
(i) $12x^2-7x-1$
(ii) $2x^2+7x+3$
(iii) $6x^2+5x-6$
(iv) $3x^2-x-4$

Q. Factorise the following using appropriate identities:
(i) $9x^2+6xy+y^2$
(ii) $4y^2-4y+1$
(iii) $x^2-\frac{y^2}{100}$

Q. Find the value of $k$, if $x-1$ is a factor of $p\left(x\right)$ in each of the following cases:
(i) $p\left(x\right)=x^2+x+k$
(ii) $p\left(x\right)=2x^2+kx+\sqrt{2}$
(iii) $p\left(x\right)=k{x}^2-\sqrt{2}x+1$
(iv) $p\left(x\right)=k{x}^2-3x+k$

Q. Find the remainder when $x^3+3x^2+3x+1$ is divided by
(i) $x+1$
(ii) $x-\frac{1}{2}$
(iii) $x$
(iv) $x+\pi$
(v) $5+2x$

Q. Find the remainder when $x^3-a{x}^2+6x-a$ is divided by $x-a$.