Q. The graphs of $y=p\left(x\right)$ are given in the figure, for some polynomials $p\left(x\right)$. Find the number of zeroes of $p\left(x\right)$, in each case.

Q.

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(i) $x^2-2x-8$

(ii) $4s^2-4s+1$

(iii) $6x^2-3-7x$
(iv) $4u^2+8u$

(v) $t^2-15$

(vi) $3x^2-x-4$

Q. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
(i) $\frac{1}{4}$, $-1$ (ii) $\sqrt{2},\frac{1}{3}$ (iii) $0,\sqrt{5}$
(iv) $1,1$ (v) $\frac{-1}{4},\frac{1}{4}$ (vi) $4,1$

Q. On dividing $x^3-3x^2+x+2$ by a polynomial g(x), the quotient and remainder were $(x-2)$ and $(-2x+4)$, respectively. Find $g\left(x\right).$

Q. Obtain all other zeroes of $3x^4+6x^3-2x^2-10x-5$, if two of its zeroes are $\sqrt{\frac{5}{3}}$ and $-\sqrt{\frac{5}{3}}$.

Q. Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:
(i) $t^2-3,2t^4+3t^3-2t^2-9t-12$
(ii) $x^2+3x+1,3x^4+5x^3-7x^2+2x+2$
(iii) $x^3-3x+1,x^5-4x^3+x^2+3x+1$

Q. Divide the polynomial $p\left(x\right)$ by the polynomial $g\left(x\right)$ and find the quotient and remainder in each of the following:
(i) $p\left(x\right)=x^3-3x^2+5x-3,g\left(x\right)=x^2-2$
(ii) $p\left(x\right)=x^4-3x^2+4x+5,g\left(x\right)=x^2+1-x$
(iii) $p\left(x\right)=x^4-5x+6,g\left(x\right)=2-x^2$

Q. Give examples of polynomials $p\left(x\right),g\left(x\right),q\left(x\right)$ and $r\left(x\right)$, which satisfy the division algorithm and
(i) deg $p\left(x\right)=$deg $q\left(x\right)$ (ii) deg $q\left(x\right)=$deg $r\left(x\right)$ (iii) deg $r\left(x\right)=0$

Q. If two zeroes of the polynomial $x^4-6x^3-26x^2+138x-35$ are $2\pm \sqrt{3}$, find the other zeroes.

Q. If the zeroes of the polynomial $x^3-3x^2+x+1$ are $a-b,a,a+b$ find $a$ and $b$.

Q. If the polynomial $x^2-2x+k$ is a factor of $x^4-6x^3+16x^2-26x+10-a$, then find the value of $k$ and $a.$

Q. Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as $2,7,14$ respectively.

Q. Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:
(i) $2x^3+x^2-5x+2;\frac{1}{2},1,-2$