Expect problems and eat them for breakfast. – Alfred A. Montapert

Rule to find quadratic polynomial with sum and product of roots

NCERT Exercise 2.1 Question 2 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) $$4, 1$$

Division of polynomials

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

Question 2. 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) $$x^3-3x+1, x^5-4x^3+x^2+3x+1$$

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

Question 4. 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(x).