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).