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).