Watch video on Division Algorithm for 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)
\( t^3 – 3, 2t^4 + 3t^3 – 2t^2 – 9t – 12 \)
(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 of \( 3x^4 + 6x^3 – 2x^2 – 10x – 5 \), if two of its zeroes are \(\sqrt{5
\over 3} and - \sqrt {5 \over 3} \)
Question 4. On dividing \(
x^3 – 3x^2 + x + 2 \) by a polynomial g(x), the quotient and remainder were \(x
– 2 \quad and \quad –2x + 4 \), respectively. Find g(x).
Question 5. Give
examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division
algorithm
(i) deg p(x) = deg q(x)
(ii) deg q(x) = deg r(x)
(iii) deg r(x) = 0
Answer: 1.(i) Quotient= \( x-3 \) and remainder =\( 7x-9 \) (ii) Quotient= \( x^2 +x -3 \) and remainder =\( 8 \) (iii) Quotient= \( - x^2 -2 \) and remainder =\( - 5x + 10 \) 2.(i) Yes, (ii) Yes, (iii) No 3. -1,1 4. \( x^2 - x + 1 \) 5. Try yourself.