\(\textcolor{blue}{\textbf{ EXERCISE 2.1}}\)
1. The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.\(\textcolor{blue}{\textbf{ EXERCISE 2.2}}\)
1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.(i) \(\large x^2 – 2x -8 \)
(ii) \(\large 4 s^2 – 4s +1 \)
(iii) \(\large 6 x^2 -3 -7x \)
(iv) \(\large 4u^2 + 8u \)
(v) \(\large t^2 -15 \)
(vi) \(\large 3x^2 –x- 4 \)
2. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
(i) \(\large \frac{1}{4}, -1\) (ii) \(\large \sqrt{2}, \frac{1}{3} \) (iii) \(\large 0, \sqrt{5} \)
(iv) \(\large 1, 1\) (v) \(\large \frac{-1}{4}, \frac{1}{4} \) (vi) \(\large 4, 1 \)
\(\textcolor{blue}{\textbf{ EXERCISE 2.3}}\)
1. Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following :(i) \(\large p(x) = x^3 – 3 x^2 + 5x – 3, g(x) = x^2 -2 \)
(ii) \(\large p(x) = x^4 – 3 x^2 + 4x +5 , g(x) = x^2 +1 – x \)
(iii) \(\large p(x) = x^4 – 5 x + 6 , g(x) = 2- x^2 \)
2. Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:
(i) \(\large t^2 – 3, 2t^4 + 3t^2 – 9t -12 \)
(ii) \(\large x^2 +3x +1, 3x^4 + 5x^3 – 7x^2 +2x +2 \)
(iii) \(\large x^3 -3x +1, x^5 – 4 x^3 + x^2 +3x +1 \)
3. Obtain all other zeroes of 3×4 + 6×3 – 2×2 – 10x – 5, if two of its zeroes are 5 5 and – 3 3 × 4. On dividing x3 – 3×2 + x + 2 by a polynomial g(x), the quotient and remainder were x – 2 and –2x + 4, respectively. Find g(x). 5. Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and (i) deg p(x) = deg q(x) (ii) deg q(x) = deg r(x) (iii) deg r(x) = 0