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) \( p(x) = x^4 – 5x + 6 \), \( g(x) = 2 – x^2 \)

2. Check in which case 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) \( x^2 – 3x + 1 \), \( x^5 – 4x^3 + x^2 + 3x + 1 \)

3. Obtain all other zeros of \( 3x^4 + 6x^3 – 2x^2 – 10x – 5 \), if two of its zeros are \( \sqrt{\frac{5}{3}} \) and \( -\sqrt{\frac{5}{3}} \).

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 \( g(x) \).

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) \( \deg r(x) = 0 \)

1. Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients.

(i) \( x^2 – 2x – 8 \)

(ii) \( 4s^2 – 4s + 1 \)

(iii) \( 6x^2 – 3 – 7x \)

(iv) \( 4u^2 + 8u \)

(v) \( t^2 – 15 \)

(vi) \( 3x^2 – x – 4 \)

2. Find the quadratic polynomial in each case, with the given numbers as the sum and product of its zeros 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) \( 4, 1 \)

3. Find the quadratic polynomial, for the zeros \( \alpha, \beta \) given in each case.

(i) 2, -1

(ii) \( \sqrt{3}, -\sqrt{3} \)

(iii) \( \frac{1}{4}, -1 \)

(iv) \( \frac{1}{2}, \frac{3}{2} \)

4. Verify that 1, -1, and -3 are the zeros of the cubic polynomial \( x^3 – 3x^2 + x + 3 \) and check the relationship between zeros and the coefficients.

    1 | 1  -3   1   3
      |     1  -2  -1
    ------------------
        1  -2  -1   2
            

   -3 | 1  -3   1   3
      |    -3  18 -57
    ------------------
        1  -6  19 -54
            

    1 | 1  -3   1   3
      |     1  -2   1
    ------------------
        1  -2  -1   4
            

1. The graphs of \( y = p(x) \) are given in the figure below, for some polynomials \( p(x) \). In each case, find the number of zeros of \( p(x) \).

2. Find the zeros of the given polynomials.

(i) \( p(x) = 3x \)

(ii) \( p(x) = x^2 + 5x + 6 \)

(iii) \( p(x) = (x + 2)(x + 3) \)

(iv) \( p(x) = x^2 – 16 \)

3. Draw the graphs of the given polynomial and find the zeros. Justify the answers.

(i) \( p(x) = x^2 – x – 12 \)

(ii) \( p(x) = x^2 – 6x + 9 \)

(iii) \( p(x) = x^2 – 4x + 5 \)

(iv) \( p(x) = x^2 + 3x – 4 \)

(v) \( p(x) = x^2 – 1 \)

4. Why are \( \frac{1}{4} \) and -1 zeros of the polynomial \( p(x) = 4x^2 + 3x – 1 \)?

1. In \( p(x) = 5x^2 – 6x + 7x – 6 \), what is the

(i) coefficient of \( x^2 \)

(ii) degree of \( p(x) \)

(iii) constant term

2. State which of the following statements are true and which are false? Give reasons for your choice.

(i) The degree of the polynomial \( \sqrt{2} x^2 – 3x + 1 \) is \( \sqrt{2} \).

(ii) The coefficient of \( x^2 \) in the polynomial \( p(x) = 3x^3 – 4x^2 + 5x + 7 \) is 2.

(iii) The degree of a constant term is zero.

(iv) \( \frac{1}{x^2 – 5x + 6} \) is a quadratic polynomial.

(v) The degree of a polynomial is one more than the number of terms in it.

3. If \( p(t) = t^3 – 1 \), find the values of \( p(1), p(-1), p(0), p(2), p(-2) \).

4. Check whether -2 and 2 are the zeros of the polynomial \( x^4 – 16 \).

5. Check whether 3 and -2 are the zeros of the polynomial \( p(x) \) when \( p(x) = x^2 – x – 6 \).