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Mathematics · Textbook solutions

Polynomials

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 18 questions

2.2 Geometrical Meaning of the Zeroes of a Polynomial

2 q

Solved Examples

Worked · 1
  1. 2.1 Eg.1
    Look at the graphs in Fig. 2.9 given below. Each is the graph of y=p(x)y = p(x), where p(x)p(x) is a polynomial. For each of the graphs, find the number of zeroes of p(x)p(x).

Exercise 2.1

Practice · 1
  1. Ex 2.1 Q1
    The graphs of y=p(x)y = p(x) are given in Fig. 2.10 below, for some polynomials p(x)p(x). Find the number of zeroes of p(x)p(x), in each case.

2.3 Relationship between Zeroes and Coefficients of a Polynomial

16 q

Solved Examples

Worked · 4
  1. 2.2 Eg.2
    Find the zeroes of the quadratic polynomial x2+7x+10x^2 + 7x + 10, and verify the relationship between the zeroes and the coefficients.
  2. 2.2 Eg.3
    Find the zeroes of the polynomial x2−3x^2 - 3 and verify the relationship between the zeroes and the coefficients.
  3. 2.2 Eg.4
    Find a quadratic polynomial, the sum and product of whose zeroes are −3-3 and 22, respectively.
  4. 2.2 Eg.5
    Verify that 3, −1, −133,\ -1,\ -\dfrac{1}{3} are the zeroes of the cubic polynomial p(x)=3x3−5x2−11x−3p(x) = 3x^3 - 5x^2 - 11x - 3, and then verify the relationship between the zeroes and the coefficients.

Exercise 2.2

Practice · 12
  1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
    Ex 2.2 Q1 (i)
    x2−2x−8x^2 - 2x - 8
  2. Ex 2.2 Q1 (ii)
    4s2−4s+14s^2 - 4s + 1
  3. Ex 2.2 Q1 (iii)
    6x2−3−7x6x^2 - 3 - 7x
  4. Ex 2.2 Q1 (iv)
    4u2+8u4u^2 + 8u
  5. Ex 2.2 Q1 (v)
    t2−15t^2 - 15
  6. Ex 2.2 Q1 (vi)
    3x2−x−43x^2 - x - 4
  7. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
    Ex 2.2 Q2 (i)
    14, −1\dfrac{1}{4},\ -1
  8. Ex 2.2 Q2 (ii)
    2, 13\sqrt{2},\ \dfrac{1}{3}
  9. Ex 2.2 Q2 (iii)
    0, 50,\ \sqrt{5}
  10. Ex 2.2 Q2 (iv)
    1, 11,\ 1
  11. Ex 2.2 Q2 (v)
    −14, 14-\dfrac{1}{4},\ \dfrac{1}{4}
  12. Ex 2.2 Q2 (vi)
    4, 14,\ 1