Polynomials Study Material Class 10

Theory: Zeros of a Polynomial

A polynomial in one variable is an expression consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. A zero of a polynomial is a number ‘x’ such that when we substitute it into the polynomial, the value of the polynomial becomes zero. Mathematically, if p(x) is a polynomial, then x = a is a zero if p(a) = 0.

Zeros are important because they are the solutions to polynomial equations and give information about the graph of the polynomial. For a linear polynomial like p(x) = ax + b, there is exactly one zero, which can be found by solving p(x) = 0.

The graph of a polynomial touches or crosses the x-axis at the point corresponding to the zero.

Below is a simple diagram showing a linear polynomial and its zero:

In this graph, the line passes through the origin, meaning x = 0 is a zero of the polynomial.

Solved Problems

Problem 1: Find the zero of p(x) = 2x + 6.
Solution: Set p(x) = 0 2x + 6 = 0 2x =  − 6 x =  − 3

Problem 2: Find the zero of p(x) = 5x − 10.
Solution: 5x − 10 = 0 5x = 10 x = 2

Problem 3: Verify that x = 2 is a zero of p(x) = x2 − 4x + 4.
Solution: p(2) = (2)2 − 4(2) + 4 = 4 − 8 + 4 = 0. Thus, x = 2 is a zero.

Problem 4: Find the zeros of p(x) = x2 − 5x + 6.
Solution: Factorizing: p(x) = (x−2)(x−3) Zeros are x = 2 and x = 3.

Problem 5: Check if x = 1 is a zero of p(x) = x3 − 3x2 + 2x.
Solution: p(1) = 1 − 3 + 2 = 0 Thus, x = 1 is a zero.

Problem 6: Find zero of p(x) = 4x + 9.
Solution: 4x + 9 = 0 4x =  − 9 $x=-\frac{9}{4}$

Problem 7: Determine the zeros of p(x) = x2 − 9.
Solution: x2 − 9 = 0 (x−3)(x+3) = 0 x = 3 or x =  − 3

Problem 8: Find the zero of p(x) = 7x + 21.
Solution: 7x + 21 = 0 7x =  − 21 x =  − 3

Problem 9: Find the zeros of p(x) = x2 + 2x − 8.
Solution: p(x) = (x−2)(x+4) Zeros are x = 2 and x =  − 4.

Problem 10: Verify if x =  − 2 is a zero of p(x) = x2 + 4x + 4.
Solution: p(−2) = (−2)2 + 4(−2) + 4 = 4 − 8 + 4 = 0 Thus, x =  − 2 is a zero.

Self Practice Problems (Exercise)

  1. Find the zero of p(x) = 3x − 15.

  2. Find the zero of p(x) = 7x + 14.

  3. Find the zero of p(x) = 5x − 20.

  4. Find the zeros of p(x) = x2 − 7x + 12.

  5. Find the zeros of p(x) = x2 + 6x + 8.

  6. Find the zeros of p(x) = x2 − 2x − 8.

  7. Verify if x = 1 is a zero of p(x) = x2 − 2x + 1.

  8. Find the zeros of p(x) = x2 − 4.

  9. Find the zero of p(x) = 6x + 18.

  10. Find the zeros of p(x) = x2 − x − 6.

  11. Check if x = 2 is a zero of p(x) = 2x2 − 4x.

  12. Find the zeros of p(x) = x2 + x − 12.

  13. Find the zero of p(x) = 8x + 16.

  14. Verify if x =  − 1 is a zero of p(x) = x3 + 1.

  15. Find the zeros of p(x) = x2 − 1.

Answers for Exercise

  1. x = 5

  2. x =  − 2

  3. x = 4

  4. x = 3, 4

  5. x =  − 2,  − 4

  6. x = 4,  − 2

  7. Yes

  8. x = 2,  − 2

  9. x =  − 3

  10. x = 3,  − 2

  11. Yes

  12. x = 3,  − 4

  13. x =  − 2

  14. Yes

  15. x = 1,  − 1

Theory: Relationship between Zeros and Coefficients

A quadratic polynomial is a polynomial of degree two and is generally written as: p(x) = ax2 + bx + c,  a ≠ 0 Let α and β be the zeros (roots) of p(x). Then the following important relationships hold: $$\text{Sum of zeros } (\alpha + \beta) = -\frac{b}{a}$$ $$\text{Product of zeros } (\alpha \times \beta) = \frac{c}{a}$$

These relationships are extremely useful for forming quadratic polynomials when zeros are given, and vice versa.

The graph of a quadratic polynomial is a parabola. If the parabola cuts the x-axis at two points, these points represent the zeros.

Here, the parabola touches the x-axis at the points corresponding to the zeros α and β.

Solved Problems

Problem 1: Find the sum and product of zeros of p(x) = x2 − 5x + 6.
Solution: $$\text{Sum} = -\frac{-5}{1} = 5, \quad \text{Product} = \frac{6}{1} = 6$$

Problem 2: Find the sum and product of zeros of p(x) = 2x2 + 3x − 5.
Solution: $$\text{Sum} = -\frac{3}{2}, \quad \text{Product} = \frac{-5}{2}$$

Problem 3: Form a quadratic polynomial whose zeros are 2 and 3.
Solution: Sum = 2 + 3 = 5,  Product = 2 × 3 = 6 Polynomial: x2 − 5x + 6

Problem 4: Form a quadratic polynomial whose zeros are $-\frac{1}{2}$ and $\frac{3}{4}$.
Solution: $$\text{Sum} = -\frac{1}{2} + \frac{3}{4} = \frac{1}{4}, \quad \text{Product} = -\frac{3}{8}$$ Polynomial: $x^2 – \frac{1}{4}x – \frac{3}{8}$

Problem 5: Find the sum and product of zeros of p(x) = 3x2 − 7x + 2.
Solution: $$\text{Sum} = -\frac{-7}{3} = \frac{7}{3}, \quad \text{Product} = \frac{2}{3}$$

Problem 6: If the sum of zeros is 4 and the product is 3, find the polynomial.
Solution: Required Polynomial: x2 − 4x + 3

Problem 7: Form the quadratic polynomial whose sum and product of zeros are 0 and $-\frac{9}{4}$.
Solution: Polynomial: $x^2 + \frac{9}{4}$

Problem 8: Find the quadratic polynomial with zeros $-\sqrt{3}$ and $\sqrt{3}$.
Solution: Sum = 0,  Product =  − 3 Polynomial: x2 + 3

Problem 9: Form the polynomial with zeros α = 2, β =  − 5.
Solution: Sum = 2 + (−5) =  − 3,  Product = 2 × (−5) =  − 10 Polynomial: x2 + 3x − 10

Problem 10: The product of zeros is 1 and sum is -2. Find the polynomial.
Solution: Polynomial: x2 + 2x + 1

Self Practice Problems (Exercise)

  1. Find sum and product of zeros of p(x) = x2 − 4x + 3.

  2. Find sum and product of zeros of p(x) = 2x2 − 3x + 1.

  3. Form a quadratic polynomial whose zeros are 1 and 2.

  4. Find the quadratic polynomial with zeros $-\frac{1}{3}$ and $\frac{2}{5}$.

  5. Find the sum and product of zeros of p(x) = 5x2 + 2x − 1.

  6. Form a quadratic polynomial with zeros $\sqrt{2}$ and $-\sqrt{2}$.

  7. Form a quadratic polynomial whose sum of zeros is  − 5 and product is 6.

  8. Find the polynomial whose zeros are 3 and  − 1.

  9. Find sum and product of zeros of p(x) = 4x2 + 8x + 3.

  10. Form the polynomial if sum of zeros = 0 and product = -16.

  11. Find the quadratic polynomial whose zeros are $\frac{1}{2}$ and $-\frac{3}{2}$.

  12. Find the quadratic polynomial whose zeros are 0 and 5.

  13. Find sum and product of zeros of p(x) = 6x2 − 5x + 1.

  14. Form a polynomial with zeros  − 7 and 2.

  15. Find sum and product of zeros of p(x) = x2 + 5x + 6.

Answers for Exercise

  1. Sum = 4, Product = 3

  2. Sum = $\frac{3}{2}$, Product = $\frac{1}{2}$

  3. x2 − 3x + 2

  4. $x^2 + \frac{1}{15}x – \frac{2}{15}$

  5. Sum = $\frac{3}{5}$, Product = $-\frac{1}{5}$

  6. x2 − 0x − 2

  7. x2 + 5x + 6

  8. x2 − 2x − 3

  9. Sum =  − 2, Product = $\frac{3}{4}$

  10. x2 − 0x − 16

  11. $x^2 + \frac{1}{2}x – \frac{3}{4}$

  12. x(x−5)

  13. Sum = $\frac{5}{6}$, Product = $\frac{1}{6}$

  14. x2 + 5x − 14

  15. Sum =  − 5, Product = 6

Instructions:

  • Choose the correct option.

  • Each question carries equal marks.

MCQs

2

  1. The degree of the polynomial 3x2 − 5x + 2 is:

    • 1

    • 2

    • 0

    • 3

  2. If α and β are zeros of x2 − 3x + 2, then α + β is:

    • 2

    • 3

    • 1

    • 5

  3. The product of zeros of 2x2 − 5x + 3 is:

    • $\frac{5}{2}$

    • $\frac{3}{2}$

    • $\frac{3}{2}$

    • $-\frac{3}{2}$

  4. Zeros of x2 − 5x + 6 are:

    • 2, 3

    • -2, -3

    • 1, 6

    • 5, 1

  5. For the polynomial 2x2 + 3x − 2, sum of zeros is:

    • $\frac{3}{2}$

    • $-\frac{3}{2}$

    • 2

    • $-\frac{2}{3}$

  6. The graph of a quadratic polynomial is a:

    • Line

    • Circle

    • Parabola

    • Ellipse

  7. Number of zeros of a quadratic polynomial is:

    • 1

    • 2

    • 3

    • 0

  8. If the sum of zeros is 0, then the polynomial is:

    • x2 + c

    • x2 − c

    • c(x2+1)

    • None

  9. Find the product of zeros of x2 + 2x + 1:

    • 2

    • 1

    • -1

    • 0

  10. For polynomial x2 − 7x + 10, zeros are:

    • 5 and 2

    • 7 and 10

    • 1 and 10

    • 2 and 5

  11. A polynomial of degree 2 is called:

    • Linear

    • Cubic

    • Quadratic

    • Constant

  12. Find the sum of zeros of 3x2 − 5x + 2:

    • $\frac{5}{3}$

    • $-\frac{5}{3}$

    • $\frac{2}{3}$

    • $-\frac{2}{3}$

  13. If α and β are zeros, αβ means:

    • Sum

    • Product

    • Division

    • Subtraction

  14. Find the zeros of x2 − 1:

    • 1 and 1

    • 1 and -1

    • -1 and 0

    • 0 and 1

  15. Product of zeros of 5x2 + 2x − 3:

    • $\frac{-3}{5}$

    • $\frac{2}{5}$

    • $\frac{3}{5}$

    • $\frac{-2}{5}$

  16. A quadratic polynomial with sum 0 and product 16 is:

    • x2 + 16

    • x2 − 16

    • x2 + 4

    • x2 − 4

  17. Find the sum of zeros of 2x2 − x − 3:

    • $\frac{1}{2}$

    • $\frac{-1}{2}$

    • $\frac{3}{2}$

    • $\frac{-3}{2}$

  18. Find the polynomial if zeros are 1 and -1:

    • x2 + 1

    • x2 − 1

    • x2

    • x2 − 2

  19. Find product of zeros of 3x2 − x − 2:

    • $\frac{-2}{3}$

    • $\frac{2}{3}$

    • $-\frac{1}{3}$

    • $-\frac{3}{2}$

  20. The graph of x2 − 4 touches x-axis at:

    • 2 and -2

    • 4 and -4

    • 1 and -1

    • None

  21. Find zeros of x2 + 4x + 3:

    • -1, -3

    • -2, -2

    • 1, 3

    • 1, -3

  22. Sum of zeros of 2x2 + 5x + 2 is:

    • $\frac{5}{2}$

    • $-\frac{5}{2}$

    • $\frac{2}{5}$

    • $-\frac{2}{5}$

  23. Polynomial with zeros 3 and 2 is:

    • x2 + 5x + 6

    • x2 − 5x + 6

    • x2 + 6x + 5

    • x2 − 6x + 5

  24. Find sum and product of zeros of x2 − 2x + 1:

    • 2 and 1

    • -2 and -1

    • 2 and 1

    • 2 and -1

  25. If α + β = 5 and αβ = 6, the polynomial is:

    • x2 − 5x + 6

    • x2 + 5x + 6

    • x2 − 6x + 5

    • x2 + 6x + 5

Answers:

2

  1. (b)

  2. (b)

  3. (c)

  4. (a)

  5. (b)

  6. (c)

  7. (b)

  8. (c)

  9. (b)

  10. (d)

  11. (c)

  12. (b)

  13. (b)

  14. (b)

  15. (a)

  16. (a)

  17. (b)

  18. (b)

  19. (a)

  20. (a)

  21. (a)

  22. (a)

  23. (b)

  24. (a)

  25. (a)

Instructions:

  • Section A contains 8 Multiple Choice Questions (MCQs).

  • Section B contains 6 Short Answer Questions.

  • Section C contains 4 Long Answer Questions.

  • Section D contains 1 Case Study-based Question.

  • Choose the correct answers for MCQs.

  • Provide brief but clear explanations for Short and Long Answer Questions.

Section A: Multiple Choice Questions (MCQs)

2

  1. The sum of the zeros of 2x2 − 5x + 3 is:

    • $\frac{5}{2}$

    • $\frac{-5}{2}$

    • $\frac{2}{3}$

    • $-\frac{2}{3}$

  2. Zeros of the polynomial x2 + 7x + 10 are:

    • 5 and 2

    • 1 and 6

    • -1 and -6

    • 7 and 10

  3. For the quadratic polynomial 3x2 − 4x + 1, the product of the zeros is:

    • $\frac{1}{3}$

    • $\frac{-1}{3}$

    • $-\frac{1}{3}$

    • $\frac{-1}{2}$

  4. The zeros of x2 − 9 are:

    • 3 and -3

    • 4 and 2

    • 1 and -1

    • 9 and -9

  5. The degree of the polynomial 4x3 − x2 + 2x − 7 is:

    • 1

    • 2

    • 3

    • 4

  6. Which of the following is a quadratic polynomial?

    • 3x3 + 5x − 6

    • x2 − 4x + 4

    • x4 − 5x2 + 6

    • 2x3 − 7x + 1

  7. The sum of the zeros of x2 − 2x − 3 is:

    • 3

    • -3

    • 2

    • -2

  8. The zeros of the polynomial x2 − 3x + 2 are:

    • 2 and 1

    • -2 and -1

    • 3 and 2

    • 1 and -2

Section B: Short Answer Questions

  1. Find the sum and product of the zeros of the quadratic polynomial x2 + 5x + 6.

  2. Find the zeros of the polynomial 3x2 − 6x + 2.

  3. Write the quadratic polynomial whose sum and product of zeros are 3 and -8, respectively.

  4. If one zero of the polynomial x2 + 5x − 6 is 1, find the other zero.

  5. Factorize the polynomial x2 + 7x + 10 completely.

  6. Find the degree of the polynomial 5x4 + 2x2 − 3x + 1 and determine if it is a polynomial in standard form.

Section C: Long Answer Questions

  1. Find the quadratic polynomial whose zeros are 2 and -3. Verify your answer by using the relationship between zeros and coefficients.

  2. A quadratic polynomial is given by x2 + 4x + 4. Show how to find its zeros and explain the method used.

  3. Explain the relationship between the zeros and coefficients of a quadratic polynomial in detail. Use examples for better understanding.

  4. Factorize the polynomial x2 − 5x + 6 and find its zeros. Also, verify the factorization using the relationship between the zeros and coefficients.

Section D: Case Study-Based Question

Case Study:

A polynomial p(x) = x2 − 4x + 4 represents the height (in meters) of a ball thrown upward at a particular time x. The ball reaches the maximum height at x = 2 seconds, and the height decreases afterward.

Answer the following questions based on the case study:

  1. What is the maximum height the ball reaches?

  2. How many seconds after the ball was thrown does it reach the maximum height?

  3. What is the value of the height of the ball at x = 0?

  4. Can we say the polynomial is a quadratic equation? Why or why not?

  5. What are the roots (zeros) of the polynomial, and what do they represent in the context of this case study?

Answers to Sections A, B, C, and D

2 Section A:
1. (b)
2. (a)
3. (c)
4. (a)
5. (c)
6. (b)
7. (d)
8. (a)
Section B:
1. Sum = -5, Product = 6
2. Zeros = $\frac{2}{3}, 1$
3. Polynomial = x2 − 3x − 8
4. Other zero = 3
5. Factorization: (x+2)(x+5)
6. Degree = 4, Polynomial is in standard form
Section C:
1. Polynomial = x2 − 4x + 6
2. Zeros = 3 and -1
3. Explanation of relationship with examples
4. Factorization: (x−2)(x−3)
Section D:
1. Maximum height = 4 meters
2. Time = 2 seconds
3. Height at x = 0 = 4 meters
4. Yes, it is a quadratic polynomial
5. Zeros = 2, 2 (time when ball touches the ground)

Quick Revision Theory Table

Concept Explanation
Concept Explanation
Polynomials A polynomial is an algebraic expression made up of terms, where each term consists of a variable raised to a non-negative integer power, multiplied by a constant coefficient.
Degree of Polynomial The highest power of the variable in a polynomial is called its degree. For example, in 4x3 + 2x2 − 3x + 1, the degree is 3.
Zeros of a Polynomial Zeros of a polynomial are the values of the variable for which the polynomial equals zero. The roots or solutions of the polynomial equation are the zeros.
Quadratic Polynomial A quadratic polynomial is a polynomial of degree 2. It is in the form ax2 + bx + c.
Relationship between Zeros and Coefficients For a quadratic polynomial ax2 + bx + c, the sum of the zeros is $-\frac{b}{a}$ and the product of the zeros is $\frac{c}{a}$.
Factorization of Polynomials Factorizing a polynomial involves writing it as a product of its factors. For example, x2 − 5x + 6 can be factorized as (x−2)(x−3).
Sum and Product of Zeros The sum of the zeros of a quadratic polynomial ax2 + bx + c is $-\frac{b}{a}$, and the product is $\frac{c}{a}$.
Roots of a Polynomial Roots or zeros are the solutions of the polynomial equation. They are the points where the graph of the polynomial intersects the x-axis.
Graph of a Quadratic Polynomial The graph of a quadratic polynomial is a parabola. The shape of the parabola is determined by the sign of the leading coefficient (positive opens upwards, negative opens downwards).
Real Zeros of a Polynomial A polynomial may have real or complex zeros. Real zeros can be found through factorization or using the quadratic formula.
Factor Theorem The Factor Theorem states that if a polynomial f(x) has a zero at x = a, then (x−a) is a factor of the polynomial.
Remainder Theorem The Remainder Theorem states that when a polynomial f(x) is divided by (x−a), the remainder is f(a).
Relationship Between Roots The roots of a quadratic polynomial ax2 + bx + c can be found using the quadratic formula: $\frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$.
Synthetic Division Synthetic division is a shortcut method for dividing a polynomial by a linear binomial of the form (x−a). It is used for simplifying polynomial division.
Common Mistakes Common mistakes include sign errors, forgetting to factor out the leading coefficient, and incorrect usage of the quadratic formula.