How to Find Zeros of a Polynomial—Step-by-Step Method with Examples
The concept of zeros of polynomial is fundamental in algebra and essential in solving equations and understanding graphs. Grasping this concept is key for school exams, competitive tests, and practical problem-solving with polynomials in real life.
Understanding Zeros of Polynomial
A zero of a polynomial is a value of the variable at which the polynomial expression becomes zero. In simple terms, if we have a polynomial f(x), its zero is any value "a" for which f(a) = 0. This is also called a root or solution of the polynomial. The degree of the polynomial tells us the maximum number of zeros it can have. Concepts like roots of polynomial, x-intercepts, and real and complex zeros are all connected to this topic.
What is Zero of a Polynomial?
A zero of a polynomial is a value of x for which the polynomial equals zero. In other words, if f(x) = 0, then x is called a zero or root of the polynomial f(x).
Formula Used in Zeros of Polynomial
The standard formula to find the zero of a linear polynomial P(x) = ax + b is:
If P(a) = 0, then a = -b/a
For quadratics (ax2 + bx + c = 0), use the quadratic formula:
x = [-b ± √(b2 - 4ac)] / (2a)
For higher degrees, techniques like factor theorem, remainder theorem, and factorisation are used.
Key Table: Zeros and Their Interpretation
Here’s a table showing the relationship between degree, maximum zeros, and their types:
| Degree of Polynomial | Max Number of Zeros | Example Polynomial |
|---|---|---|
| 1 (Linear) | 1 | 2x+3 |
| 2 (Quadratic) | 2 | x2−3x+2 |
| 3 (Cubic) | 3 | x3−6x2+11x−6 |
| n | n | Custom |
This table helps you see how the number of zeros depends directly on the degree of the polynomial.
Step-by-Step: How to Find Zeros of a Polynomial
Follow these steps to find the zeros of any polynomial:
1. Write the polynomial in standard form.2. Set the polynomial equal to zero: P(x) = 0.
3. For linear polynomials (e.g., ax + b), simply solve for x:
x = -b/a
4. For quadratic polynomials (e.g., ax2 + bx + c):
x = [-b ± √(b2 - 4ac)] / 2a
5. For cubic and higher degree polynomials:
6. Substitute the values back into the original polynomial to verify.
Let’s illustrate this with a worked example below.
Worked Example – Finding Zeros
Example 1: Find the zeros of the polynomial P(x) = x2 – 3x + 2
1. Write the polynomial: x2 – 3x + 2.2. Set P(x) = 0: x2 – 3x + 2 = 0.
3. Factorise: (x – 1)(x – 2) = 0.
4. Set each factor to zero: x – 1 = 0 ⇒ x = 1
x – 2 = 0 ⇒ x = 2
5. Check: P(1) = 1 – 3 + 2 = 0 and P(2) = 4 – 6 + 2 = 0.
Final Answer: The zeros of this polynomial are x = 1 and x = 2.
Example 2: Find the zero of the linear polynomial P(x) = 4x + 5.
1. Set P(x) = 0: 4x + 5 = 02. Rearrange: 4x = –5
3. x = –5/4
Final Answer: The zero is x = –5/4.
Linking Zeros and Graphs
Graphically, the zeros of polynomial represent the points where the graph crosses the x-axis. This is also called the x-intercept. For example, in the quadratic above, the graph of y = x2 – 3x + 2 touches the x-axis at x = 1 and x = 2. See more about the geometrical meaning of zeroes here.
Common Mistakes to Avoid
- Mixing up zeros (solutions) with factors of the polynomial expression.
- Forgetting to set P(x) = 0 before solving for zeros.
- Not checking the number of possible zeros based on the polynomial’s degree.
- Skipping the verification step after finding zeros.
Real-World Applications
Knowing how to find the zeros of polynomial is useful in many scenarios, such as finding break-even points in business, predicting projectile paths in physics, and modelling roots in engineering. Vedantu gives students the tools to see where mathematics meets everyday problem-solving.
Practice Problems – Zeros of Polynomial
- Find the zeros of P(x) = 2x – 4.
- How many zeros does the polynomial x3 – 4x2 + 4x have? Find their values.
- Determine the zeros of P(x) = x2 + x – 6.
- If x = 0 is a zero of a polynomial, write possible degrees the polynomial could have.
Summary – Key Takeaways
- The zero of a polynomial is the value making the polynomial zero.
- Number of zeros cannot exceed the degree of the polynomial.
- To find zeros: set the polynomial equal to zero, solve by factorisation or formula, then verify.
- Zeros are the same as roots and x-intercepts (graphically).
- Practice stepwise methods for accuracy in board or entrance exams.
Further Learning – Linked Topics
- Polynomial
- Polynomial Equations
- Relationship between Zeroes and Coefficients of Polynomials
- Geometrical Meaning of Zeroes of the Polynomial
- Factor Theorem
- Remainder Theorem
- Quadratics
- Multiplying Polynomials
- Degree of Polynomial
- Polynomials in One Variable
- Roots of Polynomial Equation
We explored zeros of polynomial: how to define, find, and use them, with stepwise examples for board exam success. Continue practicing on Vedantu for more concepts and revision help!
FAQs on Zeros of a Polynomial Explained with Easy Examples
1. What is a zero of a polynomial?
A zero of a polynomial is a value of x for which the polynomial expression equals zero. Formally, if f(x) = 0 for some number x = k, then k is called a zero or root of the polynomial f(x). Understanding zeros is essential for solving polynomial equations and analyzing their graphs, especially in Class 9 and 10 Maths.
2. How to find zeros of polynomial functions?
To find the zeros of a polynomial function, follow these steps:
1. Set the polynomial equal to zero, i.e., f(x) = 0.
2. Solve the resulting equation by factoring, using the quadratic formula, or other methods depending on the polynomial’s degree.
3. The solutions obtained are the zeros or roots of the polynomial.
This process is crucial for understanding the behavior and graph of the polynomial function.
3. What is the formula for zeros of a polynomial?
For a linear polynomial in one variable, P(x) = ax + b, the zero is found by using the formula: x = -\frac{b}{a}, where a ≠ 0. For quadratic and higher-degree polynomials, zeros can be found using factorization, the factor theorem, or formulas such as the quadratic formula. These formulas help locate the points where the polynomial evaluates to zero.
4. What are the zeros of the polynomial 6x² − 3 − 7x?
To find the zeros of 6x² − 7x − 3, rewrite the polynomial as 6x² − 7x − 3 = 0.
1. Use factorization or the quadratic formula: x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where a=6, b=-7, and c=-3.
2. Calculate discriminant: b^2 - 4ac = (-7)^2 - 4(6)(-3) = 49 + 72 = 121.
3. Calculate roots: x = \frac{7 \pm 11}{12}.
Thus, zeros are x = \frac{18}{12} = \frac{3}{2} and x = \frac{-4}{12} = -\frac{1}{3}.
5. What are zeros and factors of a polynomial?
The zeros of a polynomial are the values of x that make the polynomial equal to zero. Factors of a polynomial are expressions that multiply together to give the polynomial. According to the factor theorem, if k is a zero of the polynomial f(x), then (x - k) is a factor of f(x). This relationship is fundamental for finding zeros through factorization.
6. What is a zero of a polynomial class 10 notes?
In Class 10 Maths, a zero of a polynomial is defined as a value of x for which the polynomial evaluates to zero. Notes often emphasize:
• The connection between zeros and factors.
• The role of zeros in solving polynomial equations.
• How zeros help in sketching polynomial graphs.
This concept is included in the NCERT syllabus and is key for board exam preparation.
7. Why can a polynomial have complex zeros?
A polynomial can have complex zeros because not all polynomial equations have real solutions. According to the Fundamental Theorem of Algebra, every polynomial of degree n has exactly n roots in the complex number system (including repeated roots). Complex zeros occur especially when the polynomial does not cross the x-axis in the real plane but has roots in the form of complex conjugates.
8. Why is every zero also called a root?
Every zero of a polynomial is called a root because both terms describe the value of x for which the polynomial equals zero. The term root comes from the concept of solving the equation f(x) = 0. Thus, zeros and roots are interchangeable concepts that reflect solutions of polynomial equations.
9. How do multiplicity and repeated zeros affect the polynomial graph?
The multiplicity of a zero refers to the number of times it appears as a root of the polynomial. Repeated zeros with:
• Odd multiplicity cause the graph to cross the x-axis at the zero.
• Even multiplicity cause the graph to touch the x-axis and turn back.
Multiplicity affects the shape of the graph near the zeros and is essential for graph interpretation in Class 10 Maths.
10. Why do students confuse zeros with factors?
Students often confuse zeros with factors because both concepts are closely linked but distinct. Zeros are values for which the polynomial equals zero, while factors are polynomial expressions that multiply to give the polynomial. The factor theorem connects them: if k is a zero, then (x - k) is a factor. Clarifying this link helps reduce confusion.
11. Why is finding zeros important for graphing polynomials?
Finding the zeros of a polynomial is key to graphing because zeros represent the points where the graph crosses or touches the x-axis (x-intercepts). Knowing zeros helps in:
• Plotting accurate graphs.
• Understanding the polynomial’s behavior.
• Determining intervals of positive or negative values.
This makes zero-finding a fundamental skill for Class 9 and 10 Maths graphing problems.
