How to Find Zeros of a Polynomial with Formula and 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 Understanding Zeros of a Polynomial
1. What are zeros of a polynomial?
The zeros of a polynomial are the values of the variable that make the polynomial equal to 0. In other words, if f(x) is a polynomial, then any value of x that satisfies f(x) = 0 is called a zero or root.
- For example, if f(x) = x − 3, then setting x − 3 = 0 gives x = 3.
- So, 3 is the zero of the polynomial.
- Graphically, zeros are the points where the graph cuts or touches the x-axis.
2. How do you find the zeros of a polynomial?
To find the zeros of a polynomial, set the polynomial equal to 0 and solve the resulting equation.
- Step 1: Write the equation f(x) = 0.
- Step 2: Factor the polynomial if possible.
- Step 3: Set each factor equal to 0 and solve.
- Example: For f(x) = x² − 9, factor as (x − 3)(x + 3) = 0.
- Thus, the zeros are x = 3 and x = −3.
3. What is the formula to find zeros of a quadratic polynomial?
The quadratic formula to find the zeros of ax² + bx + c is x = (−b ± √(b² − 4ac)) / 2a. This formula works for any quadratic equation where a ≠ 0.
- Example: For 2x² + 3x − 2 = 0
- a = 2, b = 3, c = −2
- Discriminant = b² − 4ac = 9 + 16 = 25
- x = (−3 ± 5)/4
- Zeros are 1/2 and −2.
4. What is the relationship between zeros and factors of a polynomial?
The Factor Theorem states that if (x − a) is a factor of a polynomial, then a is a zero of that polynomial. This means factors and zeros are directly related.
- If f(a) = 0, then (x − a) is a factor.
- If (x − a) divides f(x) exactly, then a is a zero.
- Example: If f(x) = x² − 4, then (x − 2) is a factor, so 2 is a zero.
5. How many zeros can a polynomial have?
A polynomial of degree n can have at most n zeros. This is based on the Fundamental Theorem of Algebra.
- A linear polynomial (degree 1) has 1 zero.
- A quadratic polynomial (degree 2) has at most 2 zeros.
- A cubic polynomial (degree 3) has at most 3 zeros.
- Some zeros may be real or complex numbers.
6. What are real and complex zeros of a polynomial?
Real zeros are real number solutions, while complex zeros include imaginary numbers involving i where i² = −1. Both satisfy f(x) = 0.
- Example: x² − 4 = 0 has real zeros 2 and −2.
- Example: x² + 4 = 0 has complex zeros 2i and −2i.
- Quadratic equations with negative discriminant have complex zeros.
7. How do zeros of a polynomial relate to its graph?
The zeros of a polynomial are the x-intercepts of its graph. They are the points where the graph meets the x-axis.
- If the graph crosses the x-axis, the zero has odd multiplicity.
- If the graph touches and turns back, the zero has even multiplicity.
- For example, the graph of x² − 1 crosses at x = −1 and x = 1.
8. What is the multiplicity of a zero?
The multiplicity of a zero is the number of times a particular zero is repeated as a factor. It shows how the graph behaves at that zero.
- If (x − 2)² is a factor, then 2 has multiplicity 2.
- Even multiplicity: graph touches and turns.
- Odd multiplicity: graph crosses the x-axis.
9. Can a polynomial have no real zeros?
Yes, a polynomial can have no real zeros if all its solutions are complex. This happens when the graph does not intersect the x-axis.
- Example: f(x) = x² + 1
- Solving x² + 1 = 0 gives x² = −1
- Zeros are i and −i, which are not real numbers.
10. What is the Fundamental Theorem of Algebra in relation to zeros?
The Fundamental Theorem of Algebra states that every polynomial of degree n has exactly n complex zeros, counting multiplicity. This includes both real and complex roots.
- A quadratic has 2 zeros.
- A cubic has 3 zeros.
- Some zeros may repeat or be complex conjugates.
