Zeroes of a Polynomial
A zero of a polynomial is a value of the variable that makes the polynomial equal to zero. Finding zeroes is one of the central problems in algebra and forms the basis for solving polynomial equations.
In Class 9 Mathematics (NCERT Chapter 2: Polynomials), zeroes of polynomials connect the concepts of evaluation, the Remainder Theorem, and the Factor Theorem into a unified framework. This topic builds upon the understanding of polynomial types and degree studied earlier.
If p(x) is a polynomial and p(a) = 0, then a is called a zero (or root) of p(x). The number of zeroes a polynomial can have is determined by its degree.
The concept of zeroes has deep connections with graphing. On the graph of y = p(x), zeroes correspond to the x-intercepts — the points where the curve meets the x-axis. This graphical interpretation provides visual insight into the algebraic concept.
Zeroes of polynomials are used extensively in science, engineering, economics, and data analysis. Wherever a quantity becomes zero — a projectile hitting the ground, profit breaking even, or a chemical reaction reaching equilibrium — polynomial zeroes provide the mathematical solution.
What is Zeroes of a Polynomial?
Definition: A real number a is called a zero of the polynomial p(x) if p(a) = 0.
Equivalently, a is a zero of p(x) if (x − a) is a factor of p(x).
Key facts:
- A polynomial of degree n can have at most n zeroes.
- A linear polynomial (degree 1) has exactly one zero.
- A quadratic polynomial (degree 2) has at most two zeroes.
- A cubic polynomial (degree 3) has at most three zeroes.
- A constant polynomial (non-zero) has no zeroes.
- The zero polynomial (p(x) = 0) has every real number as a zero.
Zeroes of a Polynomial Formula
Finding Zeroes:
1. Linear Polynomial: p(x) = ax + b, a ≠ 0
Zero = −b/a
2. Quadratic Polynomial: p(x) = ax² + bx + c
Set p(x) = 0 and solve using:
- Factorisation (splitting the middle term)
- Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a
- Completing the square
3. Cubic Polynomial: p(x) = ax³ + bx² + cx + d
- Use the Factor Theorem to find one zero by trial.
- Divide p(x) by (x − a) to get a quadratic quotient.
- Solve the quadratic to find the remaining zeroes.
Relationship between zeroes and coefficients (Quadratic):
If α and β are the zeroes of ax² + bx + c, then:
α + β = −b/a and αβ = c/a
Derivation and Proof
Derivation: Zero of a Linear Polynomial
For p(x) = ax + b, where a ≠ 0:
- Set p(x) = 0
- ax + b = 0
- ax = −b
- x = −b/a
Verification using the Factor Theorem:
If a is a zero of p(x), then (x − a) is a factor. Conversely, if (x − a) is a factor, then p(a) = 0, so a is a zero.
Derivation: Number of zeroes equals degree
The Fundamental Theorem of Algebra establishes that:
- Every non-constant polynomial has at least one zero (over the complex numbers).
- A polynomial of degree n can be written as: p(x) = a(x − r₁)(x − r₂)...(x − rₙ)
- Each factor (x − rᵢ) gives one zero, so there are exactly n zeroes (counting multiplicity).
- Over real numbers, some zeroes may be complex, so the number of real zeroes is at most n.
Graphical interpretation:
- Zeroes of p(x) are the x-coordinates where the graph y = p(x) meets the x-axis.
- A linear polynomial graph (straight line) crosses the x-axis at exactly one point.
- A quadratic polynomial graph (parabola) may cross the x-axis at 0, 1, or 2 points.
- A cubic polynomial graph crosses the x-axis at 1, 2, or 3 points.
Types and Properties
Types of Polynomials and Their Zeroes:
1. Constant Polynomial (degree 0)
- p(x) = c, where c ≠ 0
- Has no zero, since p(x) never equals 0.
- Exception: the zero polynomial p(x) = 0 has every real number as a zero.
2. Linear Polynomial (degree 1)
- p(x) = ax + b, where a ≠ 0
- Has exactly one zero: x = −b/a
- Example: p(x) = 3x − 6 has zero x = 2.
3. Quadratic Polynomial (degree 2)
- p(x) = ax² + bx + c, where a ≠ 0
- Has at most two zeroes.
- The discriminant D = b² − 4ac determines the nature of zeroes:
- D > 0: Two distinct real zeroes
- D = 0: Two equal real zeroes (one repeated zero)
- D < 0: No real zeroes
4. Cubic Polynomial (degree 3)
- p(x) = ax³ + bx² + cx + d, where a ≠ 0
- Has at most three zeroes.
- A cubic always has at least one real zero.
5. Polynomial with Repeated Zeroes
- p(x) = (x − 2)² = x² − 4x + 4 has zero x = 2 with multiplicity 2.
- The graph touches the x-axis at x = 2 but does not cross it.
Solved Examples
Example 1: Example 1: Zero of a linear polynomial
Problem: Find the zero of p(x) = 4x − 12.
Solution:
Given:
- p(x) = 4x − 12
Set p(x) = 0:
- 4x − 12 = 0
- 4x = 12
- x = 12/4 = 3
Verification: p(3) = 4(3) − 12 = 12 − 12 = 0 ✔
Answer: The zero of p(x) is x = 3.
Example 2: Example 2: Zero of a linear polynomial with fractional answer
Problem: Find the zero of p(x) = 5x + 3.
Solution:
Given:
- p(x) = 5x + 3
Set p(x) = 0:
- 5x + 3 = 0
- 5x = −3
- x = −3/5
Verification: p(−3/5) = 5(−3/5) + 3 = −3 + 3 = 0 ✔
Answer: The zero of p(x) is x = −3/5.
Example 3: Example 3: Zeroes of a quadratic polynomial by factorisation
Problem: Find the zeroes of p(x) = x² − 5x + 6.
Solution:
Given:
- p(x) = x² − 5x + 6
Factorise by splitting the middle term:
- Find two numbers whose product = 6 and sum = −5.
- The numbers are −2 and −3.
- p(x) = x² − 2x − 3x + 6 = x(x − 2) − 3(x − 2)
- p(x) = (x − 2)(x − 3)
Set each factor to 0:
- x − 2 = 0 ⇒ x = 2
- x − 3 = 0 ⇒ x = 3
Verification: p(2) = 4 − 10 + 6 = 0 ✔; p(3) = 9 − 15 + 6 = 0 ✔
Answer: The zeroes are x = 2 and x = 3.
Example 4: Example 4: Zeroes of a quadratic polynomial with equal roots
Problem: Find the zeroes of p(x) = x² − 6x + 9.
Solution:
Given:
- p(x) = x² − 6x + 9
Recognise the perfect square:
- p(x) = (x − 3)²
- Set p(x) = 0: (x − 3)² = 0
- x − 3 = 0 ⇒ x = 3
Check discriminant: D = (−6)² − 4(1)(9) = 36 − 36 = 0. Since D = 0, there is one repeated zero.
Answer: The zero is x = 3 (repeated, with multiplicity 2).
Example 5: Example 5: Verify a given zero
Problem: Verify whether x = −1 is a zero of p(x) = 2x³ + x² − 5x − 2.
Solution:
Substitute x = −1 into p(x):
- p(−1) = 2(−1)³ + (−1)² − 5(−1) − 2
- p(−1) = 2(−1) + 1 + 5 − 2
- p(−1) = −2 + 1 + 5 − 2
- p(−1) = 2
Since p(−1) = 2 ≠ 0, x = −1 is not a zero of p(x).
Answer: No, x = −1 is not a zero of the given polynomial.
Example 6: Example 6: Find zeroes and verify sum-product relationship
Problem: Find the zeroes of p(x) = 2x² − 7x + 3 and verify the relationship between zeroes and coefficients.
Solution:
Factorise:
- 2x² − 7x + 3 = 2x² − 6x − x + 3
- = 2x(x − 3) − 1(x − 3)
- = (2x − 1)(x − 3)
Zeroes:
- 2x − 1 = 0 ⇒ x = 1/2
- x − 3 = 0 ⇒ x = 3
Verification:
- Sum of zeroes = 1/2 + 3 = 7/2 = −(−7)/2 = −b/a ✔
- Product of zeroes = (1/2)(3) = 3/2 = c/a ✔
Answer: Zeroes are x = 1/2 and x = 3. Relationships verified.
Example 7: Example 7: Zeroes of a cubic polynomial
Problem: Find all zeroes of p(x) = x³ − 4x² + x + 6.
Solution:
Step 1: Find one zero by trial
- Try x = −1: p(−1) = −1 − 4 − 1 + 6 = 0 ✔
- So x = −1 is a zero, and (x + 1) is a factor.
Step 2: Divide p(x) by (x + 1)
- x³ − 4x² + x + 6 = (x + 1)(x² − 5x + 6)
Step 3: Factorise the quadratic
- x² − 5x + 6 = (x − 2)(x − 3)
All factors: p(x) = (x + 1)(x − 2)(x − 3)
Answer: The zeroes are x = −1, x = 2, x = 3.
Example 8: Example 8: Polynomial with no real zeroes
Problem: Show that p(x) = x² + 4 has no real zeroes.
Solution:
Set p(x) = 0:
- x² + 4 = 0
- x² = −4
Since x² is always non-negative for real x, x² = −4 has no real solution.
Discriminant check:
- a = 1, b = 0, c = 4
- D = 0² − 4(1)(4) = −16 < 0
Since D < 0, the polynomial has no real zeroes.
Answer: p(x) = x² + 4 has no real zeroes.
Example 9: Example 9: Find a polynomial given its zeroes
Problem: Find a quadratic polynomial whose zeroes are 4 and −3.
Solution:
Given zeroes: α = 4, β = −3
Using the sum and product:
- Sum = α + β = 4 + (−3) = 1
- Product = αβ = 4 × (−3) = −12
Form the polynomial:
- p(x) = x² − (sum)x + (product)
- p(x) = x² − x − 12
Verification:
- p(4) = 16 − 4 − 12 = 0 ✔
- p(−3) = 9 + 3 − 12 = 0 ✔
Answer: The polynomial is p(x) = x² − x − 12.
Example 10: Example 10: Zeroes from a graph
Problem: The graph of a polynomial y = p(x) crosses the x-axis at x = −2, x = 0, and x = 5. What are the zeroes and what is the minimum degree of p(x)?
Solution:
Zeroes from the graph:
- The graph crosses the x-axis at x = −2, x = 0, x = 5.
- At these points, p(x) = 0, so these are the zeroes.
Minimum degree:
- There are 3 zeroes, so the degree must be at least 3.
- One such polynomial: p(x) = x(x + 2)(x − 5) = x³ − 3x² − 10x
Answer: Zeroes are −2, 0, 5. Minimum degree is 3.
Real-World Applications
Applications of Zeroes of Polynomials:
- Solving Equations: Finding zeroes is equivalent to solving polynomial equations. Every time a scientist, engineer, or economist writes an equation and sets it to zero, they are finding the zeroes of a polynomial. This is the most direct and common application.
- Graphing Polynomials: Zeroes determine where the graph of y = p(x) crosses or touches the x-axis. By finding the zeroes and a few additional points, the complete shape of a polynomial curve can be sketched accurately.
- Projectile Motion (Physics): The height of a projectile thrown upward is modelled by a quadratic polynomial h(t) = −4.9t² + v₀t + h₀. The zeroes of this polynomial give the times when the object is at ground level — the launch moment and the landing moment.
- Profit and Break-Even Analysis (Economics): Profit functions in business are often polynomials of revenue minus cost. Zeroes represent break-even points where the company neither profits nor loses money. These are critical for business planning.
- Signal Processing (Electronics): In electrical engineering, the transfer function of a circuit is a polynomial ratio. The zeroes of the numerator determine which frequencies are blocked, and the zeroes of the denominator determine resonance frequencies.
- Structural Engineering: Polynomial equations model the deflection of beams under load. Zeroes of the deflection polynomial indicate points of zero bending — the supports and inflection points of the beam.
- Chemistry: Equilibrium constants in chemical reactions lead to polynomial equations. Finding zeroes gives the equilibrium concentrations of reactants and products.
- Computer Science: Root-finding algorithms (such as Newton-Raphson and bisection methods) are fundamental computational tools that rely on iteratively approximating zeroes of polynomials.
Key Points to Remember
- A zero of polynomial p(x) is a value a such that p(a) = 0.
- Zeroes are also called roots of the polynomial equation p(x) = 0.
- A polynomial of degree n has at most n real zeroes.
- A linear polynomial ax + b has exactly one zero: x = −b/a.
- A quadratic polynomial ax² + bx + c has at most two zeroes. The discriminant D = b² − 4ac determines their nature.
- If D > 0: two distinct real zeroes; D = 0: one repeated zero; D < 0: no real zeroes.
- For a quadratic with zeroes α and β: sum of zeroes = −b/a, product of zeroes = c/a. These are called Vieta's formulas.
- Zeroes of p(x) correspond to x-intercepts of the graph y = p(x). The graph crosses the axis at simple zeroes and touches the axis at repeated zeroes.
- The Factor Theorem states that (x − a) is a factor of p(x) if and only if p(a) = 0. This theorem connects zeroes with factors directly.
- The Remainder Theorem states that when p(x) is divided by (x − a), the remainder is p(a). If p(a) = 0, the remainder is zero, confirming (x − a) is a factor.
- Every cubic polynomial has at least one real zero, since it must cross the x-axis. This is because odd-degree polynomials change sign from −∞ to +∞.
- The zero polynomial p(x) = 0 has every real number as its zero. It is the only polynomial with infinitely many zeroes.
Practice Problems
- Find the zero of p(x) = 7x − 21.
- Find the zeroes of p(x) = x² − 9 by factorisation.
- Find the zeroes of p(x) = 6x² − x − 2 and verify the relationship between zeroes and coefficients.
- Verify whether x = 2 is a zero of p(x) = x³ − 3x² + 4.
- Find all zeroes of p(x) = x³ − 6x² + 11x − 6.
- Find a quadratic polynomial whose zeroes are −5 and 7.
- Show that p(x) = x² + x + 1 has no real zeroes.
- If one zero of p(x) = 3x² − 8x + k is 2, find the value of k and the other zero.
Frequently Asked Questions
Q1. What is a zero of a polynomial?
A zero of a polynomial p(x) is a value a such that p(a) = 0. It is the value of x that makes the polynomial equal to zero. For example, x = 3 is a zero of p(x) = x − 3 because p(3) = 0.
Q2. How many zeroes can a polynomial of degree n have?
A polynomial of degree n can have at most n real zeroes. It may have fewer real zeroes (some zeroes could be complex numbers), but never more than n.
Q3. What is the difference between a zero and a root?
The terms zero and root are used interchangeably. A zero of the polynomial p(x) is the same as a root of the equation p(x) = 0.
Q4. How do you find the zero of a linear polynomial?
For a linear polynomial p(x) = ax + b, set p(x) = 0 and solve: ax + b = 0, so x = −b/a. This is the only zero.
Q5. Can a polynomial have no zeroes?
A non-zero constant polynomial (like p(x) = 5) has no zeroes. A quadratic or higher-degree polynomial may have no real zeroes if its discriminant is negative (e.g., p(x) = x² + 1), but it will have complex zeroes.
Q6. What does it mean if a zero has multiplicity 2?
A zero with multiplicity 2 means the factor (x − a) appears twice in the factorisation, i.e., (x − a)² divides p(x). Graphically, the curve touches the x-axis at x = a but does not cross it.
Q7. How are zeroes of a polynomial related to its graph?
The zeroes of p(x) are the x-coordinates where the graph of y = p(x) intersects or touches the x-axis. Each x-intercept corresponds to a zero of the polynomial.
Q8. What is the relationship between zeroes and coefficients of a quadratic polynomial?
For p(x) = ax² + bx + c with zeroes α and β: sum of zeroes (α + β) = −b/a, and product of zeroes (αβ) = c/a. These relationships are derived from Vieta's formulas.
Q9. Does the zero polynomial have zeroes?
Yes. The zero polynomial p(x) = 0 is zero for every value of x. So every real number is a zero of the zero polynomial. Its degree is undefined.
Q10. Is finding zeroes of polynomials part of the CBSE Class 9 syllabus?
Yes. Zeroes of polynomials are covered in Chapter 2 (Polynomials) of the CBSE Class 9 Mathematics textbook. Students learn to find zeroes by substitution and factorisation, and verify using the Factor Theorem.
Related Topics
- Value of a Polynomial
- Remainder Theorem
- Factor Theorem
- Polynomials in One Variable
- Degree of a Polynomial
- Types of Polynomials
- Factorisation of Polynomials
- Algebraic Identities (Extended)
- (a + b)³ and (a - b)³ Identities
- a³ + b³ and a³ - b³ Identities
- Zeroes of Quadratic Polynomial
- Relationship Between Zeroes and Coefficients
- Sum and Product of Zeroes
- Graph of Quadratic Polynomial
