Polynomials
Polynomials are algebraic expressions made up of variables and coefficients, combined using addition, subtraction, and multiplication, where the variable has only whole number exponents.
The word comes from the Greek words poly (many) and nomial (terms). In Class 9, students study the classification of polynomials, their zeroes, and standard algebraic identities.
Mastering polynomials sets the stage for understanding quadratic equations, calculus, and algebraic structures in higher classes.
What is Polynomials?
Definition: A polynomial in one variable x is an expression of the form:
p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻₁ + ... + a₂x² + a₁x + a₀
Where:
- aₙ, aₙ₋₁, ..., a₁, a₀ are real numbers called coefficients
- n is a non-negative integer called the degree (provided aₙ ≠ 0)
Conditions for an expression to be a polynomial:
- Exponents of the variable must be whole numbers (0, 1, 2, 3, ...). Expressions like x^(1/2) or x^(−1) are NOT polynomials.
- Coefficients must be real numbers (integers, fractions, or decimals).
- The variable must not appear in the denominator. So 1/x + 3 is not a polynomial.
Key terms:
- Term: Each part separated by + or − signs. In 3x² + 5x − 7, the terms are 3x², 5x, and −7.
- Leading coefficient: The coefficient of the term with the highest degree.
- Constant term: The term without any variable (a₀).
- Zero polynomial: p(x) = 0, where all coefficients are zero. Its degree is not defined.
- Constant polynomial: p(x) = 5 has degree 0.
Polynomials Formula
Key Formulas and Identities:
1. Standard Algebraic Identities:
- (a + b)² = a² + 2ab + b²
- (a − b)² = a² − 2ab + b²
- (a + b)(a − b) = a² − b²
- (a + b)³ = a³ + 3a²b + 3ab² + b³
- (a − b)³ = a³ − 3a²b + 3ab² − b³
- a³ + b³ = (a + b)(a² − ab + b²)
- a³ − b³ = (a − b)(a² + ab + b²)
2. Important identity:
a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)
3. Zero of a polynomial:
- A real number k is a zero of polynomial p(x) if p(k) = 0.
4. Maximum number of zeroes:
- A polynomial of degree n can have at most n zeroes.
5. Value of a polynomial:
- The value of p(x) at x = a is obtained by substituting a for x, denoted p(a).
Derivation and Proof
Derivation of Key Algebraic Identities:
Derivation of (a + b)² = a² + 2ab + b²:
- Write (a + b)² as (a + b)(a + b).
- Expand using the distributive property: a(a + b) + b(a + b)
- = a² + ab + ba + b²
- = a² + 2ab + b²
Derivation of (a + b)³ = a³ + 3a²b + 3ab² + b³:
- Write (a + b)³ as (a + b)(a + b)².
- Use (a + b)² = a² + 2ab + b².
- Multiply: (a + b)(a² + 2ab + b²)
- = a³ + 2a²b + ab² + a²b + 2ab² + b³
- = a³ + 3a²b + 3ab² + b³
Derivation of a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca):
- Start with the right-hand side and expand.
- Multiply each term of (a + b + c) with each term of (a² + b² + c² − ab − bc − ca).
- After expanding, all cross-terms cancel out in pairs (e.g., ab² and −ab², ac² and −ac²).
- The remaining terms are: a³ + b³ + c³ − 3abc.
Special case: If a + b + c = 0, then a³ + b³ + c³ = 3abc. This is frequently used in simplification problems.
Types and Properties
Polynomials are classified in two ways:
Classification by Number of Terms:
- Monomial: One term. Examples: 5x³, −7, 2y
- Binomial: Two terms. Examples: x + 5, 3x² − 2x
- Trinomial: Three terms. Examples: x² + 3x + 2, 2y³ − y + 7
- Polynomials with more than three terms have no special name.
Classification by Degree:
- Constant polynomial (Degree 0): p(x) = c (non-zero constant). Graph is a horizontal line.
- Linear polynomial (Degree 1): p(x) = ax + b, a ≠ 0. Graph is a straight line. Has exactly one zero: x = −b/a.
- Quadratic polynomial (Degree 2): p(x) = ax² + bx + c, a ≠ 0. Graph is a parabola. Can have 0, 1, or 2 real zeroes.
- Cubic polynomial (Degree 3): p(x) = ax³ + bx² + cx + d, a ≠ 0. Can have 1, 2, or 3 real zeroes.
- Biquadratic polynomial (Degree 4): p(x) = ax⁴ + bx³ + cx² + dx + e, a ≠ 0. Can have up to 4 real zeroes.
The degree determines the maximum number of zeroes and the general shape of the graph.
Solved Examples
Example 1: Example 1: Identifying polynomials
Problem: Which of the following are polynomials?
- (a) 4x³ − 2x + 7
- (b) x^(1/2) + 3x − 1
- (c) 5/x + 2
- (d) √2 · x² − 3x + 1
Solution:
- (a) All exponents are whole numbers (3, 1, 0). IS a polynomial of degree 3.
- (b) The term x^(1/2) has a fractional exponent. NOT a polynomial.
- (c) 5/x = 5x^(−1) has a negative exponent. NOT a polynomial.
- (d) All exponents are whole numbers. √2 is a real coefficient. IS a polynomial of degree 2.
Example 2: Example 2: Finding the degree and classifying polynomials
Problem: Find the degree and classify each polynomial by number of terms:
- (a) 7x⁵ − 3x² + 1
- (b) 4y
- (c) x² + 5x + 6
- (d) 9
Solution:
- (a) Degree = 5, three terms ⇒ trinomial
- (b) Degree = 1, one term ⇒ linear monomial
- (c) Degree = 2, three terms ⇒ quadratic trinomial
- (d) Degree = 0, one term ⇒ constant monomial
Example 3: Example 3: Finding the zero of a linear polynomial
Problem: Find the zero of the polynomial p(x) = 3x − 12.
Solution:
A zero is the value of x for which p(x) = 0.
- Set p(x) = 0: 3x − 12 = 0
- 3x = 12
- x = 4
Verification: p(4) = 3(4) − 12 = 12 − 12 = 0 ✓
Answer: The zero of p(x) = 3x − 12 is x = 4.
Example 4: Example 4: Finding the value of a polynomial
Problem: If p(x) = 2x³ − 3x² + x − 5, find p(2) and p(−1).
Solution:
Finding p(2):
- p(2) = 2(2)³ − 3(2)² + 2 − 5
- = 2(8) − 3(4) + 2 − 5
- = 16 − 12 + 2 − 5 = 1
Finding p(−1):
- p(−1) = 2(−1)³ − 3(−1)² + (−1) − 5
- = 2(−1) − 3(1) − 1 − 5
- = −2 − 3 − 1 − 5 = −11
Answer: p(2) = 1 and p(−1) = −11.
Example 5: Example 5: Verifying zeroes of a quadratic polynomial
Problem: Verify whether x = 1 and x = 2 are zeroes of p(x) = x² − 3x + 2.
Solution:
Check x = 1:
- p(1) = (1)² − 3(1) + 2 = 1 − 3 + 2 = 0 ✓
Check x = 2:
- p(2) = (2)² − 3(2) + 2 = 4 − 6 + 2 = 0 ✓
Answer: Both x = 1 and x = 2 are zeroes of p(x) = x² − 3x + 2.
Example 6: Example 6: Expanding using algebraic identities
Problem: Expand (2x + 3y)³ using the identity for (a + b)³.
Solution:
Using: (a + b)³ = a³ + 3a²b + 3ab² + b³, with a = 2x, b = 3y
- (2x + 3y)³ = (2x)³ + 3(2x)²(3y) + 3(2x)(3y)² + (3y)³
- = 8x³ + 3(4x²)(3y) + 3(2x)(9y²) + 27y³
- = 8x³ + 36x²y + 54xy² + 27y³
Answer: (2x + 3y)³ = 8x³ + 36x²y + 54xy² + 27y³
Example 7: Example 7: Factorisation using identities
Problem: Factorise 8x³ − 27 using the identity for a³ − b³.
Solution:
Given:
- 8x³ = (2x)³ and 27 = 3³
Using: a³ − b³ = (a − b)(a² + ab + b²), with a = 2x, b = 3
- 8x³ − 27 = (2x − 3)((2x)² + (2x)(3) + 3²)
- = (2x − 3)(4x² + 6x + 9)
Answer: 8x³ − 27 = (2x − 3)(4x² + 6x + 9)
Example 8: Example 8: Using the identity a³ + b³ + c³ − 3abc
Problem: If x + y + z = 0, find the value of x³ + y³ + z³.
Solution:
Using the identity:
- a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)
Since x + y + z = 0:
- The right side becomes 0.
- x³ + y³ + z³ − 3xyz = 0
- x³ + y³ + z³ = 3xyz
Answer: When x + y + z = 0, x³ + y³ + z³ = 3xyz.
Example 9: Example 9: Finding zeroes of a cubic polynomial
Problem: Verify that x = 1, x = 2, and x = 3 are zeroes of p(x) = x³ − 6x² + 11x − 6.
Solution:
- p(1): 1 − 6 + 11 − 6 = 0 ✓
- p(2): 8 − 24 + 22 − 6 = 0 ✓
- p(3): 27 − 54 + 33 − 6 = 0 ✓
Since p(x) is cubic (degree 3) with 3 zeroes:
- p(x) = (x − 1)(x − 2)(x − 3)
Verification by expanding:
- (x − 1)(x − 2) = x² − 3x + 2
- (x² − 3x + 2)(x − 3) = x³ − 6x² + 11x − 6 ✓
Example 10: Example 10: Evaluating using identities
Problem: Without actual multiplication, evaluate 104 × 96.
Solution:
Using: (a + b)(a − b) = a² − b²
- 104 = 100 + 4 and 96 = 100 − 4
- 104 × 96 = (100 + 4)(100 − 4)
- = 100² − 4²
- = 10000 − 16
- = 9984
Answer: 104 × 96 = 9984
Real-World Applications
Applications of Polynomials:
- Physics and Engineering: The equation of motion s = ut + (1/2)at² is a quadratic polynomial in time. Engineers use polynomials to design curves for roads, bridges, and roller coasters.
- Economics and Business: Revenue, cost, and profit functions are often polynomials. For example, R(x) = px is a linear polynomial.
- Computer Science: Polynomial algorithms are fundamental in computational complexity. Polynomial interpolation is used in cryptography and error-correcting codes.
- Architecture and Design: Polynomial curves (Bezier curves) are used in CAD for creating smooth shapes in cars, aircraft, and fonts.
- Medicine and Biology: Growth curves, drug concentration over time, and epidemiological models are described by polynomial functions.
- Statistics: Polynomial regression fits curved lines to data points for prediction and trend analysis.
Key Points to Remember
- A polynomial has variables with whole number exponents and real number coefficients.
- The degree is the highest power of the variable with a non-zero coefficient.
- By terms: monomial (1), binomial (2), trinomial (3).
- By degree: constant (0), linear (1), quadratic (2), cubic (3), biquadratic (4).
- A zero of p(x) is a value k such that p(k) = 0.
- A polynomial of degree n can have at most n zeroes.
- The zero polynomial p(x) = 0 has no defined degree.
- Identities like (a + b)², (a − b)², (a + b)³ are essential for expanding and factorising.
- If a + b + c = 0, then a³ + b³ + c³ = 3abc.
- Expressions with negative or fractional exponents are NOT polynomials.
Practice Problems
- Classify the following as monomials, binomials, or trinomials: (a) 5x² − 3x + 1, (b) 7y, (c) x² + 9, (d) −3.
- Find the zeroes of the polynomial p(x) = x² − 5x + 6.
- If p(x) = x³ + 3x² − x − 3, verify whether x = 1 and x = −1 are zeroes of p(x).
- Expand (3a − 2b)³ using the algebraic identity for (a − b)³.
- Factorise: x³ + 8y³ using the sum of cubes identity.
- Without actual division, find the value of 103 × 97 using a suitable algebraic identity.
- Find the value of p(3) for the polynomial p(x) = 4x³ − 5x² + 2x − 1.
- If a + b + c = 5 and a² + b² + c² = 29, find the value of ab + bc + ca.
Frequently Asked Questions
Q1. What is a polynomial in maths?
A polynomial is an algebraic expression with variables and coefficients, combined using addition, subtraction, and multiplication. The exponents must be non-negative integers. For example, 3x² + 5x − 2 is a polynomial, but 3/x + 2 is not.
Q2. What is the degree of a polynomial?
The degree is the highest power of the variable with a non-zero coefficient. In 4x³ − 2x + 7, the degree is 3. A constant like 5 has degree 0. The zero polynomial has no defined degree.
Q3. What is the difference between a monomial, binomial, and trinomial?
A monomial has one term (e.g., 5x²), a binomial has two terms (e.g., x + 3), and a trinomial has three terms (e.g., x² + 5x + 6). Classification is based on the number of terms separated by addition or subtraction.
Q4. What is the zero of a polynomial?
A zero (or root) of p(x) is a value of x for which p(x) = 0. For p(x) = x − 5, the zero is x = 5. A polynomial of degree n has at most n zeroes. Zeroes are the x-intercepts of the graph.
Q5. Is x^(1/2) + 2x a polynomial?
No. All exponents must be non-negative whole numbers. The term x^(1/2) has a fractional exponent (1/2), which disqualifies the entire expression.
Q6. What are algebraic identities used in polynomials?
The main identities in Class 9 are: (a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b²; (a + b)(a − b) = a² − b²; (a + b)³ and (a − b)³ expansions; and a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca).
Q7. How many zeroes can a quadratic polynomial have?
A quadratic (degree 2) can have at most 2 zeroes: exactly 2 distinct real zeroes, 1 repeated zero, or no real zeroes (when discriminant b² − 4ac < 0). The zeroes correspond to x-intercepts of the parabola.
Q8. What is the difference between a polynomial and an algebraic expression?
All polynomials are algebraic expressions, but not all algebraic expressions are polynomials. A polynomial requires all exponents to be non-negative whole numbers. Expressions like 3/x or √x are algebraic expressions but not polynomials.
Q9. What is the importance of polynomials in real life?
Polynomials model projectile motion in physics, revenue functions in business, growth patterns in biology, and orbital paths in astronomy. In computer graphics, polynomial curves create smooth shapes. In construction, they help calculate areas and volumes.
Q10. Can a polynomial have decimal or fraction coefficients?
Yes. Coefficients can be any real numbers, including decimals and fractions. For example, (1/2)x² + 0.3x − 7 is a valid polynomial of degree 2. The restriction applies to exponents (must be whole numbers), not coefficients.
Related Topics
- Degree of a Polynomial
- Zeroes of a Polynomial
- Value of a Polynomial
- Types of Algebraic Expressions
- Types of Polynomials
- Remainder Theorem
- Factor Theorem
- 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
