Polynomials

Polynomials are mathematical expressions that include variables, coefficients, and exponents, combined with addition, subtraction, and multiplication. They can have one term, known as a monomial, or several terms, such as binomials and trinomials. Polynomials are classified by their degree, which is the highest exponent in the expression. 

In algebra, polynomials help illustrate relationships. In calculus, they are used to solve equations, derivatives, and integrals. Polynomials also play important roles in physics for modeling motion, in economics for analyzing costs and revenues, and in engineering for designing and assessing systems. This makes them important tools for solving both theoretical and practical problems.

 

Table of Contents:

• Definition
• Types of Polynomials
• Operations on Polynomials
• Real-Life Applications of Polynomials
• Properties of Polynomials
• Solved Example
• Conclusion
• FAQs on Polynomials

 

Definition

A polynomial is an expression in algebra that consists of variables raised to non-negative integers powers and added, subtracted, and multiplied.

They are organized and classified based on their degree. This degree is determined by the highest power of the variable. Polynomials are important in mathematics. They serve as a foundation for expressing patterns, modeling relationships, and solving different problems in both pure and applied contexts.

General Form:
P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀
Where:

• aₙ, aₙ₋₁, ..., a₀ are real numbers (coefficients)

• x is a variable

• n is a non-negative integer (degree of the polynomial)

Polynomials never contain variables in the denominator, negative exponents, or roots with variables.

Examples of Polynomials:

• 3x² + 5x - 7

• x³ + 2x - 1

• -4x⁴ + 7

Non-Examples (Not Polynomials):

• x⁻² + 1 (contains a negative exponent)

• √x + 2 (contains a root of a variable)

• 1/x + 3 (variable in denominator)

Task:
Write 5 correct and 5 incorrect examples of polynomials and describe why they are or aren’t polynomials.

Correct Examples:

• 2x + 3 → Polynomial (linear)

• x² - 4x + 4 → Polynomial (quadratic)

• 7x³ → Polynomial (cubic monomial)

• -5x⁴ + 3x² → Polynomial (quartic)

• 8 → Polynomial (constant)

Incorrect Examples:

• x⁻¹ + 2 → Not a polynomial (negative exponent)

• √x + 1 → Not a polynomial (root involved)

• 1/x² + x → Not a polynomial (variable in denominator)

• x + log(x) → Not a polynomial (logarithmic term)

• sin(x) + x² → Not a polynomial (trigonometric term)

 

Types of Polynomials

There are two types of polynimials 1) Based on number of terms 2) Based on Degree

Based on Number of Terms:

• Monomial, (1 term)
  An algebraic expression with just one term, which can include a constant, a variable, or their product.  
  Example:  6x
  
  
• Binomial, (2 terms)
  An expression made up of exactly two terms linked by a plus or minus sign.  
  Example:  x+4
  
  
• Trinomial, (3 terms)
  An expression that contains three distinct terms combined using addition or subtraction.  
  Example: x² + 2x + 3
  
  
• Polynomial, (More than 3 terms) 
  Any expression with four or more terms is classified as a polynomial in general form.  
  Example:  x³ + x² + x + 1

 

Based on Degree:

• Zero Polynomial → 0 
  A polynomial where all coefficients are zero; degree is undefined.

• Constant Polynomial → degree 0 (example, 5)
  Contains only a constant term without variables.

• Linear Polynomial → degree 1 (example., 2x + 3)
  Highest power of the variable is 1.

• Quadratic Polynomial → degree 2 (example., x² - 5x + 6)
  Highest power of the variable is 2.

• Cubic Polynomial → degree 3 (example., x³ + 2x² - 1)
  Highest power of the variable is 3.

• Quartic Polynomial → degree 4 (example, x⁴ - 2x² + 1)
  Highest power of the variable is 4.

 

Task: Classify the following

• 4x³ + 2 → Cubic Polynomial (degree 3)

• -3 → Constant Polynomial (degree 0)

• 7x → Linear Polynomial (degree 1)

• x² + x + 1 → Quadratic Polynomial (degree 2)

 

Operations on Polynomials

Addition and Subtraction:

• Combine like terms – Add or subtract the coefficients of terms that have the same variable raised to the same power.

• Arrange in descending powers – After combining, write the terms in order from the highest power to the lowest for clarity.

Example:
(3x² + 2x + 1) + (x² - x + 5) = 4x² + x + 6

Multiplication:

• Use the distributive property – Multiply each term of the first polynomial by each term of the second polynomial, then combine like terms.

Example:
(2x)(3x² + x - 4) = 6x³ + 2x² - 8x

Division:

• Polynomial ÷ Monomial
  Divide each term of the polynomial separately by the monomial and simplify by subtracting exponents where applicable. This method is quick and direct for single-term divisors.

Example: (6x^3 + 9x^2) / (3x) = 2x^2 + 3x

• Polynomial ÷ Polynomial
  Use long division or synthetic division to divide when both have multiple terms. Continue the process until the remainder’s degree is less than the divisor’s degree.

Example: (x^2 + 5x + 6) / (x + 2) = x + 3

 

Algebraic Identities

Important Formulas:

• (a + b)² = a² + 2ab + b²

• (a - b)² = a² - 2ab + b²

• a² - b² = (a + b)(a - b)

• (x + a)(x + b) = x² + (a + b)x + ab

Task: Expand and simplify

1. (x + 3)² = x² + 6x + 9

2. (2x - 5)² = 4x² - 20x + 25

3. (x + 4)(x - 4) = x² - 16

 

Factorization of Polynomials

4 Methods of factoring polynomials:

• Taking out the common factor: Identify and remove the greatest common factor (GCF) from all terms of the polynomial to simplify it.
• Grouping: Organize terms into groups and factor each group separately. Then factor out the common binomial.
• Using identities: Apply standard algebraic identities for quick factoring.
• Splitting the middle term (quadratics): Divide the middle term into two parts. Their coefficients should multiply to the product of the first and last coefficients. Then factor by grouping.

Examples:

1. x² + 5x + 6 → (x + 2)(x + 3)

2. 2x² + 7x + 3 → (2x + 1)(x + 3)

 

Remainder and Factor Theorems

Remainder Theorem:
If a polynomial f(x) is divided by (x - a), then the remainder = f(a).

Factor Theorem:

If f(a) = 0, then (x - a) is a factor of f(x).

Example:
For f(x) = x² - 3x + 2:
f(1) = 0 → (x - 1) is a factor

 

Zeros (Roots) of Polynomials

A zero of a polynomial is a value of x such that the polynomial equals zero.

For quadratic polynomials ax² + bx + c, use:

1. Factor method
2. Quadratic formula

x = [-b ± √(b² - 4ac)] / 2a

Task: Find the zeros of

1. x² - 5x + 6 →
   (x - 2)(x - 3) = 0 → x = 2, 3

2. 2x² - x - 3 →
   (2x + 3)(x - 1) = 0 → x = -3/2, 1

 

Graphing Polynomials 

Graphs by Degree:

• Linear → straight line

• Quadratic → parabola (U-shaped)

• Cubic → S-curve

 

Real-Life Applications of Polynomials

Polynomials are widely used in real-world scenarios:

Applications:

• Finance: Interest formulas, budgeting

• Construction: Area, volume

• Science: Equations of motion

• Shopping: Discounts and total cost

Example:
Total cost = 5x² + 10x + 300 (x = number of items)

Task:
Create 3 real-world situations that use polynomials.

• Predicting profit

• Building design equations

• Estimating travel time

 

Properties of Polynomials

Closure:
Polynomial +, -, × Polynomial = Polynomial

Commutative Property: The order of numbers does not change the result in addition or multiplication.

• a + b = b + a

• ab = ba

Associative Property: How you group numbers does not affect the sum or product.

• (a + b) + c = a + (b + c)

• (ab)c = a(bc)

Distributive Property: You can distribute multiplication over addition or subtraction.

• a(b + c) = ab + ac

Identity Element:  Adding 0 to any number does not change it, and multiplying by 1 does not change it either.

• Additive identity = 0

• Multiplicative identity = 1

Inverse: The additive inverse of $a$ is −a, and the multiplicative inverse of $a$ (when a≠0) is 1/a

• Additive inverse of a = -a

• Multiplicative inverse of a = 1/a (a ≠ 0)

 

Solved Example

Identify and Classify the Polynomial

Q: Identify the type and degree of the polynomial:
P(x) = 3x^4 - 5x^2 + 2x - 7
Type: Polynomial with 4 terms (Quartic Polynomial)
Degree: 4 (highest power of the variable)

 

Add Two Polynomials

Q: Add:
(4x^2 + 3x - 2) + (2x^2 - x + 5)
Solution:
(4x^2 + 2x^2) + (3x - x) + (-2 + 5)
= 6x^2 + 2x + 3

 

Subtract Polynomials

Q: Subtract
(6x^2 - 4x + 1) - (3x^2 + 2x - 5)
Solution:
6x^2 - 4x + 1 - 3x^2 - 2x + 5
= (6x^2 - 3x^2) + (-4x - 2x) + (1 + 5)
= 3x^2 - 6x + 6

 

Multiply Polynomials

Q: Multiply:
(x + 3)(x - 2)
Solution:
= x^2 - 2x + 3x - 6
= x^2 + x - 6

 

Use Algebraic Identity

Q: Expand using identity:
(a + b)^2
Solution using identity (a + b)^2 = a^2 + 2ab + b^2:
= a^2 + 2ab + b^

 

Factor a Quadratic Polynomial

Q: Factor:
x^2 - 5x + 6
Solution:
(x - 2)(x - 3)
(Since -2 and -3 add to -5 and multiply to 6)

 

Apply Remainder Theorem

Q: Find the remainder when
f(x) = x^3 + 4x^2 - 3x + 5 is divided by x - 2
Solution:
By Remainder Theorem, substitute x = 2:
f(2) = 2^3 + 4(2)^2 - 3(2) + 5 = 8 + 16 - 6 + 5 = 23
Remainder = 23

 

Apply Factor Theorem

Q: Check if x - 3 is a factor of
f(x) = x^3 - 7x + 6
Solution:
Substitute x = 3:
f(3) = 27 - 21 + 6 = 12
Since remainder ≠ 0, x - 3 is NOT a factor

 

Find Zeros of a Polynomial

Q: Find the zeros of
f(x) = x^2 - 7x + 10
Solution:
Factor:
f(x) = (x - 2)(x - 5)
Zeros: x = 2 and x = 5

 

Real-Life Word Problem with Polynomial

Q: A factory's cost function is:
C(x) = 5x^2 + 20x + 5000, where x is the number of items produced.
Find cost of producing 10 items.
Solution:
C(10) = 5(10)^2 + 20(10) + 5000 = 500 + 200 + 5000 = ₹5700
Total cost = ₹5700

 

Conclusion

Polynomials are basic algebraic expressions constructed with variables, constants, and positive integer exponents. Polynomials are categorized in terms of degree and number of terms and are governed by certain rules in operations such as addition, subtraction, multiplication, and division. Understanding polynomial functions, identities, factorization techniques, and theorems such as the Remainder and Factor Theorems assists in solving equations and appreciating real-world applications in finance, science, and engineering. Polynomials provide a solid platform for advanced mathematics.

 

Frequently Asked Questions on Polynomials

1. What is a Polynomial?

Answer: A polynomial is an algebraic expression made up of variables and coefficients, connected by addition, subtraction, and multiplication. The exponents of the variables must be non-negative integers.

Example of a Polynomial:

3x^2 + 5x - 7 is a polynomial. It has terms 3x^2, 5x, and -7 with non-negative integer exponents on x.

 

2. Is √2 a Polynomial?

Answer: No, √2 is not a polynomial. A polynomial must have a variable raised to a non-negative integer exponent. √2 is a constant and does not involve any variables. Therefore, it is not considered a polynomial.

 

3. What are the 4 Types of Polynomials?

Answer: 1. Monomial: A polynomial with only one term. Example: 6x

2. Binomial: A polynomial with two terms. Example: x + 4

3. Trinomial: A polynomial with three terms. Example: x^2 + 2x + 3

4. Polynomial: A general term for any polynomial with more than three terms. Example: x^3 + x^2 + x + 1

 

4. What is a Polynomial?

Answer: A polynomial is an algebraic expression that can have one or more terms, with the variables having non-negative integer exponents. The terms can involve addition, subtraction, and multiplication, but they cannot include division by a variable, negative exponents, or roots with variables.

Examples of Polynomials:

1. 2x + 3 (linear polynomial)

2. x^2 - 4x + 4 (quadratic polynomial)

3. 7x^3 (monomial polynomial)

 

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