Polynomials

Introduction to Polynomials

Polynomials find wide application in algebra and calculus to express relations and solve equations for different areas such as physics, economics, and engineering. A polynomial can have multiple terms or one term and is categorized according to the number of terms and their order.

Table of Contents:

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

 

Definition:

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

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

1. Based on Number of Terms:

• Monomial – 1 term → Example: 6x

• Binomial – 2 terms → Example: x + 4

• Trinomial – 3 terms → Example: x² + 2x + 3

• Polynomial – More than 3 terms → Example: x³ + x² + x + 1

 

2. Based on Degree:

• Zero Polynomial → 0 (no degree defined)

• Constant Polynomial → degree 0 (example, 5)

• Linear Polynomial → degree 1 (example., 2x + 3)

• Quadratic Polynomial → degree 2 (example., x² - 5x + 6)

• Cubic Polynomial → degree 3 (example., x³ + 2x² - 1)

• Quartic Polynomial → degree 4 (example, x⁴ - 2x² + 1)

 

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

1. Addition and Subtraction:

• Combine like terms

• Arrange in descending powers

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

Multiplication:

• Use the distributive property

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

Division:

• Polynomial ÷ monomial: divide each term individually

• Polynomial ÷ polynomial: use long division or synthetic division

Example:
(4x² + 8x) ÷ 4x = x + 2

 

2. 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

 

3. Factorization of Polynomials

Methods:

• Taking out the common factor

• Grouping

• Using identities

• Splitting the middle term (quadratics)

Examples:

1. x² + 5x + 6 → (x + 2)(x + 3)

2. 2x² + 7x + 3 → (2x + 1)(x + 3)

 

4. 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

 

5. 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

 

6. 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

1. Commutative Property:

• a + b = b + a

• ab = ba

2. Associative Property:

• (a + b) + c = a + (b + c)

• (ab)c = a(bc)

3. Distributive Property:

• a(b + c) = ab + ac

4. Identity Element:

• Additive identity = 0

• Multiplicative identity = 1

5. Inverse:

• Additive inverse of a = -a

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

 

Solved Example:

1. 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)

 

2. 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

 

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

 

4. Multiply Polynomials

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

 

5. 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^

 

6. 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)

 

7. 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

 

8. 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

 

9. 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

 

10. 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 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.

 

Related Links: 

Rational Numbers: Mathematics is dependent on rational numbers since they provide a basis for arithmetic, algebra, and also real applications.

Number System: A number system is a mathematical scheme for representing and expressing numbers, learn more about it!!

 

FAQs:

1. What is a Polynomial?

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?

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?

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. Which is Called Polynomial?

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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