MathIsimple
Lesson 1.2: Polynomial Addition, Subtraction & Multiplication

Master Polynomial Operations & Properties

Explore the fundamental operations of polynomials! Learn to add, subtract, and multiply polynomials using systematic methods, understand polynomial properties, and apply these skills to solve geometric and real-world problems.

Learning Objectives

Define and classify polynomials
Perform polynomial addition and subtraction
Master polynomial multiplication using FOIL
Apply polynomial operations to geometry

Core Concept 1: Polynomial Definitions & Classification

Understanding Polynomials

A polynomial is a sum of monomials, where each monomial is a product of a coefficient and variables raised to non-negative integer powers.

Example: 2x³ - x² + 7x - 4

Term 12x³(cubic term, degree 3)
Term 2-x²(quadratic term, degree 2)
Term 37x(linear term, degree 1)
Term 4-4(constant term, degree 0)
Degree: 3 (highest power)• Terms: 4• Classification: Cubic polynomial

Monomial

Single term: 3x²

One term polynomial

Binomial

Two terms: x + 3

Two term polynomial

Trinomial

Three terms: x² + 2x + 1

Three term polynomial

Core Concept 2: Polynomial Addition & Subtraction

Combining Like Terms

To add or subtract polynomials, combine like terms by adding or subtracting their coefficients.

Example: (3x² + 2x) + (x² - 5x)

Step 1:Remove parentheses: 3x² + 2x + x² - 5x
Step 2:Group like terms: (3x² + x²) + (2x - 5x)
Step 3:Combine coefficients: 4x² - 3x

Subtraction Example: (2x³ - 3x + 1) - (x³ + 2x - 4)

Step 1:Distribute negative: 2x³ - 3x + 1 - x³ - 2x + 4
Step 2:Group like terms: (2x³ - x³) + (-3x - 2x) + (1 + 4)
Step 3:Combine: x³ - 5x + 5

Like Terms

Same variables with same exponents

3x² and 5x² are like terms

Unlike Terms

Different variables or exponents

3x² and 5x are unlike terms

Core Concept 3: Polynomial Multiplication & FOIL Method

Distributive Property for Multiplication

Multiply each term in the first polynomial by each term in the second polynomial, then combine like terms.

FOIL Method: (x+3)(x−2)(x+3)(x-2)

F - First terms:

x⋅x=x2x\cdot x=x^2

O - Outer terms:

x⋅(−2)=−2xx\cdot(-2)=-2x

I - Inner terms:

3⋅x=3x3\cdot x=3x

L - Last terms:

3⋅(−2)=−63\cdot(-2)=-6

Result: x2−2x+3x−6=x2+x−6x^2-2x+3x-6=x^2+x-6

General Multiplication: (2x + 1)(x² - 3x + 2)

Step 1:2x(x2−3x+2)+1⋅(x2−3x+2)2x(x^2-3x+2)+1\cdot(x^2-3x+2)
Step 2:2x3−6x2+4x+x2−3x+22x^3-6x^2+4x+x^2-3x+2
Step 3:2x3−5x2+x+22x^3-5x^2+x+2

FOIL Method

For binomial × binomial

First, Outer, Inner, Last

Distributive Property

For any polynomial multiplication

Multiply each term by each term

Core Concept 4: Polynomial Properties & Closure

Closure Properties

Polynomials are closed under addition, subtraction, and multiplication. This means the result of these operations on polynomials is always another polynomial.

Addition Closure

(x² + 3x) + (2x - 1) = x² + 5x - 1

Result is a polynomial

Subtraction Closure

(x² + 3x) - (2x - 1) = x² + x + 1

Result is a polynomial

Multiplication Closure

(x + 2)(x - 3) = x² - x - 6

Result is a polynomial

Degree Properties

Addition/Subtraction:deg(P + Q) ≤ max(deg(P), deg(Q))
Multiplication:deg(P × Q) = deg(P) + deg(Q)

Real-World Application: Geometric Area Calculations

Problem Scenario

A rectangular garden has length (x + 4) meters and width (2x - 1) meters. Find the area of the garden and express it as a polynomial in standard form.

Solution Steps:

Area =Length × Width
Area =(x+4)(2x−1)(x+4)(2x-1)
Using FOIL:x(2x) + x(-1) + 4(2x) + 4(-1)
Area =2x² - x + 8x - 4
Area =2x2+7x−42x^2+7x-4(combine like terms)

Key Insights:

  • • The area is represented by a quadratic polynomial
  • • The coefficient of x² (2) represents the rate of area increase
  • • The constant term (-4) represents the area when x = 0
  • • This polynomial can be used to find area for any value of x

Practice Problems

Problem 1:

Add: (3x² - 2x + 5) + (x² + 4x - 3)

Show Solution

= 3x² - 2x + 5 + x² + 4x - 3

= (3x² + x²) + (-2x + 4x) + (5 - 3)

= 4x² + 2x + 2

Problem 2:

Multiply: (2x + 3)(x - 4)

Show Solution

Using FOIL:

F: 2x × x = 2x²

O: 2x × (-4) = -8x

I: 3 × x = 3x

L: 3 × (-4) = -12

= 2x² - 8x + 3x - 12 = 2x² - 5x - 12

Problem 3:

A square has side length (x + 2). Find its area and perimeter.

Show Solution

Area = (x + 2)² = x² + 4x + 4

Perimeter = 4(x + 2) = 4x + 8

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