Algebra 1 Unit 2: Polynomial Operations

Essential Question for the Unit:

"How do operations on polynomials compare to arithmetic operations?"

Unit Pacing at a Glance

  • Day 1: Adding and subtracting polynomials by combining like terms.
  • Day 2: Multiplying polynomials using the distributive property and area models.
  • Day 3: Applying polynomial operations using pictorial and symbolic models (area, perimeter).
  • Day 4: Solving mixed multi-step problems involving all polynomial operations.

1. Adding & Subtracting Polynomials

The key is to identify and combine like terms (same variable, same exponent).

Addition

Combine the coefficients of like terms.

\((3x^2 + 5x) + (2x^2 - 2x)\)

\(= (3x^2 + 2x^2) + (5x - 2x)\)

\(= 5x^2 + 3x\)

Subtraction

Distribute the negative to the second polynomial, then combine like terms.

\((8x^2 - 2x) - (5x^2 + 3x)\)

\(= 8x^2 - 2x - 5x^2 - 3x\)

\(= 3x^2 - 5x\)

2. Multiplying Polynomials

To find the product, every term in the first polynomial must be multiplied by every term in the second. Use one of these methods to stay organized.

Example: Find the product of \((x + 4)(2x - 3)\)

Method 1: Distributive Property

Distribute each term from the first polynomial to the second.

\(x(2x - 3) + 4(2x - 3)\)

\(= 2x^2 - 3x + 8x - 12\)

Final Answer: \(2x^2 + 5x - 12\)

Method 2: Area Model

Draw a box, multiply to fill it in, then combine like terms on the diagonal.

\(2x\)\(-3\)
\(x+4\)\(x\)\(2x^2\)\(-3x\)
\(+4\)\(+8x\)\(-12\)

Final Answer: \(2x^2 + 5x - 12\)

3. Applications in Geometry

Polynomials are often used to represent dimensions of geometric shapes. You can then perform operations to find perimeter, area, or volume.

Worked Example: Area of a Rectangle

Find the area of a rectangle with length \( (2x+1) \) and width \( (x+3) \).

  • Step 1: Set up the problem. Area = length × width. So, we need to calculate \((2x+1)(x+3)\).
  • Step 2: Multiply the polynomials. Using the distributive property:

    \(2x(x+3) + 1(x+3)\)

    \(= 2x^2 + 6x + x + 3\)

  • Step 3: Combine like terms.

    \(= 2x^2 + 7x + 3\)

  • Final Answer: The area is \(2x^2 + 7x + 3\).

Key Vocabulary

Polynomial An expression with one or more terms, where variables have non-negative integer exponents.
Like Terms Terms with the exact same variable part (e.g., \(3x^2\) and \(-5x^2\)).
Monomial A polynomial with exactly one term (e.g., \(7x\)).
Binomial A polynomial with exactly two terms (e.g., \(x + 5\)).
Trinomial A polynomial with exactly three terms (e.g., \(x^2 + 2x + 1\)).