Unit 2 Resources & Lessons
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Daily Lesson Slides
Day 0: Polynomials Basics
Day 1: Adding & Subtracting Polynomials
Day 2: Multiplying Polynomials
Day 3: Factoring Polynomials
Day 4: Dividing Polynomials
Polynomial Basics: Practice Worksheet
Polynomial Operations: Practice Worksheet
Intro to Factoring: Practice Worksheet
Day 1: Adding & Subtracting Polynomials
Objective: To determine sums and differences of polynomial expressions in one variable.
Vocabulary
A is an expression of one or more terms with variables raised to non-negative integer powers.
are terms that have the same variable raised to the same power.
A is a polynomial with one term. A has two terms.
Concept
Adding and subtracting polynomials is all about combining like terms. Think of it like sorting objects: you can add apples to apples, but not apples to oranges. To subtract, you add the opposite (distribute the negative).
I Do: Examples
Example 1 (Addition): Find the sum of \((3x^2 + 5x - 4) + (2x^2 - 7x + 1)\).
Step 1: Group like terms.
\((3x^2 + 2x^2) + (5x - 7x) + (-4 + 1)\)
Step 2: Combine coefficients.
\(5x^2 - 2x - 3\)
Example 2 (Subtraction): Find the difference of \((8x^2 - 2x + 9) - (5x^2 + 3x - 1)\).
Step 1: Distribute the negative to the second polynomial.
\(8x^2 - 2x + 9 - 5x^2 - 3x + 1\)
Step 2: Group like terms and combine.
\((8x^2 - 5x^2) + (-2x - 3x) + (9 + 1)\)
\(3x^2 - 5x + 10\)
We Do: Let's Try Together
You Do: Practice On Your Own
Practice
We Do 1: \((4a^2 + 2a - 1) + (a^2 - 5a + 9)\) =
We Do 2: \((7y + 4) - (2y + 2)\) =
You Do 1: Simplify: \((6x^2 - x + 3) + (3x^2 + 4x - 2)\) =
You Do 2: Simplify: \((12c^2 + 5c - 1) - (8c^2 - 2c - 6)\) =
Day 2: Multiplying Polynomials
Objective: To determine the product of polynomial expressions using the distributive property and area models.
Concept
To multiply polynomials, you must multiply every term in the first polynomial by every term in the second. We can organize this using two main strategies.
Distributive Property (FOIL for Binomials)
Distribute each term from the first polynomial to the second one. Then combine like terms.
Area Model (Box Method)
Draw a grid, write the terms of each polynomial on the sides, multiply to fill the boxes, and combine like terms.
I Do: Examples
Example: Find the product of \((x + 4)(2x - 3)\).
Method 1: Distributive Property
\(x(2x - 3) + 4(2x - 3)\)
\(2x^2 - 3x + 8x - 12\)
\(2x^2 + 5x - 12\)
Method 2: Area Model
| 2x | -3 | ||
| x + 4 | x | \(2x^2\) | \(-3x\) |
| +4 | \(+8x\) | \(-12\) | |
Combine like terms: \(2x^2 - 3x + 8x - 12\)
\(2x^2 + 5x - 12\)
We Do: Let's Try Together
You Do: Practice On Your Own
Practice
We Do 1: Use the area model for \((x + 2)(x + 5)\). Product:
We Do 2: Use distribution for \((3a - 1)(a + 4)\). Product:
You Do 1: Multiply: \((4x + 2)(3x + 5)\) =
You Do 2: Multiply: \((x - 6)(x^2 + 2x + 1)\) =
Day 3: Practice with Models
Objective: To reinforce polynomial operations using pictorial and symbolic models.
Concept: Algebra Tiles
We can represent polynomials visually. This helps confirm our symbolic work. We use different shapes for each type of term. A positive and negative of the same type form a "zero pair" and cancel out.
\(x^2\)
\(x\)
1
\(-x^2\)
\(-x\)
-1
I Do: Modeling Operations
Example 1 (Addition): Model \((2x + 3) + (x + 1)\) by combining groups of tiles.
+
=
Result: \(3x + 4\)
Example 2 (Subtraction): Model \((3x + 4) - (x + 1)\). This is like \((3x+4) + (-x-1)\).
We combine the first group with the opposite of the second, then form zero pairs.
+
→
Result: \(2x + 3\)
We Do: Practice with Models
1. The perimeter of a triangle is \(7x-10\). If two sides are \(x+5\) and \(3x-2\), what is the third side?
2. Find the area of the composite shape shown below, which is made of two rectangles.
You Do: Practice on Your Own
1. Find the area of a rectangle with length \(2x+1\) and width \(x+3\). Use any method.
2. Find the volume of a rectangular prism with length \(x\), width \(x+2\), and height \(x+5\).
Practice
We Do 1: The perimeter of a triangle is \(7x-10\). If two sides are \(x+5\) and \(3x-2\), what is the third side?
We Do 2: Find the area of the composite shape shown in the notes.
You Do 1: Find the area of a rectangle with length \(2x+1\) and width \(x+3\).
You Do 2: Find the volume of a rectangular prism with length \(x\), width \(x+2\), and height \(x+5\).
Day 4: Mixed Operations Review
Objective: To apply addition, subtraction, and multiplication skills to solve a variety of polynomial problems.
I Do: Multi-Step Problems
Problem 1: A square has a side length of \(3x+2\). A smaller square with side length \(x\) is removed. What is the area of the remaining shape?
Step 1: Find the area of the large square: \(A_{large} = (3x+2)(3x+2) = 9x^2 + 12x + 4\)
Step 2: Find the area of the small square: \(A_{small} = (x)(x) = x^2\)
Step 3: Subtract the small area from the large area: \((9x^2 + 12x + 4) - (x^2)\)
Remaining Area = \(8x^2 + 12x + 4\)
Problem 2: Subtract the product of \((x-1)\) and \((x+4)\) from \((5x^2 - 2x + 10)\).
Step 1: Find the product: \((x-1)(x+4) = x^2 + 4x - x - 4 = x^2 + 3x - 4\)
Step 2: Perform the subtraction: \((5x^2 - 2x + 10) - (x^2 + 3x - 4)\)
Step 3: Distribute the negative and combine like terms: \(5x^2 - 2x + 10 - x^2 - 3x + 4\)
Result = \(4x^2 - 5x + 14\)
We Do: Let's Try Together
You Do: Mixed Review Practice
Mixed Review Practice
We Do 1. ADD: \((10x^2 - 8x) + (3x^2 - x + 1)\) =
We Do 2. MULTIPLY: \((y + 8)(y - 3)\) =
You Do 1. SUBTRACT: \((m^2 - m + 5) - (4m^2 + m - 2)\) =
You Do 2. MULTIPLY: \((2k - 1)(k^2 + 3k - 4)\) =
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