Multiplication of Polynomials | CK-12 Foundation - Free Printable

It looks like you're working on multiplying binomials using the Multiplication Mat method, which is a visual algebraic tool often used to represent polynomial multiplication (like FOIL) using area models. The example shows how $(x + 2)(x - 3)$ is multiplied by arranging tiles on a grid and then simplifying.

Let’s go through each problem step-by-step, explaining the thinking process as if we were using the Multiplication Mat (area model), and then simplify algebraically.

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How the Multiplication Mat Works:

- The mat is divided into regions representing the product of two binomials.
- Each term in the first binomial is placed along one axis, and the second along the other.
- We fill in the grid with tiles representing the products of terms.
- Like terms are combined (e.g., positive and negative tiles cancel out).
- The final expression is the sum of all remaining tiles.

We’ll use the FOIL method (First, Outer, Inner, Last) or distributive property for clarity.

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Now let's solve each problem:

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1. $ (y + 4)(y + 1) $

Step-by-step:

- First: $ y \cdot y = y^2 $
- Outer: $ y \cdot 1 = y $
- Inner: $ 4 \cdot y = 4y $
- Last: $ 4 \cdot 1 = 4 $

Add them:
$ y^2 + y + 4y + 4 = y^2 + 5y + 4 $

✔ Answer: $ y^2 + 5y + 4 $

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2. $ (2x + 5)(x - 1) $

- First: $ 2x \cdot x = 2x^2 $
- Outer: $ 2x \cdot (-1) = -2x $
- Inner: $ 5 \cdot x = 5x $
- Last: $ 5 \cdot (-1) = -5 $

Add:
$ 2x^2 - 2x + 5x - 5 = 2x^2 + 3x - 5 $

✔ Answer: $ 2x^2 + 3x - 5 $

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3. $ (x - 3)(y - 2) $

This is a binomial times another binomial with different variables.

- First: $ x \cdot y = xy $
- Outer: $ x \cdot (-2) = -2x $
- Inner: $ -3 \cdot y = -3y $
- Last: $ -3 \cdot (-2) = 6 $

Add:
$ xy - 2x - 3y + 6 $

✔ Answer: $ xy - 2x - 3y + 6 $

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4. $ (y + 3)(y - 1) $

- First: $ y \cdot y = y^2 $
- Outer: $ y \cdot (-1) = -y $
- Inner: $ 3 \cdot y = 3y $
- Last: $ 3 \cdot (-1) = -3 $

Add:
$ y^2 - y + 3y - 3 = y^2 + 2y - 3 $

✔ Answer: $ y^2 + 2y - 3 $

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5. $ (2x - 1)(3x + 1) $

- First: $ 2x \cdot 3x = 6x^2 $
- Outer: $ 2x \cdot 1 = 2x $
- Inner: $ -1 \cdot 3x = -3x $
- Last: $ -1 \cdot 1 = -1 $

Add:
$ 6x^2 + 2x - 3x - 1 = 6x^2 - x - 1 $

✔ Answer: $ 6x^2 - x - 1 $

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6. $ (x - 1)(x - 1) $

This is $ (x - 1)^2 $

- First: $ x \cdot x = x^2 $
- Outer: $ x \cdot (-1) = -x $
- Inner: $ -1 \cdot x = -x $
- Last: $ -1 \cdot (-1) = 1 $

Add:
$ x^2 - x - x + 1 = x^2 - 2x + 1 $

✔ Answer: $ x^2 - 2x + 1 $

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7. $ (y + 1)(y + 1) $

This is $ (y + 1)^2 $

- First: $ y \cdot y = y^2 $
- Outer: $ y \cdot 1 = y $
- Inner: $ 1 \cdot y = y $
- Last: $ 1 \cdot 1 = 1 $

Add:
$ y^2 + y + y + 1 = y^2 + 2y + 1 $

✔ Answer: $ y^2 + 2y + 1 $

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8. $ (x + 1)(x + 3) $

- First: $ x \cdot x = x^2 $
- Outer: $ x \cdot 3 = 3x $
- Inner: $ 1 \cdot x = x $
- Last: $ 1 \cdot 3 = 3 $

Add:
$ x^2 + 3x + x + 3 = x^2 + 4x + 3 $

✔ Answer: $ x^2 + 4x + 3 $

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9. $ (3 + x)(2 - x) $

Rewriting: $ (x + 3)(-x + 2) $ — same thing

- First: $ x \cdot (-x) = -x^2 $
- Outer: $ x \cdot 2 = 2x $
- Inner: $ 3 \cdot (-x) = -3x $
- Last: $ 3 \cdot 2 = 6 $

Add:
$ -x^2 + 2x - 3x + 6 = -x^2 - x + 6 $

✔ Answer: $ -x^2 - x + 6 $

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10. $ (x - 2)(x - 5) $

- First: $ x \cdot x = x^2 $
- Outer: $ x \cdot (-5) = -5x $
- Inner: $ -2 \cdot x = -2x $
- Last: $ -2 \cdot (-5) = 10 $

Add:
$ x^2 - 5x - 2x + 10 = x^2 - 7x + 10 $

✔ Answer: $ x^2 - 7x + 10 $

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✔ Final Answers:

1. $ y^2 + 5y + 4 $
2. $ 2x^2 + 3x - 5 $
3. $ xy - 2x - 3y + 6 $
4. $ y^2 + 2y - 3 $
5. $ 6x^2 - x - 1 $
6. $ x^2 - 2x + 1 $
7. $ y^2 + 2y + 1 $
8. $ x^2 + 4x + 3 $
9. $ -x^2 - x + 6 $
10. $ x^2 - 7x + 10 $

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📌 Explanation of Thinking:

When using the Multiplication Mat, you:
- Place the terms of each binomial on the sides of a rectangle.
- Fill in each box with the product of the corresponding terms.
- Combine like terms (especially important when there are both positive and negative tiles).
- Remove zero pairs (like +1 and -1 tiles canceling each other).

For example, in $(x + 2)(x - 3)$, you would have:
- $ x \cdot x = x^2 $
- $ x \cdot (-3) = -3x $
- $ 2 \cdot x = 2x $
- $ 2 \cdot (-3) = -6 $

Then combine: $ x^2 -3x + 2x -6 = x^2 -x -6 $, matching the example.

This visual method helps students understand why the distributive property works and how terms combine.

Let me know if you'd like a visual breakdown of any specific problem!