Multiplication of Polynomials | CK-12 Foundation - Free Printable

You’ve uploaded a list of 10 algebraic expressions that all involve multiplying two binomials. The task is to expand each expression (i.e., multiply the binomials and simplify).

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

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

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

Combine:
\(x^2 - 3x + 2x - 6 = x^2 - x - 6\)

✔ Answer: \(x^2 - x - 6\)

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2. \((2y - 1)(y - 2)\)

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

Combine:
\(2y^2 - 4y - y + 2 = 2y^2 - 5y + 2\)

✔ Answer: \(2y^2 - 5y + 2\)

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3. \((x - 1)(x - 1)\) → This is a square!

Use \((a - b)^2 = a^2 - 2ab + b^2\)

Here, \(a = x\), \(b = 1\)

So: \(x^2 - 2x + 1\)

✔ Answer: \(x^2 - 2x + 1\)

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4. \((x - 2)(x + 2)\) → Difference of squares!

Use \((a - b)(a + b) = a^2 - b^2\)

Here, \(a = x\), \(b = 2\)

So: \(x^2 - 4\)

✔ Answer: \(x^2 - 4\)

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5. \((2x - 5)(x - 1)\)

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

Combine:
\(2x^2 - 2x - 5x + 5 = 2x^2 - 7x + 5\)

✔ Answer: \(2x^2 - 7x + 5\)

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6. \((3 - y)(2 - y)\)

Let’s rearrange to make it easier: \((-y + 3)(-y + 2)\)

Or just multiply as is:

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

Combine:
\(6 - 3y - 2y + y^2 = y^2 - 5y + 6\)

✔ Answer: \(y^2 - 5y + 6\)

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7. \((x - 2)(x - 3)\)

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

Combine:
\(x^2 - 3x - 2x + 6 = x^2 - 5x + 6\)

✔ Answer: \(x^2 - 5x + 6\)

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8. \((2x - 3)(3x - 1)\)

- First: \(2x \cdot 3x = 6x^2\)
- Outer: \(2x \cdot (-1) = -2x\)
- Inner: \(-3 \cdot 3x = -9x\)
- Last: \(-3 \cdot (-1) = 3\)

Combine:
\(6x^2 - 2x - 9x + 3 = 6x^2 - 11x + 3\)

✔ Answer: \(6x^2 - 11x + 3\)

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9. \((y - 2x)(y - x)\)

- First: \(y \cdot y = y^2\)
- Outer: \(y \cdot (-x) = -xy\)
- Inner: \(-2x \cdot y = -2xy\)
- Last: \(-2x \cdot (-x) = 2x^2\)

Combine like terms:
\(y^2 - xy - 2xy + 2x^2 = y^2 - 3xy + 2x^2\)

✔ Answer: \(2x^2 - 3xy + y^2\)
*(Note: Usually written with descending powers of x)*

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10. \((x - 1)(4 - x)\)

Rewrite second factor as \((-x + 4)\) to make FOIL easier:

\((x - 1)(-x + 4)\)

- First: \(x \cdot (-x) = -x^2\)
- Outer: \(x \cdot 4 = 4x\)
- Inner: \(-1 \cdot (-x) = x\)
- Last: \(-1 \cdot 4 = -4\)

Combine:
\(-x^2 + 4x + x - 4 = -x^2 + 5x - 4\)

✔ Answer: \(-x^2 + 5x - 4\)

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## ✔ Final Answers:

1. \(x^2 - x - 6\)
2. \(2y^2 - 5y + 2\)
3. \(x^2 - 2x + 1\)
4. \(x^2 - 4\)
5. \(2x^2 - 7x + 5\)
6. \(y^2 - 5y + 6\)
7. \(x^2 - 5x + 6\)
8. \(6x^2 - 11x + 3\)
9. \(2x^2 - 3xy + y^2\)
10. \(-x^2 + 5x - 4\)

Let me know if you’d like to see any step-by-step diagrams or explanations for specific ones!