Multiplying Polynomials Worksheet, Steps, And Examples - Free Printable

Let's solve each of the polynomial multiplication problems step by step using the distributive property (also known as the FOIL method for binomials). We will multiply each term in the first polynomial by each term in the second polynomial, then combine like terms.

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1) (x + 4)(x + 5)

Use FOIL:
- F: $ x \cdot x = x^2 $
- O: $ x \cdot 5 = 5x $
- I: $ 4 \cdot x = 4x $
- L: $ 4 \cdot 5 = 20 $

Now add:
$ x^2 + 5x + 4x + 20 = x^2 + 9x + 20 $

✔ Answer: $ x^2 + 9x + 20 $

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2) (x + 9)(x - 2)

FOIL:
- F: $ x \cdot x = x^2 $
- O: $ x \cdot (-2) = -2x $
- I: $ 9 \cdot x = 9x $
- L: $ 9 \cdot (-2) = -18 $

Add:
$ x^2 - 2x + 9x - 18 = x^2 + 7x - 18 $

✔ Answer: $ x^2 + 7x - 18 $

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3) (2x - 8)(x + 6)

Distribute each term:
- $ 2x \cdot x = 2x^2 $
- $ 2x \cdot 6 = 12x $
- $ -8 \cdot x = -8x $
- $ -8 \cdot 6 = -48 $

Add:
$ 2x^2 + 12x - 8x - 48 = 2x^2 + 4x - 48 $

✔ Answer: $ 2x^2 + 4x - 48 $

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4) (2x + 3)(4x - 5)

FOIL:
- F: $ 2x \cdot 4x = 8x^2 $
- O: $ 2x \cdot (-5) = -10x $
- I: $ 3 \cdot 4x = 12x $
- L: $ 3 \cdot (-5) = -15 $

Add:
$ 8x^2 - 10x + 12x - 15 = 8x^2 + 2x - 15 $

✔ Answer: $ 8x^2 + 2x - 15 $

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5) (5x - 6)(3x + 9)

FOIL:
- F: $ 5x \cdot 3x = 15x^2 $
- O: $ 5x \cdot 9 = 45x $
- I: $ -6 \cdot 3x = -18x $
- L: $ -6 \cdot 9 = -54 $

Add:
$ 15x^2 + 45x - 18x - 54 = 15x^2 + 27x - 54 $

✔ Answer: $ 15x^2 + 27x - 54 $

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6) (8x - 1)(2x - 3)

FOIL:
- F: $ 8x \cdot 2x = 16x^2 $
- O: $ 8x \cdot (-3) = -24x $
- I: $ -1 \cdot 2x = -2x $
- L: $ -1 \cdot (-3) = 3 $

Add:
$ 16x^2 - 24x - 2x + 3 = 16x^2 - 26x + 3 $

✔ Answer: $ 16x^2 - 26x + 3 $

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7) (3x + 2)(x² - 2x + 1)

Distribute each term in the first polynomial to every term in the second:

First, distribute $ 3x $:
- $ 3x \cdot x^2 = 3x^3 $
- $ 3x \cdot (-2x) = -6x^2 $
- $ 3x \cdot 1 = 3x $

Then, distribute $ 2 $:
- $ 2 \cdot x^2 = 2x^2 $
- $ 2 \cdot (-2x) = -4x $
- $ 2 \cdot 1 = 2 $

Now combine all terms:
$ 3x^3 - 6x^2 + 3x + 2x^2 - 4x + 2 $

Combine like terms:
- $ 3x^3 $
- $ (-6x^2 + 2x^2) = -4x^2 $
- $ (3x - 4x) = -x $
- $ +2 $

✔ Answer: $ 3x^3 - 4x^2 - x + 2 $

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8) (2x² - x)(3x - 1)

Distribute each term:

First, distribute $ 2x^2 $:
- $ 2x^2 \cdot 3x = 6x^3 $
- $ 2x^2 \cdot (-1) = -2x^2 $

Then, distribute $ -x $:
- $ -x \cdot 3x = -3x^2 $
- $ -x \cdot (-1) = x $

Now combine:
$ 6x^3 - 2x^2 - 3x^2 + x $

Combine like terms:
- $ 6x^3 $
- $ (-2x^2 - 3x^2) = -5x^2 $
- $ +x $

✔ Answer: $ 6x^3 - 5x^2 + x $

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✔ Final Answers:

1) $ x^2 + 9x + 20 $
2) $ x^2 + 7x - 18 $
3) $ 2x^2 + 4x - 48 $
4) $ 8x^2 + 2x - 15 $
5) $ 15x^2 + 27x - 54 $
6) $ 16x^2 - 26x + 3 $
7) $ 3x^3 - 4x^2 - x + 2 $
8) $ 6x^3 - 5x^2 + x $

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