Multiplying Polynomials Practice Worksheet with problems and a rectangle area question.

Let's solve each problem step by step from the worksheet titled "Multiplying Polynomials Practice."

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1. $(y - 2)(y - 4)$

Use the distributive property (FOIL method):

$$
= y(y) + y(-4) + (-2)(y) + (-2)(-4)
= y^2 - 4y - 2y + 8
= y^2 - 6y + 8
$$

✔ Answer: $ y^2 - 6y + 8 $

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2. $(4x^2)(x - 4)(x + 5)$

First, multiply the binomials $(x - 4)(x + 5)$:

$$
(x - 4)(x + 5) = x(x) + x(5) + (-4)(x) + (-4)(5)
= x^2 + 5x - 4x - 20
= x^2 + x - 20
$$

Now multiply by $4x^2$:

$$
4x^2(x^2 + x - 20) = 4x^2 \cdot x^2 + 4x^2 \cdot x + 4x^2 \cdot (-20)
= 4x^4 + 4x^3 - 80x^2
$$

✔ Answer: $ 4x^4 + 4x^3 - 80x^2 $

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3. $3a^2(8a^2 - 7a + 10)$

Distribute $3a^2$ to each term:

$$
= 3a^2 \cdot 8a^2 + 3a^2 \cdot (-7a) + 3a^2 \cdot 10
= 24a^4 - 21a^3 + 30a^2
$$

✔ Answer: $ 24a^4 - 21a^3 + 30a^2 $

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4. $(7x + 3)(2x - 4)$

Use FOIL:

$$
= 7x(2x) + 7x(-4) + 3(2x) + 3(-4)
= 14x^2 - 28x + 6x - 12
= 14x^2 - 22x - 12
$$

✔ Answer: $ 14x^2 - 22x - 12 $

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5. $(3x + 4y)^2$

This is a square of a binomial: $(a + b)^2 = a^2 + 2ab + b^2$

$$
= (3x)^2 + 2(3x)(4y) + (4y)^2
= 9x^2 + 24xy + 16y^2
$$

✔ Answer: $ 9x^2 + 24xy + 16y^2 $

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6. $-3a^2b^2(a^2 - 2y - 2y^2)$

Distribute $-3a^2b^2$ to each term:

$$
= -3a^2b^2 \cdot a^2 + (-3a^2b^2)(-2y) + (-3a^2b^2)(-2y^2)
= -3a^4b^2 + 6a^2b^2y + 6a^2b^2y^2
$$

✔ Answer: $ -3a^4b^2 + 6a^2b^2y + 6a^2b^2y^2 $

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7. $(w^2 + yz)(2x^2 + 4w^2y + 5y^2)$

We’ll distribute each term in the first polynomial to each term in the second.

Let’s expand:

$$
= w^2(2x^2) + w^2(4w^2y) + w^2(5y^2) + yz(2x^2) + yz(4w^2y) + yz(5y^2)
$$

Now compute each:

- $w^2 \cdot 2x^2 = 2w^2x^2$
- $w^2 \cdot 4w^2y = 4w^4y$
- $w^2 \cdot 5y^2 = 5w^2y^2$
- $yz \cdot 2x^2 = 2x^2yz$
- $yz \cdot 4w^2y = 4w^2y^2z$
- $yz \cdot 5y^2 = 5y^3z$

Now combine all:

$$
2w^2x^2 + 4w^4y + 5w^2y^2 + 2x^2yz + 4w^2y^2z + 5y^3z
$$

✔ Answer: $ 2w^2x^2 + 4w^4y + 5w^2y^2 + 2x^2yz + 4w^2y^2z + 5y^3z $

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8. $(3x - 2y)(2x - 2y)$

Use FOIL:

$$
= 3x(2x) + 3x(-2y) + (-2y)(2x) + (-2y)(-2y)
= 6x^2 - 6xy - 4xy + 4y^2
= 6x^2 - 10xy + 4y^2
$$

✔ Answer: $ 6x^2 - 10xy + 4y^2 $

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9. Find the expression that represents the area of the rectangle.

Given:
- Length: $ (x + 3) $
- Width: $ (2x - 5) $

Area of a rectangle = length × width

So:

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

Use FOIL:

$$
= x(2x) + x(-5) + 3(2x) + 3(-5)
= 2x^2 - 5x + 6x - 15
= 2x^2 + x - 15
$$

✔ Answer: $ 2x^2 + x - 15 $

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $ y^2 - 6y + 8 $ |
| 2 | $ 4x^4 + 4x^3 - 80x^2 $ |
| 3 | $ 24a^4 - 21a^3 + 30a^2 $ |
| 4 | $ 14x^2 - 22x - 12 $ |
| 5 | $ 9x^2 + 24xy + 16y^2 $ |
| 6 | $ -3a^4b^2 + 6a^2b^2y + 6a^2b^2y^2 $ |
| 7 | $ 2w^2x^2 + 4w^4y + 5w^2y^2 + 2x^2yz + 4w^2y^2z + 5y^3z $ |
| 8 | $ 6x^2 - 10xy + 4y^2 $ |
| 9 | $ 2x^2 + x - 15 $ |

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