Worksheet on Special Products | PDF - Free Printable

Problem: Solve the special products and write complete solutions for each.

The worksheet involves solving various algebraic expressions using special product formulas. Below, I will solve each problem step by step.

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#### 1. \((2x - 2y)^2\)

Formula Used: \((a - b)^2 = a^2 - 2ab + b^2\)

Here, \(a = 2x\) and \(b = 2y\).

\[
(2x - 2y)^2 = (2x)^2 - 2(2x)(2y) + (2y)^2
\]

\[
= 4x^2 - 8xy + 4y^2
\]

Final Answer: \(\boxed{4x^2 - 8xy + 4y^2}\)

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#### 2. \((x - y - 2z)^2\)

This is a bit more complex because it involves three terms. We can use the formula for the square of a trinomial:

\[
(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca
\]

Here, \(a = x\), \(b = -y\), and \(c = -2z\).

\[
(x - y - 2z)^2 = x^2 + (-y)^2 + (-2z)^2 + 2(x)(-y) + 2(-y)(-2z) + 2(-2z)(x)
\]

\[
= x^2 + y^2 + 4z^2 - 2xy + 4yz - 4zx
\]

Final Answer: \(\boxed{x^2 + y^2 + 4z^2 - 2xy + 4yz - 4zx}\)

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#### 3. \((x^2 - z^3)^3\)

Formula Used: \((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\)

Here, \(a = x^2\) and \(b = z^3\).

\[
(x^2 - z^3)^3 = (x^2)^3 - 3(x^2)^2(z^3) + 3(x^2)(z^3)^2 - (z^3)^3
\]

\[
= x^6 - 3x^4z^3 + 3x^2z^6 - z^9
\]

Final Answer: \(\boxed{x^6 - 3x^4z^3 + 3x^2z^6 - z^9}\)

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#### 4. \((2x^4 - x)(2x^3 + x)\)

This is a difference of squares in disguise. Let's expand it directly:

\[
(2x^4 - x)(2x^3 + x) = 2x^4 \cdot 2x^3 + 2x^4 \cdot x - x \cdot 2x^3 - x \cdot x
\]

\[
= 4x^7 + 2x^5 - 2x^4 - x^2
\]

Final Answer: \(\boxed{4x^7 + 2x^5 - 2x^4 - x^2}\)

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#### 5. \((x - 3)(x^2 + 3x + 9)\)

This is a case of the sum of cubes formula in reverse:

\[
(a - b)(a^2 + ab + b^2) = a^3 - b^3
\]

Here, \(a = x\) and \(b = 3\).

\[
(x - 3)(x^2 + 3x + 9) = x^3 - 3^3
\]

\[
= x^3 - 27
\]

Final Answer: \(\boxed{x^3 - 27}\)

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#### 6. \((3x - 2y)^3\)

Formula Used: \((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\)

Here, \(a = 3x\) and \(b = 2y\).

\[
(3x - 2y)^3 = (3x)^3 - 3(3x)^2(2y) + 3(3x)(2y)^2 - (2y)^3
\]

\[
= 27x^3 - 3(9x^2)(2y) + 3(3x)(4y^2) - 8y^3
\]

\[
= 27x^3 - 54x^2y + 36xy^2 - 8y^3
\]

Final Answer: \(\boxed{27x^3 - 54x^2y + 36xy^2 - 8y^3}\)

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#### 7. \((2x - 5y)^2\)

Formula Used: \((a - b)^2 = a^2 - 2ab + b^2\)

Here, \(a = 2x\) and \(b = 5y\).

\[
(2x - 5y)^2 = (2x)^2 - 2(2x)(5y) + (5y)^2
\]

\[
= 4x^2 - 20xy + 25y^2
\]

Final Answer: \(\boxed{4x^2 - 20xy + 25y^2}\)

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#### 8. \((x - 3y + 2z)^2\)

Use the formula for the square of a trinomial:

\[
(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca
\]

Here, \(a = x\), \(b = -3y\), and \(c = 2z\).

\[
(x - 3y + 2z)^2 = x^2 + (-3y)^2 + (2z)^2 + 2(x)(-3y) + 2(-3y)(2z) + 2(2z)(x)
\]

\[
= x^2 + 9y^2 + 4z^2 - 6xy - 12yz + 4zx
\]

Final Answer: \(\boxed{x^2 + 9y^2 + 4z^2 - 6xy - 12yz + 4zx}\)

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#### 9. \((x^2 - 2)(x^2 + 2)\)

This is a difference of squares:

\[
(a - b)(a + b) = a^2 - b^2
\]

Here, \(a = x^2\) and \(b = 2\).

\[
(x^2 - 2)(x^2 + 2) = (x^2)^2 - 2^2
\]

\[
= x^4 - 4
\]

Final Answer: \(\boxed{x^4 - 4}\)

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#### 10. \((x^2 - 2y)(x^4 + 2x^2y + 4y^2)\)

This is a case of the difference of cubes formula in reverse:

\[
(a - b)(a^2 + ab + b^2) = a^3 - b^3
\]

Here, \(a = x^2\) and \(b = 2y\).

\[
(x^2 - 2y)(x^4 + 2x^2y + 4y^2) = (x^2)^3 - (2y)^3
\]

\[
= x^6 - 8y^3
\]

Final Answer: \(\boxed{x^6 - 8y^3}\)

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#### 11. \((x^2 + x + 4)^2\)

Use the formula for the square of a trinomial:

\[
(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca
\]

Here, \(a = x^2\), \(b = x\), and \(c = 4\).

\[
(x^2 + x + 4)^2 = (x^2)^2 + x^2 + 4^2 + 2(x^2)(x) + 2(x)(4) + 2(4)(x^2)
\]

\[
= x^4 + x^2 + 16 + 2x^3 + 8x + 8x^2
\]

Combine like terms:

\[
= x^4 + 2x^3 + 9x^2 + 8x + 16
\]

Final Answer: \(\boxed{x^4 + 2x^3 + 9x^2 + 8x + 16}\)

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#### 12. \((2x - 1)^2\)

Formula Used: \((a - b)^2 = a^2 - 2ab + b^2\)

Here, \(a = 2x\) and \(b = 1\).

\[
(2x - 1)^2 = (2x)^2 - 2(2x)(1) + 1^2
\]

\[
= 4x^2 - 4x + 1
\]

Final Answer: \(\boxed{4x^2 - 4x + 1}\)

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#### 13. \((x + 3)^3\)

Formula Used: \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\)

Here, \(a = x\) and \(b = 3\).

\[
(x + 3)^3 = x^3 + 3(x^2)(3) + 3(x)(3^2) + 3^3
\]

\[
= x^3 + 9x^2 + 27x + 27
\]

Final Answer: \(\boxed{x^3 + 9x^2 + 27x + 27}\)

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#### 14. \((3x + 2)^3\)

Formula Used: \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\)

Here, \(a = 3x\) and \(b = 2\).

\[
(3x + 2)^3 = (3x)^3 + 3(3x)^2(2) + 3(3x)(2^2) + 2^3
\]

\[
= 27x^3 + 3(9x^2)(2) + 3(3x)(4) + 8
\]

\[
= 27x^3 + 54x^2 + 36x + 8
\]

Final Answer: \(\boxed{27x^3 + 54x^2 + 36x + 8}\)

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#### 15. \((x - 4y)(x^2 + 4xy + 16y^2)\)

This is a case of the difference of cubes formula in reverse:

\[
(a - b)(a^2 + ab + b^2) = a^3 - b^3
\]

Here, \(a = x\) and \(b = 4y\).

\[
(x - 4y)(x^2 + 4xy + 16y^2) = x^3 - (4y)^3
\]

\[
= x^3 - 64y^3
\]

Final Answer: \(\boxed{x^3 - 64y^3}\)

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#### 16. \((x - 4y)(x + 4y)\)

This is a difference of squares:

\[
(a - b)(a + b) = a^2 - b^2
\]

Here, \(a = x\) and \(b = 4y\).

\[
(x - 4y)(x + 4y) = x^2 - (4y)^2
\]

\[
= x^2 - 16y^2
\]

Final Answer: \(\boxed{x^2 - 16y^2}\)

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Final Answers:

1. \(\boxed{4x^2 - 8xy + 4y^2}\)
2. \(\boxed{x^2 + y^2 + 4z^2 - 2xy + 4yz - 4zx}\)
3. \(\boxed{x^6 - 3x^4z^3 + 3x^2z^6 - z^9}\)
4. \(\boxed{4x^7 + 2x^5 - 2x^4 - x^2}\)
5. \(\boxed{x^3 - 27}\)
6. \(\boxed{27x^3 - 54x^2y + 36xy^2 - 8y^3}\)
7. \(\boxed{4x^2 - 20xy + 25y^2}\)
8. \(\boxed{x^2 + 9y^2 + 4z^2 - 6xy - 12yz + 4zx}\)
9. \(\boxed{x^4 - 4}\)
10. \(\boxed{x^6 - 8y^3}\)
11. \(\boxed{x^4 + 2x^3 + 9x^2 + 8x + 16}\)
12. \(\boxed{4x^2 - 4x + 1}\)
13. \(\boxed{x^3 + 9x^2 + 27x + 27}\)
14. \(\boxed{27x^3 + 54x^2 + 36x + 8}\)
15. \(\boxed{x^3 - 64y^3}\)
16. \(\boxed{x^2 - 16y^2}\)