Worksheet on Special Products | PDF - Free Printable

Problem Overview:

The task involves solving the special products of binomials, which are expressions of the form \((a + b)^2\), \((a - b)^2\), and \((a + b)(a - b)\). These are fundamental algebraic identities that simplify the process of expanding or factoring expressions.

The given worksheet lists 16 problems, each requiring the application of one of these identities. Below, I will solve each problem step by step and explain the solution.

---

Algebraic Identities Used:

1. Square of a Binomial (Sum):
\[
(a + b)^2 = a^2 + 2ab + b^2
\]

2. Square of a Binomial (Difference):
\[
(a - b)^2 = a^2 - 2ab + b^2
\]

3. Product of a Sum and Difference:
\[
(a + b)(a - b) = a^2 - b^2
\]

---

Solutions to Each Problem:

#### 1. \((2x + y - z)^2\)
This is a square of a trinomial, but we can treat it as a binomial where \(a = 2x + y\) and \(b = z\):
\[
(2x + y - z)^2 = [(2x + y) - z]^2
\]
Using the identity \((a - b)^2 = a^2 - 2ab + b^2\):
\[
[(2x + y) - z]^2 = (2x + y)^2 - 2(2x + y)z + z^2
\]
Now expand \((2x + y)^2\):
\[
(2x + y)^2 = (2x)^2 + 2(2x)(y) + y^2 = 4x^2 + 4xy + y^2
\]
Substitute back:
\[
(2x + y - z)^2 = (4x^2 + 4xy + y^2) - 2(2x + y)z + z^2
\]
Simplify \(-2(2x + y)z\):
\[
-2(2x + y)z = -4xz - 2yz
\]
Thus:
\[
(2x + y - z)^2 = 4x^2 + 4xy + y^2 - 4xz - 2yz + z^2
\]

Final Answer:
\[
\boxed{4x^2 + 4xy + y^2 - 4xz - 2yz + z^2}
\]

#### 2. \((3x + 2y - z)^2\)
Similar to Problem 1, treat it as \([(3x + 2y) - z]^2\):
\[
(3x + 2y - z)^2 = [(3x + 2y) - z]^2
\]
Using \((a - b)^2 = a^2 - 2ab + b^2\):
\[
[(3x + 2y) - z]^2 = (3x + 2y)^2 - 2(3x + 2y)z + z^2
\]
Expand \((3x + 2y)^2\):
\[
(3x + 2y)^2 = (3x)^2 + 2(3x)(2y) + (2y)^2 = 9x^2 + 12xy + 4y^2
\]
Substitute back:
\[
(3x + 2y - z)^2 = (9x^2 + 12xy + 4y^2) - 2(3x + 2y)z + z^2
\]
Simplify \(-2(3x + 2y)z\):
\[
-2(3x + 2y)z = -6xz - 4yz
\]
Thus:
\[
(3x + 2y - z)^2 = 9x^2 + 12xy + 4y^2 - 6xz - 4yz + z^2
\]

Final Answer:
\[
\boxed{9x^2 + 12xy + 4y^2 - 6xz - 4yz + z^2}
\]

#### 3. \((x^2 + y^2 - z^2)^2\)
Treat it as \([(x^2 + y^2) - z^2]^2\):
\[
(x^2 + y^2 - z^2)^2 = [(x^2 + y^2) - z^2]^2
\]
Using \((a - b)^2 = a^2 - 2ab + b^2\):
\[
[(x^2 + y^2) - z^2]^2 = (x^2 + y^2)^2 - 2(x^2 + y^2)z^2 + (z^2)^2
\]
Expand \((x^2 + y^2)^2\):
\[
(x^2 + y^2)^2 = (x^2)^2 + 2(x^2)(y^2) + (y^2)^2 = x^4 + 2x^2y^2 + y^4
\]
Substitute back:
\[
(x^2 + y^2 - z^2)^2 = (x^4 + 2x^2y^2 + y^4) - 2(x^2 + y^2)z^2 + z^4
\]
Simplify \(-2(x^2 + y^2)z^2\):
\[
-2(x^2 + y^2)z^2 = -2x^2z^2 - 2y^2z^2
\]
Thus:
\[
(x^2 + y^2 - z^2)^2 = x^4 + 2x^2y^2 + y^4 - 2x^2z^2 - 2y^2z^2 + z^4
\]

Final Answer:
\[
\boxed{x^4 + 2x^2y^2 + y^4 - 2x^2z^2 - 2y^2z^2 + z^4}
\]

#### 4. \((2x^2 - y^2 - x)^2\)
Treat it as \([(2x^2 - y^2) - x]^2\):
\[
(2x^2 - y^2 - x)^2 = [(2x^2 - y^2) - x]^2
\]
Using \((a - b)^2 = a^2 - 2ab + b^2\):
\[
[(2x^2 - y^2) - x]^2 = (2x^2 - y^2)^2 - 2(2x^2 - y^2)x + x^2
\]
Expand \((2x^2 - y^2)^2\):
\[
(2x^2 - y^2)^2 = (2x^2)^2 - 2(2x^2)(y^2) + (y^2)^2 = 4x^4 - 4x^2y^2 + y^4
\]
Substitute back:
\[
(2x^2 - y^2 - x)^2 = (4x^4 - 4x^2y^2 + y^4) - 2(2x^2 - y^2)x + x^2
\]
Simplify \(-2(2x^2 - y^2)x\):
\[
-2(2x^2 - y^2)x = -4x^3 + 2xy^2
\]
Thus:
\[
(2x^2 - y^2 - x)^2 = 4x^4 - 4x^2y^2 + y^4 - 4x^3 + 2xy^2 + x^2
\]

Final Answer:
\[
\boxed{4x^4 - 4x^2y^2 + y^4 - 4x^3 + 2xy^2 + x^2}
\]

#### 5. \((x^2 - y^2 + x^2)^2\)
Simplify the expression inside the parentheses first:
\[
x^2 - y^2 + x^2 = 2x^2 - y^2
\]
So the problem becomes:
\[
(2x^2 - y^2)^2
\]
Using \((a - b)^2 = a^2 - 2ab + b^2\):
\[
(2x^2 - y^2)^2 = (2x^2)^2 - 2(2x^2)(y^2) + (y^2)^2
\]
Simplify each term:
\[
(2x^2)^2 = 4x^4, \quad -2(2x^2)(y^2) = -4x^2y^2, \quad (y^2)^2 = y^4
\]
Thus:
\[
(2x^2 - y^2)^2 = 4x^4 - 4x^2y^2 + y^4
\]

Final Answer:
\[
\boxed{4x^4 - 4x^2y^2 + y^4}
\]

#### 6. \((x + 2y)^2\)
Using \((a + b)^2 = a^2 + 2ab + b^2\):
\[
(x + 2y)^2 = x^2 + 2(x)(2y) + (2y)^2
\]
Simplify each term:
\[
x^2 + 2(x)(2y) = x^2 + 4xy, \quad (2y)^2 = 4y^2
\]
Thus:
\[
(x + 2y)^2 = x^2 + 4xy + 4y^2
\]

Final Answer:
\[
\boxed{x^2 + 4xy + 4y^2}
\]

#### 7. \((2x + y)^2\)
Using \((a + b)^2 = a^2 + 2ab + b^2\):
\[
(2x + y)^2 = (2x)^2 + 2(2x)(y) + y^2
\]
Simplify each term:
\[
(2x)^2 = 4x^2, \quad 2(2x)(y) = 4xy, \quad y^2 = y^2
\]
Thus:
\[
(2x + y)^2 = 4x^2 + 4xy + y^2
\]

Final Answer:
\[
\boxed{4x^2 + 4xy + y^2}
\]

#### 8. \((x - 2y)^2\)
Using \((a - b)^2 = a^2 - 2ab + b^2\):
\[
(x - 2y)^2 = x^2 - 2(x)(2y) + (2y)^2
\]
Simplify each term:
\[
x^2 - 2(x)(2y) = x^2 - 4xy, \quad (2y)^2 = 4y^2
\]
Thus:
\[
(x - 2y)^2 = x^2 - 4xy + 4y^2
\]

Final Answer:
\[
\boxed{x^2 - 4xy + 4y^2}
\]

#### 9. \((x^2 - 2)^2\)
Using \((a - b)^2 = a^2 - 2ab + b^2\):
\[
(x^2 - 2)^2 = (x^2)^2 - 2(x^2)(2) + (2)^2
\]
Simplify each term:
\[
(x^2)^2 = x^4, \quad -2(x^2)(2) = -4x^2, \quad (2)^2 = 4
\]
Thus:
\[
(x^2 - 2)^2 = x^4 - 4x^2 + 4
\]

Final Answer:
\[
\boxed{x^4 - 4x^2 + 4}
\]

#### 10. \((x^2 - 2y)^2\)
Using \((a - b)^2 = a^2 - 2ab + b^2\):
\[
(x^2 - 2y)^2 = (x^2)^2 - 2(x^2)(2y) + (2y)^2
\]
Simplify each term:
\[
(x^2)^2 = x^4, \quad -2(x^2)(2y) = -4x^2y, \quad (2y)^2 = 4y^2
\]
Thus:
\[
(x^2 - 2y)^2 = x^4 - 4x^2y + 4y^2
\]

Final Answer:
\[
\boxed{x^4 - 4x^2y + 4y^2}
\]

#### 11. \((x - 2)(x^2 + 2x + 4)\)
This is a product of a binomial and a trinomial. Expand it directly:
\[
(x - 2)(x^2 + 2x + 4) = x(x^2 + 2x + 4) - 2(x^2 + 2x + 4)
\]
Distribute \(x\) and \(-2\):
\[
x(x^2 + 2x + 4) = x^3 + 2x^2 + 4x
\]
\[
-2(x^2 + 2x + 4) = -2x^2 - 4x - 8
\]
Combine the results:
\[
(x - 2)(x^2 + 2x + 4) = (x^3 + 2x^2 + 4x) + (-2x^2 - 4x - 8)
\]
Simplify:
\[
x^3 + 2x^2 - 2x^2 + 4x - 4x - 8 = x^3 - 8
\]

Final Answer:
\[
\boxed{x^3 - 8}
\]

#### 12. \((2x - 1)(x^2 + 2x + 1)\)
Expand the product:
\[
(2x - 1)(x^2 + 2x + 1) = 2x(x^2 + 2x + 1) - 1(x^2 + 2x + 1)
\]
Distribute \(2x\) and \(-1\):
\[
2x(x^2 + 2x + 1) = 2x^3 + 4x^2 + 2x
\]
\[
-1(x^2 + 2x + 1) = -x^2 - 2x - 1
\]
Combine the results:
\[
(2x - 1)(x^2 + 2x + 1) = (2x^3 + 4x^2 + 2x) + (-x^2 - 2x - 1)
\]
Simplify:
\[
2x^3 + 4x^2 - x^2 + 2x - 2x - 1 = 2x^3 + 3x^2 - 1
\]

Final Answer:
\[
\boxed{2x^3 + 3x^2 - 1}
\]

#### 13. \((x + 3)(x^2 - 3x + 9)\)
Expand the product:
\[
(x + 3)(x^2 - 3x + 9) = x(x^2 - 3x + 9) + 3(x^2 - 3x + 9)
\]
Distribute \(x\) and \(3\):
\[
x(x^2 - 3x + 9) = x^3 - 3x^2 + 9x
\]
\[
3(x^2 - 3x + 9) = 3x^2 - 9x + 27
\]
Combine the results:
\[
(x + 3)(x^2 - 3x + 9) = (x^3 - 3x^2 + 9x) + (3x^2 - 9x + 27)
\]
Simplify:
\[
x^3 - 3x^2 + 3x^2 + 9x - 9x + 27 = x^3 + 27
\]

Final Answer:
\[
\boxed{x^3 + 27}
\]

#### 14. \((3x + 2)(3x^2 - 6x + 4)\)
Expand the product:
\[
(3x + 2)(3x^2 - 6x + 4) = 3x(3x^2 - 6x + 4) + 2(3x^2 - 6x + 4)
\]
Distribute \(3x\) and \(2\):
\[
3x(3x^2 - 6x + 4) = 9x^3 - 18x^2 + 12x
\]
\[
2(3x^2 - 6x + 4) = 6x^2 - 12x + 8
\]
Combine the results:
\[
(3x + 2)(3x^2 - 6x + 4) = (9x^3 - 18x^2 + 12x) + (6x^2 - 12x + 8)
\]
Simplify:
\[
9x^3 - 18x^2 + 6x^2 + 12x - 12x + 8 = 9x^3 - 12x^2 + 8
\]

Final Answer:
\[
\boxed{9x^3 - 12x^2 + 8}
\]

#### 15. \((2x - 3y)(4x^2 + 6xy + 9y^2)\)
Expand the product:
\[
(2x - 3y)(4x^2 + 6xy + 9y^2) = 2x(4x^2 + 6xy + 9y^2) - 3y(4x^2 + 6xy + 9y^2)
\]
Distribute \(2x\) and \(-3y\):
\[
2x(4x^2 + 6xy + 9y^2) = 8x^3 + 12x^2y + 18xy^2
\]
\[
-3y(4x^2 + 6xy + 9y^2) = -12x^2y - 18xy^2 - 27y^3
\]
Combine the results:
\[
(2x - 3y)(4x^2 + 6xy + 9y^2) = (8x^3 + 12x^2y + 18xy^2) + (-12x^2y - 18xy^2 - 27y^3)
\]
Simplify:
\[
8x^3 + 12x^2y - 12x^2y + 18xy^2 - 18xy^2 - 27y^3 = 8x^3 - 27y^3
\]

Final Answer:
\[
\boxed{8x^3 - 27y^3}
\]

#### 16. \((x - 4y)(x^2 + 4xy + 16y^2)\)
Expand the product:
\[
(x - 4y)(x^2 + 4xy + 16y^2) = x(x^2 + 4xy + 16y^2) - 4y(x^2 + 4xy + 16y^2)
\]
Distribute \(x\) and \(-4y\):
\[
x(x^2 + 4xy + 16y^2) = x^3 + 4x^2y + 16xy^2
\]
\[
-4y(x^2 + 4xy + 16y^2) = -4x^2y - 16xy^2 - 64y^3
\]
Combine the results:
\[
(x - 4y)(x^2 + 4xy + 16y^2) = (x^3 + 4x^2y + 16xy^2) + (-4x^2y - 16xy^2 - 64y^3)
\]
Simplify:
\[
x^3 + 4x^2y - 4x^2y + 16xy^2 - 16xy^2 - 64y^3 = x^3 - 64y^3
\]

Final Answer:
\[
\boxed{x^3 - 64y^3}
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ 4x^2 + 4xy + y^2 - 4xz - 2yz + z^2 \\
2. & \ 9x^2 + 12xy + 4y^2 - 6xz - 4yz + z^2 \\
3. & \ x^4 + 2x^2y^2 + y^4 - 2x^2z^2 - 2y^2z^2 + z^4 \\
4. & \ 4x^4 - 4x^2y^2 + y^4 - 4x^3 + 2xy^2 + x^2 \\
5. & \ 4x^4 - 4x^2y^2 + y^4 \\
6. & \ x^2 + 4xy + 4y^2 \\
7. & \ 4x^2 + 4xy + y^2 \\
8. & \ x^2 - 4xy + 4y^2 \\
9. & \ x^4 - 4x^2 + 4 \\
10. & \ x^4 - 4x^2y + 4y^2 \\
11. & \ x^3 - 8 \\
12. & \ 2x^3 + 3x^2 - 1 \\
13. & \ x^3 + 27 \\
14. & \ 9x^3 - 12x^2 + 8 \\
15. & \ 8x^3 - 27y^3 \\
16. & \ x^3 - 64y^3 \\
\end{aligned}
}
\]