Worksheet on Special Products | PDF - Free Printable

Problem Description:

The worksheet contains 16 problems involving the expansion of special products. These problems are based on algebraic identities such as the square of a binomial, difference of squares, and other related formulas. The task is to solve each problem by expanding the given expressions.

Solution Approach:

We will solve each problem step by step using the appropriate algebraic identities. Here are the key identities we will use:

1. Square of a Binomial:
- $(a + b)^2 = a^2 + 2ab + b^2$
- $(a - b)^2 = a^2 - 2ab + b^2$

2. Difference of Squares:
- $(a + b)(a - b) = a^2 - b^2$

3. Cube of a Binomial:
- $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$
- $(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$

4. Sum/Difference of Cubes:
- $(a + b)(a^2 - ab + b^2) = a^3 + b^3$
- $(a - b)(a^2 + ab + b^2) = a^3 - b^3$

5. General Expansion:
- Use distributive property for more complex expressions.

Solutions:

#### Problem 1: $(2x - 2y)^2$
Using the identity $(a - b)^2 = a^2 - 2ab + b^2$:
\[
(2x - 2y)^2 = (2x)^2 - 2(2x)(2y) + (2y)^2
\]
\[
= 4x^2 - 8xy + 4y^2
\]

#### Problem 2: $(x - y - 2z)^2$
This is a trinomial squared. Expand it step by step:
\[
(x - y - 2z)^2 = (x - (y + 2z))^2
\]
Using $(a - b)^2 = a^2 - 2ab + b^2$:
\[
= x^2 - 2x(y + 2z) + (y + 2z)^2
\]
\[
= x^2 - 2xy - 4xz + (y^2 + 4yz + 4z^2)
\]
\[
= x^2 - 2xy - 4xz + y^2 + 4yz + 4z^2
\]

#### Problem 3: $(x^2 - x^3)^2$
Using $(a - b)^2 = a^2 - 2ab + b^2$:
\[
(x^2 - x^3)^2 = (x^2)^2 - 2(x^2)(x^3) + (x^3)^2
\]
\[
= x^4 - 2x^5 + x^6
\]

#### Problem 4: $(2x^2 - x)(2x^2 + x)$
Using the difference of squares identity $(a - b)(a + b) = a^2 - b^2$:
\[
(2x^2 - x)(2x^2 + x) = (2x^2)^2 - x^2
\]
\[
= 4x^4 - x^2
\]

#### Problem 5: $(x - 3)(x^2 + 3x + 9)$
This is a sum of cubes in reverse form. Using $(a - b)(a^2 + ab + b^2) = a^3 - b^3$:
\[
(x - 3)(x^2 + 3x + 9) = x^3 - 3^3
\]
\[
= x^3 - 27
\]

#### Problem 6: $(3x - 2y)^3$
Using the cube of a binomial identity $(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$:
\[
(3x - 2y)^3 = (3x)^3 - 3(3x)^2(2y) + 3(3x)(2y)^2 - (2y)^3
\]
\[
= 27x^3 - 54x^2y + 36xy^2 - 8y^3
\]

#### Problem 7: $(2x - 5y)^2$
Using $(a - b)^2 = a^2 - 2ab + b^2$:
\[
(2x - 5y)^2 = (2x)^2 - 2(2x)(5y) + (5y)^2
\]
\[
= 4x^2 - 20xy + 25y^2
\]

#### Problem 8: $(x - 3y + 2z)^2$
This is a trinomial squared. Expand it step by step:
\[
(x - 3y + 2z)^2 = (x - (3y - 2z))^2
\]
Using $(a - b)^2 = a^2 - 2ab + b^2$:
\[
= x^2 - 2x(3y - 2z) + (3y - 2z)^2
\]
\[
= x^2 - 6xy + 4xz + (9y^2 - 12yz + 4z^2)
\]
\[
= x^2 - 6xy + 4xz + 9y^2 - 12yz + 4z^2
\]

#### Problem 9: $(x^2 - 2)(x^2 + 2)$
Using the difference of squares identity $(a - b)(a + b) = a^2 - b^2$:
\[
(x^2 - 2)(x^2 + 2) = (x^2)^2 - 2^2
\]
\[
= x^4 - 4
\]

#### Problem 10: $(x^2 - 2y)(x^2 + 2xy + 4y^2)$
This is a difference of cubes in reverse form. Using $(a - b)(a^2 + ab + b^2) = a^3 - b^3$:
\[
(x^2 - 2y)(x^2 + 2xy + 4y^2) = (x^2)^3 - (2y)^3
\]
\[
= x^6 - 8y^3
\]

#### Problem 11: $(x^4 + 2x + 4)^2$
This is a trinomial squared. Expand it step by step:
\[
(x^4 + 2x + 4)^2 = (x^4 + (2x + 4))^2
\]
Using $(a + b)^2 = a^2 + 2ab + b^2$:
\[
= (x^4)^2 + 2(x^4)(2x + 4) + (2x + 4)^2
\]
\[
= x^8 + 4x^5 + 8x^4 + (4x^2 + 16x + 16)
\]
\[
= x^8 + 4x^5 + 8x^4 + 4x^2 + 16x + 16
\]

#### Problem 12: $(2x - 1)^3$
Using the cube of a binomial identity $(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$:
\[
(2x - 1)^3 = (2x)^3 - 3(2x)^2(1) + 3(2x)(1)^2 - (1)^3
\]
\[
= 8x^3 - 12x^2 + 6x - 1
\]

#### Problem 13: $(x + 3)^3$
Using the cube of a binomial identity $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$:
\[
(x + 3)^3 = x^3 + 3(x^2)(3) + 3(x)(3^2) + 3^3
\]
\[
= x^3 + 9x^2 + 27x + 27
\]

#### Problem 14: $(3x + 2)^3$
Using the cube of a binomial identity $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$:
\[
(3x + 2)^3 = (3x)^3 + 3(3x)^2(2) + 3(3x)(2^2) + 2^3
\]
\[
= 27x^3 + 54x^2 + 36x + 8
\]

#### Problem 15: $(x - 4y)(x^2 + 4xy + 16y^2)$
This is a difference of cubes in reverse form. Using $(a - b)(a^2 + ab + b^2) = a^3 - b^3$:
\[
(x - 4y)(x^2 + 4xy + 16y^2) = x^3 - (4y)^3
\]
\[
= x^3 - 64y^3
\]

#### Problem 16: $(x - 4y)(x + 4y)$
Using the difference of squares identity $(a - b)(a + b) = a^2 - b^2$:
\[
(x - 4y)(x + 4y) = x^2 - (4y)^2
\]
\[
= x^2 - 16y^2
\]

Final Answers:

\[
\boxed{
\begin{array}{ll}
1. & 4x^2 - 8xy + 4y^2 \\
2. & x^2 - 2xy - 4xz + y^2 + 4yz + 4z^2 \\
3. & x^4 - 2x^5 + x^6 \\
4. & 4x^4 - x^2 \\
5. & x^3 - 27 \\
6. & 27x^3 - 54x^2y + 36xy^2 - 8y^3 \\
7. & 4x^2 - 20xy + 25y^2 \\
8. & x^2 - 6xy + 4xz + 9y^2 - 12yz + 4z^2 \\
9. & x^4 - 4 \\
10. & x^6 - 8y^3 \\
11. & x^8 + 4x^5 + 8x^4 + 4x^2 + 16x + 16 \\
12. & 8x^3 - 12x^2 + 6x - 1 \\
13. & x^3 + 9x^2 + 27x + 27 \\
14. & 27x^3 + 54x^2 + 36x + 8 \\
15. & x^3 - 64y^3 \\
16. & x^2 - 16y^2 \\
\end{array}
}
\]