Factoring Special Cases - Free Printable

Here are the step-by-step solutions for each problem on the worksheet. These problems involve "Special Cases" of factoring, which usually means using the Difference of Squares pattern or the Perfect Square Trinomial pattern.

Key Patterns to Remember:
1. Difference of Squares: $a^2 - b^2 = (a - b)(a + b)$
2. Perfect Square Trinomial: $a^2 - 2ab + b^2 = (a - b)^2$

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1) $16a^2 - 9$
* This is a difference of squares because both terms are perfect squares ($16a^2$ is $(4a)^2$ and $9$ is $3^2$).
* Using the pattern $(a-b)(a+b)$:
* First term square root: $4a$
* Second term square root: $3$
* Answer: $(4a - 3)(4a + 3)$

2) $4m^2 - 25$
* This is a difference of squares.
* Square root of $4m^2$ is $2m$.
* Square root of $25$ is $5$.
* Answer: $(2m - 5)(2m + 5)$

3) $16b^2 - 40b + 25$
* Check if this is a perfect square trinomial.
* First term: $16b^2 = (4b)^2$
* Last term: $25 = 5^2$
* Middle term check: Is $-40b$ equal to $-2 \cdot (4b) \cdot (5)$?
* $-2 \cdot 4b \cdot 5 = -40b$. Yes, it matches.
* Since the middle sign is negative, we use $(a - b)^2$.
* Answer: $(4b - 5)^2$

4) $4x^2 - 4x + 1$
* Check for perfect square trinomial.
* First term: $4x^2 = (2x)^2$
* Last term: $1 = 1^2$
* Middle term check: Is $-4x$ equal to $-2 \cdot (2x) \cdot (1)$?
* $-2 \cdot 2x \cdot 1 = -4x$. Yes, it matches.
* Use the negative pattern $(a - b)^2$.
* Answer: $(2x - 1)^2$

5) $9x^2 - 1$
* Difference of squares.
* Square root of $9x^2$ is $3x$.
* Square root of $1$ is $1$.
* Answer: $(3x - 1)(3x + 1)$

6) $n^2 - 25$
* Difference of squares.
* Square root of $n^2$ is $n$.
* Square root of $25$ is $5$.
* Answer: $(n - 5)(n + 5)$

7) $n^4 - 100$
* Difference of squares. Note that $n^4$ is $(n^2)^2$.
* Square root of $n^4$ is $n^2$.
* Square root of $100$ is $10$.
* Answer: $(n^2 - 10)(n^2 + 10)$
*(Note: $n^2 - 10$ cannot be factored further using integers because 10 is not a perfect square).*

8) $a^4 - 9$
* Difference of squares.
* Square root of $a^4$ is $a^2$.
* Square root of $9$ is $3$.
* Answer: $(a^2 - 3)(a^2 + 3)$

9) $k^4 - 36$
* Difference of squares.
* Square root of $k^4$ is $k^2$.
* Square root of $36$ is $6$.
* Answer: $(k^2 - 6)(k^2 + 6)$

10) $n^4 - 49$
* Difference of squares.
* Square root of $n^4$ is $n^2$.
* Square root of $49$ is $7$.
* Answer: $(n^2 - 7)(n^2 + 7)$

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Final Answer:
1) $(4a - 3)(4a + 3)$
2) $(2m - 5)(2m + 5)$
3) $(4b - 5)^2$
4) $(2x - 1)^2$
5) $(3x - 1)(3x + 1)$
6) $(n - 5)(n + 5)$
7) $(n^2 - 10)(n^2 + 10)$
8) $(a^2 - 3)(a^2 + 3)$
9) $(k^2 - 6)(k^2 + 6)$
10) $(n^2 - 7)(n^2 + 7)$