Algebra 1 Worksheets - Easy Hard Science - Free Printable

Problem Analysis:

The task involves two main parts:
1. Expanding expressions: Simplifying given algebraic expressions by distributing terms.
2. Factoring expressions: Rewriting quadratic expressions in their factored forms.

Let's solve each part step by step.

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Part 1: Expand

#### Expression 1: $(x + 1)(x - 1)$
This is a difference of squares, which follows the formula:
\[
(a + b)(a - b) = a^2 - b^2
\]
Here, $a = x$ and $b = 1$. Applying the formula:
\[
(x + 1)(x - 1) = x^2 - 1^2 = x^2 - 1
\]

#### Expression 2: $(x + 1)^2$
This is a perfect square trinomial, which follows the formula:
\[
(a + b)^2 = a^2 + 2ab + b^2
\]
Here, $a = x$ and $b = 1$. Applying the formula:
\[
(x + 1)^2 = x^2 + 2(x)(1) + 1^2 = x^2 + 2x + 1
\]

#### Expression 3: $(x - 1)^2$
This is also a perfect square trinomial, which follows the formula:
\[
(a - b)^2 = a^2 - 2ab + b^2
\]
Here, $a = x$ and $b = 1$. Applying the formula:
\[
(x - 1)^2 = x^2 - 2(x)(1) + 1^2 = x^2 - 2x + 1
\]

#### Expression 4: $(x - 1)(x - 1)$
This is the same as $(x - 1)^2$, so we can directly use the result from the previous step:
\[
(x - 1)(x - 1) = x^2 - 2x + 1
\]

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Part 2: Factor

#### Expression 1: $x^2 - 1$
This is a difference of squares, which factors as:
\[
a^2 - b^2 = (a + b)(a - b)
\]
Here, $a = x$ and $b = 1$. Applying the formula:
\[
x^2 - 1 = (x + 1)(x - 1)
\]

#### Expression 2: $x^2 + 2x + 1$
This is a perfect square trinomial, which factors as:
\[
a^2 + 2ab + b^2 = (a + b)^2
\]
Here, $a = x$ and $b = 1$. Applying the formula:
\[
x^2 + 2x + 1 = (x + 1)^2
\]

#### Expression 3: $x^2 - 2x + 1$
This is also a perfect square trinomial, which factors as:
\[
a^2 - 2ab + b^2 = (a - b)^2
\]
Here, $a = x$ and $b = 1$. Applying the formula:
\[
x^2 - 2x + 1 = (x - 1)^2
\]

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Final Answers:

1. Expand:
- $(x + 1)(x - 1) = \boxed{x^2 - 1}$
- $(x + 1)^2 = \boxed{x^2 + 2x + 1}$
- $(x - 1)^2 = \boxed{x^2 - 2x + 1}$
- $(x - 1)(x - 1) = \boxed{x^2 - 2x + 1}$

2. Factor:
- $x^2 - 1 = \boxed{(x + 1)(x - 1)}$
- $x^2 + 2x + 1 = \boxed{(x + 1)^2}$
- $x^2 - 2x + 1 = \boxed{(x - 1)^2}$

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Boxed Final Answer:

\[
\boxed{
\begin{aligned}
&\text{Expand:} \\
&1. (x + 1)(x - 1) = x^2 - 1 \\
&2. (x + 1)^2 = x^2 + 2x + 1 \\
&3. (x - 1)^2 = x^2 - 2x + 1 \\
&4. (x - 1)(x - 1) = x^2 - 2x + 1 \\
\\
&\text{Factor:} \\
&1. x^2 - 1 = (x + 1)(x - 1) \\
&2. x^2 + 2x + 1 = (x + 1)^2 \\
&3. x^2 - 2x + 1 = (x - 1)^2
\end{aligned}
}
\]