# Factoring 1: Algebraic Expressions

Factoring an algebraic expression is a process of identifying and extracting common factors from the terms of the expression. This technique simplifies the expression and is essential in algebra. Let's look at how to factor an expression like $2x + 6y$.

## Example: Factoring $2x + 6y$

### Step 1: Identify Common Factors

The first step is to find any common factors in each term of the expression. A common factor is a number or variable that evenly divides each term.

- For $2x + 6y$, the common factor is $2$.

### Step 2: Divide Each Term by the Common Factor

Next, divide each term in the expression by the common factor.

- Divide $2x$ by $2$: $\frac{2x}{2} = x$
- Divide $6y$ by $2$: $\frac{6y}{2} = 3y$

### Step 3: Rewrite the Expression

Now, rewrite the expression as the common factor multiplied by the new terms.

- The factored form of $2x + 6y$ is $2(x + 3y)$.

## Practice Problems

Try your hand at factoring these expressions:

1. $4x + 8$
2. $3x^2 + 6x$
3. $5y - 15$
4. $8x^3-4x^2$
5. $2x^4-6x^3+4x$

:::spoiler
<summary>Answers: </summary>

1. $4(x+2)$
2. $3x(x+2)$
3. $5(y-3)$
4. $4x^2(2x-1)$
5. $2x(x^3-3x^2+2)$
:::
## Tips for Effective Factoring

- Start by looking for the greatest common factor in all terms.
- Remember to factor out common variables as well as numbers.
- You can check your work by distributing the factor back into the parentheses to see if you get the original expression.

Enjoy practicing factoring!

# Factoring 2: Quadratic Expressions

Building on basic factoring skills, we now turn our attention to factoring quadratic expressions. A quadratic expression typically takes the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. Factoring such expressions involves finding two binomials that, when multiplied together, give the original quadratic expression.

## Factoring Quadratics: An Example

Consider the quadratic expression $x^2 + 5x + 6$.

### Step 1: Identify the Structure

The expression is in the form $ax^2 + bx + c$, where $a = 1$, $b = 5$, and $c = 6$.

### Step 2: Find Two Numbers

We need to find two numbers that multiply to $c$ (6 in this case) and add up to $b$ (5 in this case).

- After some trial and error, we find that $2$ and $3$ fit the bill since $2 \times 3 = 6$ and $2 + 3 = 5$.

### Step 3: Rewrite the Expression

The expression can be rewritten as two binomials: $(x + 2)(x + 3)$.

### Step 4: Verify by Expanding

Expand $(x + 2)(x + 3)$ to verify that it simplifies back to the original expression:
- $(x + 2)(x + 3) = x^2 + 2x + 3x + 6 = x^2 + 5x + 6$

## Practice Problems

Try factoring these quadratic expressions:

1. $x^2 + 6x + 8$
2. $x^2 + 4x - 5$
3. $x^2 - 7x + 10$

:::spoiler
<summary>Answers:</summary>

1. $(x+2)(x+4)$
2. $(x+5)(x-1)$
3. $(x-2)(x-5)$
:::

## Tips for Factoring Quadratics

- Always start by looking for a common factor in all terms (if there is one).
- For quadratics in the form $x^2 + bx + c$, find two numbers that multiply to $c$ and add up to $b$.
- If the coefficient of $x^2$ is not 1 (i.e., $a \neq 1$), the problem becomes more complex and may require techniques like the quadratic formula or completing the square.

Enjoy exploring the world of quadratic factoring!

# Factoring 3: Quadratics with a ≠ 1

In this section, we'll tackle factoring quadratic expressions of the form $ax^2 + bx + c$ where $a \neq 1$. This type of factoring is a bit more complex but follows a systematic approach.

## Factoring Example: $2x^2 + 5x + 3$

### Step 1: Multiply 'a' and 'c'

Multiply the coefficients 'a' and 'c'.
- For $2x^2 + 5x + 3$, $a = 2$ and $c = 3$, so $ac = 6$.

### Step 2: Find Two Numbers

Find two numbers that multiply to $ac = 6$ and add up to 'b' (5 in this case).
- Numbers 2 and 3 work since $2 \times 3 = 6$ and $2 + 3 = 5$.

### Step 3: Rewrite the Middle Term

Rewrite the middle term (5x) using these numbers.
- $2x^2 + 5x + 3$ becomes $2x^2 + 2x + 3x + 3$.

### Step 4: Factor by Grouping

Group and factor.
- Group terms: $(2x^2 + 2x) + (3x + 3)$.
- Factor out common elements: $2x(x + 1) + 3(x + 1)$.

### Step 5: Factor Out the Common Binomial

Extract the common binomial.
- The expression simplifies to $(x + 1)(2x + 3)$.

## Practice Problems

Factor these quadratic expressions:

1. $3x^2 + 11x + 10$
2. $4x^2 + 4x - 3$
3. $5x^2 - 14x - 3$

:::spoiler
<summary>Answers:</summary>

1. $(3x + 5)(x + 2)$
2. $(2x - 1)(2x + 3)$
3. $(5x + 1)(x - 3)$
:::

## Tips for Factoring Quadratics with a ≠ 1

- Begin by multiplying 'a' and 'c'.
- Identify two numbers that multiply to $ac$ and sum to 'b'.
- Rewrite and then factor by grouping.
- Regular practice is key to mastering this method.

Enjoy your factoring journey!

# Traditional Method of Factoring Quadratics

Factoring quadratic expressions, which are polynomials of the form $ax^2 + bx + c$, is a crucial skill in algebra. The traditional method involves identifying binomials that multiply to give the original quadratic expression. This approach is effective for quadratics where the coefficient of $x^2$ (denoted as 'a') is either 1 or not 1.

## Example: Factoring $2x^2 + 7x + 3$

### Step 1: Find Factor Pairs

1. **Factor Pairs of $2x^2$:** The only factor pair is $2x$ and $x$.

2. **Factor Pairs of $3$:** The factor pairs are $1$ and $3$, and in reverse order, $3$ and $1$.

3. **Construct Multiplication Tables based on the Factor Pairs above:**
- Consider possible binomial pairs: $(2x+1)(x+3)$ and $(2x+3)(x+1)$.
- We'll expand each pair to identify which one reconstructs the original quadratic.

### Step 2: Expansion and Verification

1. **Expand $(2x+1)(x+3)$:**
- Multiply: $2x \times x = 2x^2$
- Multiply: $2x \times 3 = 6x$
- Multiply: $1 \times x = x$
- Multiply: $1 \times 3 = 3$
- Combine: $2x^2 + 6x + x + 3 = 2x^2 + 7x + 3$

Here's Vertical Multiplication that shows it as well:

| | | |
|---------------|---------------|---------------|
| | $2x$ | $1$ |
| | $x$ | $3$ |
| || |
| | $6x$ | $3$ |
| $2x^2$ | $x$ | |
| || |
| $2x^2$ | $7x$ | $3$ |

2. **Expand $(2x+3)(x+1)$ (for comparison):**
- Multiply: $2x \times x = 2x^2$
- Multiply: $2x \times 1 = 2x$
- Multiply: $3 \times x = 3x$
- Multiply: $3 \times 1 = 3$
- Combine: $2x^2 + 2x + 3x + 3 = 2x^2 + 5x + 3$

Here's Vertical Multiplication that shows it as well:

| | | |
|---------------|---------------|---------------|
| | $2x$ | $3$ |
| | $x$ | $1$ |
|------ |------ |---- |
| | $2x$ | $3$ |
| $2x^2$ | $3x$ | |
|------ |-----|------ |
| $2x^2$ | $5x$ | $3$ |

### Conclusion

Only $(2x+1)(x+3)$, when expanded out, gives the original quadratic $2x^2+7x+3$. Therefore, the factored form of $2x^2 + 7x + 3$ is $(2x+1)(x+3)$.

This traditional method, involving a bit of trial and error, is a reliable way to factor quadratics, especially when the leading coefficient is not 1. It's important to note that this is one of several methods for factoring quadratics, others include completing the square and using the quadratic formula.

## Practice Problems

Try these additional problems to practice the traditional method of factoring quadratics:

1. Factor $x^2 + 5x + 6$.

2. Factor $3x^2 + 11x + 6$.

3. Factor $4x^2 + 4x + 1$.

4. Factor $x^2 - 3x - 18$.

5. Factor $2x^2 - 5x - 3$.
6. Factor $9x^2-16$.

:::spoiler
<summary>Answers:</summary>

1. $(x+2)(x+3)$
2. $(3x+2)(x+3)$
3. $(2x+1)^2$
4. $(x-6)(x+3)$
5. $(2x+1)(x-3)$
6. $(3x+4)(3x-4)$
:::
Remember, the first step is to find factor pairs for the $x^2$ term and the constant term. Then, use these factors to construct binomial pairs and expand them to see which pair gives the original quadratic expression.