## Factoring and Expanding (x - 7)(x - 10)

This expression represents the product of two binomials: (x - 7) and (x - 10). We can explore both **factoring** and **expanding** this expression.

### Expanding the Expression

Expanding the expression means multiplying the two binomials together. We can use the **FOIL** method (First, Outer, Inner, Last) to do this:

**First:**x * x = x²**Outer:**x * -10 = -10x**Inner:**-7 * x = -7x**Last:**-7 * -10 = 70

Combining the terms, we get:
**x² - 10x - 7x + 70**

Simplifying further:
**x² - 17x + 70**

Therefore, the expanded form of (x - 7)(x - 10) is **x² - 17x + 70**.

### Factoring the Expression

Factoring the expression means finding the two binomials that multiply together to give us the original expression. In this case, we already know the factored form: **(x - 7)(x - 10)**.

However, let's assume we only have the expanded form (x² - 17x + 70) and want to factor it. Here's how we can do it:

**Find two numbers that multiply to give the constant term (70) and add up to the coefficient of the x term (-17).**In this case, the numbers are -7 and -10.**Rewrite the middle term (-17x) using these two numbers.**This gives us: x² - 7x - 10x + 70**Factor by grouping.**- Group the first two terms and the last two terms: (x² - 7x) + (-10x + 70)
- Factor out the greatest common factor from each group: x(x - 7) - 10(x - 7)
- Notice that both groups share a common factor of (x - 7). Factor that out: (x - 7)(x - 10)

Therefore, the factored form of x² - 17x + 70 is **(x - 7)(x - 10)**.

### Applications

Understanding how to factor and expand expressions like (x - 7)(x - 10) is crucial in various mathematical contexts, including:

**Solving quadratic equations:**Factoring a quadratic equation can help find its roots.**Graphing quadratic functions:**The factored form provides insights into the x-intercepts of the function's graph.**Simplifying expressions:**Factoring and expanding can be used to simplify complex algebraic expressions.