(x---10).jpeg "(x - 7)(x - 10)")
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.