(x-7).jpeg "(x-8)(x-7)")
Factoring and Expanding (x - 8)(x - 7)
This expression represents the product of two binomials: (x - 8) and (x - 7). We can approach this in two ways:
Expanding the Expression
To expand this, we use the FOIL method:
- First: Multiply the first terms of each binomial: x * x = x²
- Outer: Multiply the outer terms of the binomials: x * -7 = -7x
- Inner: Multiply the inner terms of the binomials: -8 * x = -8x
- Last: Multiply the last terms of each binomial: -8 * -7 = 56
Now we combine the terms:
x² - 7x - 8x + 56 = x² - 15x + 56
Factoring the Expression
If we are given the expanded form, x² - 15x + 56, we can factor it back into the original binomials. Here's how:
- Find two numbers that multiply to 56 and add up to -15.
- In this case, the numbers are -8 and -7.
- Rewrite the middle term (-15x) using these two numbers: x² - 8x - 7x + 56
- Group the terms: (x² - 8x) + (-7x + 56)
- Factor out the greatest common factor from each group: x(x - 8) - 7(x - 8)
- Factor out the common binomial (x - 8): (x - 8)(x - 7)
Therefore, the factored form of x² - 15x + 56 is (x - 8)(x - 7).
These processes, expanding and factoring, are essential in algebra and provide valuable tools for solving equations and simplifying expressions.