(x-7).jpeg "(2x+1)(x-7)")
Expanding and Simplifying (2x+1)(x-7)
In mathematics, expanding and simplifying expressions is a crucial skill. Let's explore how to do this with the expression (2x+1)(x-7).
Understanding the Concept
The expression (2x+1)(x-7) represents the product of two binomials. To expand it, we need to multiply each term in the first binomial by each term in the second binomial.
The FOIL Method
A common technique for expanding binomials is the FOIL method:
- First: Multiply the first terms of each binomial.
- Outer: Multiply the outer terms of the binomials.
- Inner: Multiply the inner terms of the binomials.
- Last: Multiply the last terms of each binomial.
Let's apply this to our expression:
- First: (2x) * (x) = 2x²
- Outer: (2x) * (-7) = -14x
- Inner: (1) * (x) = x
- Last: (1) * (-7) = -7
Now, we have: 2x² - 14x + x - 7
Combining Like Terms
The final step is to combine the like terms:
2x² - 13x - 7
Therefore, the expanded and simplified form of (2x+1)(x-7) is 2x² - 13x - 7.
Further Applications
This process of expanding and simplifying is essential for various mathematical operations such as:
- Solving equations: By expanding and simplifying an equation, we can manipulate it to solve for the unknown variable.
- Graphing functions: The expanded form of an expression can be used to graph the corresponding function.
- Calculus: Expanding and simplifying expressions is a common step in solving derivatives and integrals.
By mastering this technique, you gain a powerful tool for tackling various mathematical problems.