(4x%2B1).jpeg "(4x-1)(4x+1)")
Factoring the Difference of Squares: (4x-1)(4x+1)
The expression (4x-1)(4x+1) represents a special case of multiplication known as the difference of squares. Understanding this pattern can help simplify algebraic expressions and solve equations more efficiently.
Recognizing the Pattern
The difference of squares pattern is characterized by two binomials:
- One binomial is the sum of two terms.
- The other binomial is the difference of the same two terms.
In our example, (4x-1) and (4x+1) fit this pattern perfectly:
- 4x is the common term.
- 1 is the other term.
Applying the Formula
The difference of squares formula states:
(a + b)(a - b) = a² - b²
Where:
- a represents the first term in each binomial.
- b represents the second term in each binomial.
Applying this to our example:
- a = 4x
- b = 1
Therefore, we can simplify (4x-1)(4x+1) as follows:
(4x - 1)(4x + 1) = (4x)² - (1)²
Simplifying the Expression
Simplifying further:
(4x)² - (1)² = 16x² - 1
Therefore, the simplified form of (4x-1)(4x+1) is 16x² - 1.
Conclusion
Understanding the difference of squares pattern allows us to efficiently factor and simplify expressions like (4x-1)(4x+1). This knowledge is crucial for solving equations, simplifying complex algebraic expressions, and gaining a deeper understanding of mathematical patterns.