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Factoring and Expanding the Expression (4x-1)(4x+1)(x-4)
This article will explore how to factor and expand the expression (4x-1)(4x+1)(x-4). We will also discuss the applications of this type of expression in mathematics.
Factoring the Expression
The expression (4x-1)(4x+1)(x-4) can be factored by recognizing the difference of squares pattern in the first two factors.
Difference of Squares: a² - b² = (a+b)(a-b)
Applying this to our expression:
- (4x-1)(4x+1) = (4x)² - (1)² = 16x² - 1
Now our expression becomes:
(16x² - 1)(x-4)
We can further expand this by using the distributive property (FOIL):
16x²(x-4) - 1(x-4)
This gives us:
16x³ - 64x² - x + 4
Expanding the Expression
We can also expand the expression (4x-1)(4x+1)(x-4) directly by using the distributive property.
Step 1: Expand the first two factors:
(4x-1)(4x+1) = 16x² - 1
Step 2: Multiply the result from step 1 by the third factor:
(16x² - 1)(x-4) = 16x³ - 64x² - x + 4
Therefore, the expanded form of (4x-1)(4x+1)(x-4) is 16x³ - 64x² - x + 4.
Applications of Factoring and Expanding
Factoring and expanding expressions are fundamental skills in algebra and have various applications in different areas of mathematics, such as:
- Solving equations: Factoring allows us to find the roots (solutions) of polynomial equations.
- Simplifying expressions: Factoring can simplify complex expressions and make them easier to work with.
- Calculus: Expanding expressions is crucial for differentiating and integrating functions.
- Graphing: Factoring can help us identify the x-intercepts of a polynomial function.
Conclusion
The expression (4x-1)(4x+1)(x-4) provides an excellent example of how to factor and expand polynomial expressions. Recognizing patterns like the difference of squares is a valuable tool in simplifying and manipulating algebraic expressions. These skills are essential for solving equations, simplifying expressions, and understanding various mathematical concepts.