(x-1%2F4)(16x-1).jpeg "(x+1/4)(x-1/4)(16x-1)")
(x+1/4)(x-1/4)(16x-1): A Comprehensive Analysis
Introduction
In this article, we will delve into the fascinating world of algebraic expressions, specifically exploring the properties and behavior of the expression (x+1/4)(x-1/4)(16x-1). This expression is comprised of three factors, each with its unique characteristics, which, when combined, reveal intriguing patterns and relationships.
Factorization and Simplification
Let's start by analyzing each factor individually:
Factor 1: (x+1/4)
This factor can be rewritten as x + 0.25. It represents a linear expression with a positive slope and a y-intercept of 0.25. The graph of this expression would be a straight line with a shallow slope.
Factor 2: (x-1/4)
This factor can be rewritten as x - 0.25. It also represents a linear expression, but with a negative slope and a y-intercept of -0.25. The graph of this expression would be a straight line with a shallow slope, mirrored about the x-axis.
Factor 3: (16x-1)
This factor represents a linear expression with a steep slope and a y-intercept of -1. The graph of this expression would be a straight line with a higher slope compared to the first two factors.
The Product of the Factors
Now that we've analyzed each factor individually, let's examine the product of these factors:
(x+1/4)(x-1/4)(16x-1)
Expanding this product, we get:
16x^3 - 16x^2 - x + 1
Simplifying the expression, we arrive at:
16x^3 - 16x^2 - x + 1
This resulting expression is a cubic polynomial with a leading coefficient of 16.
Graphical Representation
The graph of (x+1/4)(x-1/4)(16x-1) would be a cubic curve with a positive leading coefficient, implying that the curve opens upward. The zeros of the function would be the solutions to the equation (x+1/4)(x-1/4)(16x-1) = 0.
Conclusion
In this article, we've explored the properties of the expression (x+1/4)(x-1/4)(16x-1), analyzing each factor individually and examining the product of these factors. We've seen how the combination of these factors results in a cubic polynomial with unique characteristics. This exploration has provided a deeper understanding of algebraic expressions and their behavior.
