(x-2)(x-1).jpeg "(x+6)(x-2)(x-1)")
Expanding and Understanding (x+6)(x-2)(x-1)
This expression represents the product of three linear factors: (x+6), (x-2), and (x-1). Let's explore how to expand it and understand its significance.
Expanding the Expression
To expand the expression, we can use the distributive property multiple times.
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First Expansion: We can start by multiplying the first two factors: (x+6)(x-2) = x(x-2) + 6(x-2) = x² - 2x + 6x - 12 = x² + 4x - 12
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Second Expansion: Now, we multiply the result from step 1 by the third factor: (x² + 4x - 12)(x-1) = x²(x-1) + 4x(x-1) - 12(x-1) = x³ - x² + 4x² - 4x - 12x + 12
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Simplifying: Finally, we combine the like terms to obtain the expanded form: x³ + 3x² - 16x + 12
Significance of the Expanded Form
The expanded form, x³ + 3x² - 16x + 12, represents a cubic polynomial. It describes a curve with a shape similar to a "S" and has the following key characteristics:
- Degree: The highest power of the variable (x³) determines the degree of the polynomial, which is 3.
- Roots: The values of x where the polynomial equals zero are called its roots. These roots correspond to the x-intercepts of the curve. In our case, the original factored form (x+6)(x-2)(x-1) directly tells us the roots: x = -6, x = 2, and x = 1.
- End Behavior: The end behavior of the curve is determined by the leading term (x³). As x approaches positive or negative infinity, the curve will extend towards positive or negative infinity respectively.
Conclusion
Expanding the expression (x+6)(x-2)(x-1) reveals a cubic polynomial with specific roots and end behavior. Understanding this relationship between factored and expanded forms is essential for analyzing polynomials and their corresponding curves in various mathematical contexts.