(x-5)(x-5).jpeg "(x-5)(x-5)(x-5)")
Understanding (x-5)(x-5)(x-5)
The expression (x-5)(x-5)(x-5) represents a cubic polynomial in its factored form. This means that it is the product of three identical linear factors: (x-5). Let's break down what this means and how to work with it.
Factoring and Expanding
Factoring involves breaking down a polynomial into a product of simpler expressions. In this case, we have already been given the factored form.
Expanding the expression means multiplying out the factors to get the standard polynomial form. Here's how we can do that:
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First, focus on two factors: (x-5)(x-5) = x^2 - 10x + 25 (using the FOIL method: First, Outer, Inner, Last)
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Multiply the result by the remaining factor: (x^2 - 10x + 25)(x-5) = x^3 - 15x^2 + 75x - 125
Therefore, the expanded form of (x-5)(x-5)(x-5) is x^3 - 15x^2 + 75x - 125.
Key Concepts
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Roots and Zeroes: The expression (x-5) equals zero when x=5. Since this factor appears three times, the polynomial has a triple root at x=5. This means the graph of the polynomial will touch the x-axis at x=5 but not cross it.
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Degree and Shape: The expanded form shows the polynomial has a degree of 3 (highest power of x is 3). Cubic polynomials typically have an "S" shaped curve.
Applications
Understanding factored forms like (x-5)(x-5)(x-5) is essential in various mathematical contexts, including:
- Solving Equations: Setting the expression equal to zero and solving for x will give us the root (x=5).
- Graphing Functions: The factored form helps us determine the x-intercepts and the general shape of the function's graph.
- Calculus: The factored form can be used to find derivatives and integrals of the function.
Conclusion
(x-5)(x-5)(x-5) is a compact representation of a cubic polynomial. By understanding how to factor and expand it, we gain insights into its properties, its roots, and its graphical behavior. These insights are valuable for solving equations, analyzing functions, and applying concepts in various areas of mathematics.