End Behavior Of Polynomial Function P(x)=-2(3-x)(2x+5)^2

In the realm of mathematics, understanding the behavior of functions is paramount, especially when dealing with polynomial functions. These functions, characterized by their algebraic expressions involving variables raised to non-negative integer powers, play a crucial role in modeling real-world phenomena and solving mathematical problems. One of the key aspects of analyzing polynomial functions is determining their end behavior, which describes how the function behaves as the input variable (x) approaches positive and negative infinity. In this comprehensive article, we will delve into the intricacies of end behavior, using the specific example of the polynomial function

$p(x) = -2(3-x)(2x+5)^2$

as a case study. Our aim is to dissect the function's structure, identify its leading term, and ultimately decipher its end behavior, thereby gaining a deeper understanding of its overall characteristics.

Deciphering the Polynomial Function: Unveiling its Structure

Before we embark on the journey of unraveling the end behavior of our polynomial function, it is imperative that we first dissect its structure and identify its key components. The function in question is:

$p(x) = -2(3-x)(2x+5)^2$

At first glance, this function might appear somewhat complex, but upon closer inspection, we can discern its underlying structure. The function is expressed as a product of three factors: a constant term (-2), a linear term (3-x), and a quadratic term (2x+5)^2. To gain a clearer understanding of the function's behavior, it is often beneficial to expand the expression and rewrite it in its standard polynomial form. Let's embark on this expansion process:

First, we need to expand the quadratic term (2x+5)^2. This can be achieved by applying the binomial theorem or simply by multiplying the term by itself:

$(2x+5)^2 = (2x+5)(2x+5) = 4x^2 + 20x + 25$

Now that we have expanded the quadratic term, we can substitute it back into the original expression:

$p(x) = -2(3-x)(4x^2 + 20x + 25)$

Next, we need to multiply the linear term (3-x) by the expanded quadratic term:

$(3-x)(4x^2 + 20x + 25) = 12x^2 + 60x + 75 - 4x^3 - 20x^2 - 25x$

Combining like terms, we get:

$-4x^3 - 8x^2 + 35x + 75$

Finally, we multiply the entire expression by the constant term -2:

$p(x) = -2(-4x^3 - 8x^2 + 35x + 75) = 8x^3 + 16x^2 - 70x - 150$

Thus, we have successfully expanded the polynomial function and rewritten it in its standard form:

$p(x) = 8x^3 + 16x^2 - 70x - 150$

This standard form representation provides valuable insights into the function's characteristics, particularly its degree and leading coefficient.

Identifying the Leading Term: The Key to End Behavior

Now that we have successfully expanded the polynomial function and expressed it in its standard form, the next crucial step in deciphering its end behavior is to identify the leading term. The leading term is the term with the highest power of the variable (x) in the polynomial. In our case, the polynomial function is:

$p(x) = 8x^3 + 16x^2 - 70x - 150$

By examining the terms of the polynomial, we can readily identify the leading term as:

$8x^3$

The leading term plays a pivotal role in determining the end behavior of the polynomial function. The degree of the leading term, which is the exponent of the variable, and the leading coefficient, which is the coefficient of the leading term, collectively dictate how the function behaves as x approaches positive and negative infinity.

In our specific example, the leading term is $8x^3$. The degree of this term is 3, which is an odd number, and the leading coefficient is 8, which is a positive number. These two pieces of information are the key to unlocking the secrets of the function's end behavior.

Unveiling the End Behavior: A Tale of Two Infinities

With the leading term firmly in our grasp, we are now poised to unveil the end behavior of the polynomial function. As we discussed earlier, the end behavior describes how the function behaves as the input variable (x) approaches positive and negative infinity. The degree and leading coefficient of the leading term hold the key to this behavior.

In our case, the leading term is $8x^3$. The degree is 3 (odd), and the leading coefficient is 8 (positive). Let's analyze how these characteristics influence the end behavior:

• As x approaches positive infinity (x → ∞):

  When x becomes an extremely large positive number, the leading term $8x^3$ will also become an extremely large positive number. This is because a positive number raised to an odd power remains positive, and multiplying it by a positive coefficient (8 in this case) further amplifies its positivity. Therefore, as x approaches positive infinity, the function p(x) also approaches positive infinity.

• As x approaches negative infinity (x → -∞):

  When x becomes an extremely large negative number, the leading term $8x^3$ will become an extremely large negative number. This is because a negative number raised to an odd power becomes negative, and multiplying it by a positive coefficient (8 in this case) preserves its negativity. Therefore, as x approaches negative infinity, the function p(x) also approaches negative infinity.

In summary, the end behavior of the polynomial function $p(x) = 8x^3 + 16x^2 - 70x - 150$ can be described as follows:

• As x gets larger and larger towards positive values, the value of the function also gets larger and larger towards positive values.
• As x gets larger and larger towards negative values, the value of the function also gets larger and larger towards negative values.

This behavior is characteristic of polynomial functions with odd degrees and positive leading coefficients. They tend to rise towards positive infinity on the right side of the graph and fall towards negative infinity on the left side of the graph.

Conclusion: The Significance of End Behavior

In this comprehensive exploration, we have successfully deciphered the end behavior of the polynomial function $p(x) = -2(3-x)(2x+5)^2$. By expanding the function, identifying its leading term, and analyzing its degree and leading coefficient, we were able to determine how the function behaves as x approaches positive and negative infinity. This understanding of end behavior is crucial for several reasons:

• Graphing the function: End behavior provides valuable information about the overall shape of the function's graph. It helps us sketch the general trend of the function as x moves away from the origin.
• Solving equations: End behavior can assist in determining the existence and approximate location of real roots (x-intercepts) of the polynomial equation.
• Modeling real-world phenomena: Polynomial functions are often used to model real-world phenomena, and understanding their end behavior allows us to make predictions about the long-term behavior of the system being modeled.

In conclusion, the end behavior of a polynomial function is a fundamental aspect of its characterization. By mastering the techniques discussed in this article, you will be well-equipped to analyze and understand the behavior of a wide range of polynomial functions, thereby expanding your mathematical prowess and problem-solving capabilities.

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