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Chapter 3: Problem 7

Find the degree, the leading term, the leading coefficient, the constant term and the end behavior of the given polynomial. \(P(x)=(x-1)(x-2)(x-3)(x-4)\)

Short Answer

Expert verified
Degree: 4, Leading Term: \(x^4\), Leading Coefficient: 1, Constant Term: 24, End Behavior: Both \(x\to\infty\) and \(x\to- \infty\), \(P(x)\to\infty\).

Step by step solution

01

Expand the Polynomial

First, we need to expand the given polynomial expression. We have:\(P(x) = (x - 1)(x - 2)(x - 3)(x - 4)\).Start by expanding the first two terms: \((x - 1)(x - 2) = x^2 - 3x + 2\).Next, expand the last two terms: \((x - 3)(x - 4) = x^2 - 7x + 12\).Finally, multiply these two resulting quadratics together:\((x^2 - 3x + 2)(x^2 - 7x + 12)\). Execute the multiplication to get:\(x^4 - 10x^3 + 35x^2 - 50x + 24\).
02

Identify the Degree

The degree of a polynomial is the highest power of the variable in the expanded polynomial. Here, the highest power of \(x\) is 4, so the degree of the polynomial \(P(x)\) is 4.
03

Determine the Leading Term

The leading term of a polynomial is the term with the highest power of the variable. From the expansion, the leading term of \(P(x)\) is \(x^4\).
04

Find the Leading Coefficient

The leading coefficient is the coefficient of the leading term. Since the leading term is \(x^4\), the leading coefficient is 1.
05

Identify the Constant Term

The constant term is the term in the polynomial that does not have any variables, i.e., the numerical term. In the expanded form, the constant term of \(P(x)\) is 24.
06

Describe the End Behavior

For a polynomial of degree 4 with a positive leading coefficient (1), as \(x\) approaches infinity, \(P(x)\) also approaches infinity. Similarly, as \(x\) approaches negative infinity, \(P(x)\) also approaches infinity. This behavior is typical for even-degree polynomials with positive leading coefficients.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Leading Term
The leading term of a polynomial is a crucial component when analyzing its structure. It is the term that contains the highest power of the variable, and it gives significant information about the polynomial. In our example, after expanding the given polynomial \(P(x)=(x-1)(x-2)(x-3)(x-4)\), we have the term \(x^4\), which is the leading term here. Identifying the leading term is essential as it determines the degree of the polynomial, which hints at many features of the polynomial including its end behavior.
Leading Coefficient
Alongside the leading term, the leading coefficient plays a pivotal role. It is the coefficient of the leading term, providing insights into the numerical aspect of the polynomial's growth. In the case of \(P(x)=x^4 - 10x^3 + 35x^2 - 50x + 24\), the leading coefficient is 1. This number is key, as it determines the initial "speed" or "amplitude" at which the polynomial grows or shrinks. The sign of this coefficient (positive or negative) significantly impacts the end behavior, particularly whether the polynomial's tails go to positive or negative infinity.
Constant Term
The constant term in a polynomial is the term without any variable factors, commonly seen as the "standalone" number in an equation. It's crucial because it often represents the polynomial's y-intercept, providing specific values when the variable equals zero. Here, in \(P(x)=x^4 - 10x^3 + 35x^2 - 50x + 24\), the constant term is 24. Although it does not influence the end behavior or the degree, it is important for particular solutions and intersections with axes in coordinate geometry.
End Behavior of Polynomials
The end behavior of a polynomial refers to how the graph behaves as the variable approaches positive or negative infinity. This is predominantly affected by both the degree and the leading coefficient. For even-degree polynomials with positive leading coefficients, such as the given polynomial \(x^4\), both "ends" of the graph will point upwards as \(x\) goes to both negative and positive infinity. This typical characteristic is a useful predictor of the graph's shape in these limitless zones, differentiating it from odd-degree or negative coefficient polynomials which behave distinctly.

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Most popular questions from this chapter

Find the real zeros of \(f(x)=x^{3}-\frac{1}{12} x^{2}-\frac{7}{72} x+\frac{1}{72}\) by first finding a polynomial \(q(x)\) with integer coefficients such that \(q(x)=N \cdot f(x)\) for some integer \(N\). (Recall that the Rational Zeros Theorem required the polynomial in question to have integer coefficients.) Show that \(f\) and \(q\) have the same real zeros.

You are given a polynomial and one of its zeros. Use the techniques in this section to find the rest of the real zeros and factor the polynomial. \(2 x^{3}-x^{2}-10 x+5, \quad c=\frac{1}{2}\)

While developing their newest game, Sasquatch Attack!, the makers of the PortaBoy (from Example 2.1.5) revised their cost function and now use \(C(x)=.03 x^{3}-4.5 x^{2}+225 x+250\), for \(x \geq 0\). As before, \(C(x)\) is the cost to make \(x\) PortaBoy Game Systems. Market research indicates that the demand function \(p(x)=-1.5 x+250\) remains unchanged. Use a graphing utility to find the production level \(x\) that maximizes the profit made by producing and selling \(x\) PortaBoy game systems.

Find the real zeros of the given polynomial and their corresponding multiplicities. Use this information along with a sign chart to provide a rough sketch of the graph of the polynomial. Compare your answer with the result from a graphing utility. \(P(x)=(x-1)(x-2)(x-3)(x-4)\)

An electric circuit is built with a variable resistor installed. For each of the following resistance values (measured in kilo-ohms, \(k \Omega\) ), the corresponding power to the load (measured in milliwatts, \(m W\) ) is given in the table below. 17 $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \text { Resistance: }(k \Omega) & 1.012 & 2.199 & 3.275 & 4.676 & 6.805 & 9.975 \\ \hline \text { Power: }(m W) & 1.063 & 1.496 & 1.610 & 1.613 & 1.505 & 1.314 \\\ \hline \end{array} $$ (a) Make a scatter diagram of the data using the Resistance as the independent variable and Power as the dependent variable. (b) Use your calculator to find quadratic (2nd degree), cubic (3rd degree) and quartic (4th degree) regression models for the data and judge the reasonableness of each. (c) For each of the models found above, find the predicted maximum power that can be delivered to the load. What is the corresponding resistance value? (d) Discuss with your classmates the limitations of these models - in particular, discuss the end behavior of each.
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