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Chapter 4: Problem 15

Find a polynomial function of degree 4 with \(-1\) as a zero of multiplicity 3 and 0 as a zero of multiplicity 1.

Short Answer

Expert verified
The polynomial function is \( P(x) = x^4 + 3x^3 + 3x^2 + x \).

Step by step solution

01

Identify the Multiplicities

Given the polynomial function's zeros and their multiplicities: -1 (with multiplicity 3) and 0 (with multiplicity 1). This means the polynomial will have factors corresponding to these zeros raised to their respective multiplicities.
02

Write the Polynomial Factors

For a zero \(-1\) with multiplicity 3, the factor will be \( (x + 1)^3 \). For a zero 0 with multiplicity 1, the factor will be \( x \). Thus, the polynomial can be written as: \[ P(x) = a(x + 1)^3 \times x \], where \( a \) is a constant.
03

Form the Polynomial Function

Combine the factors to form the polynomial function: \[ P(x) = a(x + 1)^3 \times x \]
04

Expand the Polynomial (Optional)

To expand the polynomial: \((x + 1)^3 = x^3 + 3x^2 + 3x + 1\). Thus: \[ P(x) = a(x^4 + 3x^3 + 3x^2 + x) \]
05

Choose the Constant 'a'

If no specific value for \( a \) is given, we can assume \( a = 1 \) for simplicity. Thus, \[ P(x) = x^4 + 3x^3 + 3x^2 + x \]

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Key Concepts

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

Multiplicity
Multiplicity of a zero in a polynomial function refers to the number of times a particular zero appears in the polynomial. For instance, if a polynomial has a zero at \(-1\) with multiplicity 3, this means that \(-1\) is a root of the polynomial repeated 3 times. This is generally expressed as \( (x+1)^3 \). Similarly, if zero is a root with multiplicity 1, it is written as \( x \). Typically, higher multiplicity zeros mean the graph will touch the x-axis at that zero without crossing it.
Zero of a Polynomial
A zero (or root) of a polynomial function is a value of \( x \) that makes the function equal to zero. For example, in the polynomial given in the exercise, we see zeros at \(-1\) and \( 0\). This means if you substitute \(-1\) or \(0\) into the polynomial function, the result will be zero. Zeros help you understand where the graph of the polynomial will intersect the x-axis.
Polynomial Expansion
Polynomial expansion involves distributing and combining like terms to transform a polynomial into its standard form. In this exercise, we start with \( (x + 1)^3 \times x \) and expand \( (x + 1)^3 \) using binomial expansion formula which results in \( x^3 + 3x^2 + 3x + 1 \). Then we multiply the expanded form by \( x \) to get \( x^4 + 3x^3 + 3x^2 + x \). This expanded form shows the polynomial in its simplest terms.
Degree of a Polynomial
The degree of a polynomial is the highest power of the variable \( x \) in the polynomial expression. It tells us the maximum number of zeros and the basic shape of the graph. For example, the polynomial \( x^4 + 3x^3 + 3x^2 + x \) has a degree of 4 because the highest power of \( x \) is \( 4 \). Higher-degree polynomials have more complex graphs with more twists and turns.

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Most popular questions from this chapter

List the critical values of the related function. Then solve the inequality. $$\frac{1}{x-3} \leq 0$$

Make a hand-drawn graph. Be sure to label all the asymptotes. List the domain and the \(x\) -intercepts and the \(y\) -intercepts. Check your work using \(a\) graphing calculator. $$f(x)=\frac{5 x^{4}}{x^{4}+1}$$

What does Descartes' rule of signs tell you about the number of positive real zeros and the number of negative real zeros of the function? $$C(x)=7 x^{6}+3 x^{4}-x-10$$

Solve. $$x^{5}+24 > 3 x^{3}+8 x^{2}$$

Graph the function. $$f(x)=\frac{2 x^{3}+x^{2}-8 x-4}{x^{3}+x^{2}-9 x-9}$$
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