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Chapter 3: Problem 4

Determine the zeros and multiplicities for each polynomial. $$f(x)=5 x(x+6)^{2}(5 x-1)^{5}$$

Short Answer

Expert verified
Zeros: \(0\) (multiplicity 1), \(-6\) (multiplicity 2), \(\frac{1}{5}\) (multiplicity 5).

Step by step solution

01

Identify the polynomial factors

Consider each factor of the polynomial separately: \(5x\), \((x+6)^{2}\), and \((5x-1)^{5}\).
02

Find the zeros of each factor

Set each factor equal to zero and solve for \(x\): 1. For \(5x = 0\), the zero is \(x = 0\). 2. For \((x+6)^{2} = 0\), the zero is \(x = -6\). 3. For \((5x-1)^{5} = 0\), the zero is \(x = \frac{1}{5}\).
03

Determine the multiplicities

Examine the exponents of each factor to determine the multiplicities: 1. The zero \(x = 0\) is from the factor \(5x\), and has a multiplicity of 1. 2. The zero \(x = -6\) is from the factor \((x+6)^{2}\), and has a multiplicity of 2. 3. The zero \(x = \frac{1}{5}\) is from the factor \((5x-1)^{5}\), and has a multiplicity of 5.

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Key Concepts

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

Finding Zeros
To find the zeros of a polynomial, you need to set the polynomial equal to zero and solve for the variable. This process can seem daunting, but breaking it down into manageable steps can help.

Let's consider the polynomial provided: f(x)=5 x (x+6)^{2} (5 x-1)^{5}.

Notice that this polynomial is already factored, which makes it easier to identify and solve each factor separately.
Polynomial Factors
Each factor of a polynomial can give you a zero. Here, the polynomial is already neatly factored into three parts: 5x, (x+6)^{2}, and (5x-1)^{5}.

Solving each part helps us to find the corresponding zero:
  • For 5x = 0, we solve for x to get x = 0.
  • For (x+6)^{2} = 0, we solve by taking the square root of both sides to get x = -6.
  • For (5x-1)^{5} = 0, we solve by isolating x to get x = 1/5.

By breaking down the polynomial into its factors, you can more easily tackle each part and find the zeros efficiently.
Multiplicity
Multiplicity refers to the number of times a particular zero appears in a polynomial. In our example, each factor comes with an exponent, indicating the multiplicity of the zero.

Here鈥檚 a breakdown:
  • The zero x = 0 comes from the factor 5x, which has an exponent of 1. Therefore, this zero has a multiplicity of 1.
  • The zero x = -6 comes from the factor (x+6)^{2}, which has an exponent of 2. This means x = -6 has a multiplicity of 2.
  • The zero x = 1/5 comes from the factor (5x-1)^{5}, which is raised to the power of 5, giving it a multiplicity of 5.
Understanding multiplicities is key for graphing and analyzing polynomials as they indicate the behavior of the polynomial at each zero.

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Most popular questions from this chapter

a. Factor the polynomial over the set of real numbers. b. Factor the polynomial over the set of complex numbers. $$f(x)=x^{4}+2 x^{3}+x^{2}+8 x-12$$

The procedure to solve a polynomial or rational inequality may be applied to all inequalities of the form \(f(x)>0, f(x)<0,\) \(f(x) \geq 0,\) and \(f(x) \leq 0 .\) That is, find the real solutions to the related equation and determine restricted values of \(x .\) Then determine the sign of \(f(x)\) on each interval defined by the boundary points. Use this process to solve the inequalities. $$ \frac{1}{\sqrt{x-3}-5} \leq 0 $$

Find all fourth roots of 1 , by solving the equation \(x^{4}=1\).

The number of adults in U.S. prisons and jails for the years \(1980-2008\) is shown in the graph. (Source: U.S. Department of Justice, www.justice.gov) The variable \(t\) represents the number of years since 1980 . The function defined by \(P(t)=-0.091 t^{3}+3.48 t^{2}+15.4 t+335\) represents the number of adults in prison \(P(t)\) (in thousands). The function defined by \(J(t)=23.0 t+159\) represents the number of adults in jail \(J(t)\) (in thousands). a. Write the function defined by \(N(t)=(P+J)(t)\) and interpret its meaning in context. b. Write the function defined by \(R(t)=\left(\frac{J}{N}\right)(t)\) and interpret its meaning in the context of this problem. c. Evaluate \(R(25)\) and interpret its meaning in context. Round to 3 decimal places.

Explain why a polynomial with real coefficients of degree 3 must have at least one real zero.
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