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Chapter 4: Problem 16

Find a polynomial function of degree 5 with \(-\frac{1}{2}\) as a zero of multiplicity \(2,0\) as a zero of multiplicity 1 and 1 as a zero of multiplicity 2.

Short Answer

Expert verified
The polynomial function is \f(x) = k(x^5 - x^4 - \frac{3}{4} x^3 + \frac{3}{4} x^2 - \frac{1}{4} x)\.

Step by step solution

01

- Identify the roots and their multiplicities

The roots provided are \(-\frac{1}{2}\) with multiplicity 2, 0 with multiplicity 1, and 1 with multiplicity 2.
02

- Write the polynomial factors

Express the polynomial as the product of its factors based on the known roots and their multiplicities. The factors are \(x + \frac{1}{2}\)^2, \(x\), and \(x - 1\)^2.
03

- Form the polynomial

Combine the factors into a single polynomial function. This gives \[f(x) = k \big(x + \frac{1}{2}\big)^2 x (x - 1)^2 \] where \(k\) is a constant.
04

- Expand the polynomial

Expand each binomial and then multiply all terms together: \[(x + \frac{1}{2})^2 = x^2 + x + \frac{1}{4}\], \(x\), and \[(x - 1)^2 = x^2 - 2x + 1\]. Multiply them to get the polynomial in standard form.
05

- Find the expanded form

Multiply the three parts together: \[f(x) = k (x^2 + x + \frac{1}{4}) x (x^2 - 2x + 1) = k(x^5 - x^4 - \frac{3}{4}x^3 + \frac{3}{4}x^2 - \frac{1}{4}x) \].

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Key Concepts

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

Degree of Polynomial
The degree of a polynomial is a critical aspect in understanding polynomial functions. It's defined as the highest power of the variable in the polynomial equation.
For example, if you have a polynomial function like \[ f(x) = x^5 - 4x^3 + 3x + 7 \]
the degree is 5 because the highest exponent of x is 5.
In our exercise, we needed a polynomial of degree 5.
We identified the zeros and their multiplicities to determine the polynomial's form. By multiplying all the factors, we ensured the final polynomial had a degree 5, since the sum of the multiplicities (2+1+2) adds up to the degree.
Roots and Multiplicities
Roots (or zeros) of a polynomial are the values of x that make the polynomial equal to zero.
Multiplicity refers to how often a particular root appears. For example, if \[ (x + 1)^2 \] is a factor, then -1 is a root with multiplicity 2.
In our problem, the roots given were \[ -\frac{1}{2}, 0, \text{ and } 1 \]. Here, \[ -\frac{1}{2} \text{ and } 1 \] have a multiplicity of 2, and 0 has a multiplicity of 1. This means the polynomial touches the x-axis but does not cross it at \[ -\frac{1}{2} \text{and } 1 \], and crosses it at 0.
Polynomial Expansion
Polynomial expansion involves expressing a polynomial in its extended form.
This means you start with binomial factors and expand them to form a singular polynomial equation.
In our exercise, we started with the factors \[ (x + \frac{1}{2})^2, x, (x - 1)^2 \].
Our goal was to expand these binomials and then multiply them together.
For example, we expanded \[ (x + \frac{1}{2})^2 \] to \[ x^2 + x + \frac{1}{4} \], and \[ (x - 1)^2 \] to \[ x^2 - 2x + 1 \].
This method allows us to see all individual terms in the polynomial before combining them into a standard form.
Binomial Multiplication
Binomial multiplication is crucial for expanding polynomial factors.
It involves multiplying the terms of two binomials together to form a new polynomial.
For example, multiply \[ (x + a)(x + b) \], which results in \[ x^2 + (a + b)x + ab \].
In our exercise, we multiplied our expanded factors:
\[ (x^2 + x + \frac{1}{4}) x (x^2 - 2x + 1) \]
Step by step:
- First, we multiplied \[ (x^2 + x + \frac{1}{4}) \] by \[ x \]
- Then we multiplied the resulting polynomial by \[ (x^2 - 2x + 1) \]
This process ensures each term is correctly calculated and combined, leading to the final polynomial form:
\[ x^5 - x^4 - \frac{3}{4}x^3 + \frac{3}{4}x^2 - \frac{1}{4}x \].

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Most popular questions from this chapter

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. $$f(x)=\frac{x^{2}-x-2}{x+2}$$

Solve. $$2|x|^{2}-|x|+2 \leq 5$$

Solve. Find exact solutions. $$7 x^{2}+4 x=3$$

Factor the polynomial function \(f(x) .\) Then solve the equation \(f(x)=0\) $$f(x)=x^{4}+11 x^{3}+41 x^{2}+61 x+30$$

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? $$R(x)=3 x^{5}-5 x^{3}-4 x$$
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