Problem 43
Question
Find an expression for a polynomial \(p(x)\) with real coefficients that satisfies the given conditions. There may be more than one possible answer. Degree \(4 ; x=1\) and \(x=\frac{1}{3}\) are both zeros of multiplicity 2
Step-by-Step Solution
Verified Answer
\(\ p(x) = x^4 -\frac{8}{3}x^3 +\frac{14}{9}x^2 -\frac{4}{3}x + \frac{1}{9}\)
1Step 1: Write down the factors
To find the polynomial, use the fact that if 'a' is a zero of the polynomial, then '(x - a)' is a factor of the polynomial. In this case, both '1' and '\(\frac{1}{3}\)' are zeros of multiplicity 2, thus \((x - 1)^2\) and \(\(x - \frac{1}{3})^2\) are factors of the polynomial.
2Step 2: Create the polynomial
The polynomial \(p(x)\) will then be the product of these two factors. This gives us \( p(x) = (x - 1)^2 \cdot \(x - \frac{1}{3})^2\).
3Step 3: Simplify
To simplify this polynomial, expand \( (x - 1)^2\) and \(\(x -\frac{1}{3})^2\), and then multiply the results. This gives us \( p(x) = (x^2 - 2x + 1) \cdot \(x^2 -\frac{2}{3}x +\frac{1}{9})\). In order to further expand this, distribute the terms of the first equation into the second equation, giving us four terms to add together. After simplifying, we find \(p(x) = x^4 -\frac{8}{3}x^3 +\frac{14}{9}x^2 -\frac{4}{3}x + \frac{1}{9}\).
Key Concepts
Polynomial ZerosMultiplicity of ZerosSimplifying Polynomials
Polynomial Zeros
Polynomial zeros, also known as roots or solutions, are the values for which the polynomial function yields zero. To understand polynomial zeros, suppose we have a function \(p(x)\). If after substituting a value \(a\) into the polynomial we obtain \(p(a) = 0\), then \(a\) is called a zero of the polynomial. For a polynomial of degree 4, such as in our exercise, there can be up to four real zeros.
When given a zero of the polynomial, we can write down a corresponding factor. For instance, if \(x = 1\) is a zero, then \(x - 1\) is a factor of the polynomial. Factors are thus the building blocks of a polynomial, and every zero can be directly associated with a factor.
When given a zero of the polynomial, we can write down a corresponding factor. For instance, if \(x = 1\) is a zero, then \(x - 1\) is a factor of the polynomial. Factors are thus the building blocks of a polynomial, and every zero can be directly associated with a factor.
Multiplicity of Zeros
The multiplicity of a zero refers to the number of times that zero appears as a root of the polynomial. In other words, it's the number of times a particular factor is repeated in the product that gives the polynomial. For instance, a zero of multiplicity 2 means that its corresponding factor will appear twice in the polynomial's expression.
If \(p(x)\) has a zero of multiplicity 2 at \(x = a\), the factor \(x - a\) will be squared in the polynomial expression, represented as \( (x - a)^2\). In the example from the exercise, both \(x = 1\) and \(x = \frac{1}{3}\) have a multiplicity of 2, which leads to the polynomial having the factors \( (x - 1)^2\) and \( (x - \frac{1}{3})^2\). High multiplicity affects the graph of the polynomial; it will touch the x-axis and turn around at the zero of a higher multiplicity instead of crossing it.
If \(p(x)\) has a zero of multiplicity 2 at \(x = a\), the factor \(x - a\) will be squared in the polynomial expression, represented as \( (x - a)^2\). In the example from the exercise, both \(x = 1\) and \(x = \frac{1}{3}\) have a multiplicity of 2, which leads to the polynomial having the factors \( (x - 1)^2\) and \( (x - \frac{1}{3})^2\). High multiplicity affects the graph of the polynomial; it will touch the x-axis and turn around at the zero of a higher multiplicity instead of crossing it.
Simplifying Polynomials
Simplifying polynomials is the process of expressing a polynomial in its simplest form, where all like terms have been combined and the expression has been made as compact as possible. Simplification may involve expanding products of binomials, combining like terms, and reducing expressions. In our example, simplifying the polynomial \( p(x) = (x - 1)^2 \cdot (x - \frac{1}{3})^2\) requires expanding the squared factors and multiplying the resulting expressions.
The expansion of the factors allows us to see the 'hidden' terms that would eventually combine to form the simplified polynomial. After distributing and combining like terms, the polynomial is expressed in standard form, \( a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0\), where the terms are ordered by decreasing powers of \(x\). This is crucial not only for understanding the structure of the polynomial but also for performing operations like differentiation, integration, and solving polynomial equations.
The expansion of the factors allows us to see the 'hidden' terms that would eventually combine to form the simplified polynomial. After distributing and combining like terms, the polynomial is expressed in standard form, \( a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0\), where the terms are ordered by decreasing powers of \(x\). This is crucial not only for understanding the structure of the polynomial but also for performing operations like differentiation, integration, and solving polynomial equations.
Other exercises in this chapter
Problem 43
Use Descartes' Rule of Signs to determine the number of positive and negative zeros of \(p\). You need not find the zeros. $$p(x)=4 x^{4}-5 x^{3}+6 x-3$$
View solution Problem 43
Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$h(x)=\frac{-2 x}{(x-1)(x+4)}$$
View solution Problem 43
Let \(x-\frac{1}{2}\) be a factor of a polynomial function \(p(x) .\) Find \(p\left(\frac{1}{2}\right).\)
View solution Problem 43
Sketch the polynomial function using transformations. $$h(x)=-\frac{1}{2}(x+1)^{3}-2$$
View solution