Problem 9
Question
State the degree of each polynomial equation. Find all of the real and imaginary roots to each equation. State the multiplicity of a root when it is greater than 1. $$x^{4}+2 x^{3}+x^{2}=0$$
Step-by-Step Solution
Verified Answer
The polynomial has a degree of 4. The roots are 0 (multiplicity 2) and -1 (multiplicity 2).
1Step 1: Identify the Degree of the Polynomial
The degree of a polynomial is the highest power of the variable present in the equation. Here, the highest power of x in the equation \( x^{4} + 2x^{3} + x^{2} = 0 \) is 4. Therefore, the degree of this polynomial is 4.
2Step 2: Factor the Polynomial
To find the roots, factor the polynomial equation. First, factor out the common term \( x^2 \): \[ x^2 (x^2 + 2x + 1) = 0 \]Next, factor the quadratic term \( x^2 + 2x + 1 \): \[ x^2 (x + 1)^2 = 0 \]
3Step 3: Find the Roots
Set each factor equal to zero to find the roots: \[ x^2 = 0 \] This gives root \( x = 0 \) with multiplicity 2. \[ (x + 1)^2 = 0 \] This gives root \( x = -1 \) with multiplicity 2.
4Step 4: State the Roots and Their Multiplicities
The polynomial \( x^4 + 2x^3 + x^2 = 0 \) has the following roots and multiplicities:- Root 0 with multiplicity 2- Root -1 with multiplicity 2
Key Concepts
Polynomial FactorizationRoots of PolynomialMultiplicity of Roots
Polynomial Factorization
Polynomial factorization involves breaking down a polynomial into simpler polynomials that, when multiplied together, give the original polynomial. This is crucial for finding the roots of the polynomial more easily.
For our exercise with the polynomial equation, we start by factoring out the greatest common term. In the equation given:
\[\begin{aligned}&x^4 + 2x^3 + x^2 = 0\end{aligned}\]
A common factor of \(\begin{aligned}&x^2\end{aligned}\) is identifiable. Thus, we factor it out:
\[\begin{aligned}&x^2 (x^2 + 2x + 1) = 0\end{aligned}\]
The next step is to factor the quadratic expression \(\begin{aligned}&(x^2 + 2x + 1)\end{aligned}\). This represents a perfect square trinomial and can be factored easily as:
\[\begin{aligned}&(x + 1)^2\end{aligned}\]
This simplifies the original polynomial into:
\[\begin{aligned}&x^2 (x + 1)^2 = 0\end{aligned}\]
Factorization simplifies the polynomial, making it easier to identify the roots and their multiplicities.
For our exercise with the polynomial equation, we start by factoring out the greatest common term. In the equation given:
\[\begin{aligned}&x^4 + 2x^3 + x^2 = 0\end{aligned}\]
A common factor of \(\begin{aligned}&x^2\end{aligned}\) is identifiable. Thus, we factor it out:
\[\begin{aligned}&x^2 (x^2 + 2x + 1) = 0\end{aligned}\]
The next step is to factor the quadratic expression \(\begin{aligned}&(x^2 + 2x + 1)\end{aligned}\). This represents a perfect square trinomial and can be factored easily as:
\[\begin{aligned}&(x + 1)^2\end{aligned}\]
This simplifies the original polynomial into:
\[\begin{aligned}&x^2 (x + 1)^2 = 0\end{aligned}\]
Factorization simplifies the polynomial, making it easier to identify the roots and their multiplicities.
Roots of Polynomial
The roots of a polynomial are the values of x that satisfy the equation, making it equal to zero. For the factored polynomial:
\[\begin{aligned}&x^2 (x + 1)^2 = 0\end{aligned}\]
We find the roots by setting each factor equal to zero:
\[\begin{aligned}&x^2 = 0\end{aligned}\]
This gives us the root \(\begin{aligned}&x = 0\end{aligned}\), which is repeated two times because it is a square term.
Similarly, for the factor:
\[\begin{aligned}&(x + 1)^2 = 0\end{aligned}\]
We get the root \(\begin{aligned}&x = -1\end{aligned}\), also repeated two times.
Hence, the polynomial \(\begin{aligned}&x^4 + 2x^3 + x^2 = 0\end{aligned}\) has:
Both these roots appear with a multiplicity of 2.
\[\begin{aligned}&x^2 (x + 1)^2 = 0\end{aligned}\]
We find the roots by setting each factor equal to zero:
\[\begin{aligned}&x^2 = 0\end{aligned}\]
This gives us the root \(\begin{aligned}&x = 0\end{aligned}\), which is repeated two times because it is a square term.
Similarly, for the factor:
\[\begin{aligned}&(x + 1)^2 = 0\end{aligned}\]
We get the root \(\begin{aligned}&x = -1\end{aligned}\), also repeated two times.
Hence, the polynomial \(\begin{aligned}&x^4 + 2x^3 + x^2 = 0\end{aligned}\) has:
- Root \(\begin{aligned}&x = 0\end{aligned}\)
- Root \(\begin{aligned}&x = -1\end{aligned}\)
Both these roots appear with a multiplicity of 2.
Multiplicity of Roots
Multiplicity refers to the number of times a particular root appears in a polynomial equation. If a root satisfies the equation at multiple counts, e.g., it appears twice, thrice, or more, its multiplicity is greater than 1.
For the given polynomial equation:
\[\begin{aligned}&x^4 + 2x^3 + x^2 = 0\end{aligned}\]
After factorizing, we end up with:
\[\begin{aligned}&x^2 (x + 1)^2 = 0\end{aligned}\]
The root \(\begin{aligned}&x = 0\end{aligned}\) corresponds to \(\begin{aligned}&x^2\end{aligned}\), indicating it appears twice.
Similarly, the root \(\begin{aligned}&x = -1\end{aligned}\) corresponds to \(\begin{aligned}&(x + 1)^2\end{aligned}\), indicating it also appears twice.
Therefore,
For the given polynomial equation:
\[\begin{aligned}&x^4 + 2x^3 + x^2 = 0\end{aligned}\]
After factorizing, we end up with:
\[\begin{aligned}&x^2 (x + 1)^2 = 0\end{aligned}\]
The root \(\begin{aligned}&x = 0\end{aligned}\) corresponds to \(\begin{aligned}&x^2\end{aligned}\), indicating it appears twice.
Similarly, the root \(\begin{aligned}&x = -1\end{aligned}\) corresponds to \(\begin{aligned}&(x + 1)^2\end{aligned}\), indicating it also appears twice.
Therefore,
- Root \(\begin{aligned}&x = 0\end{aligned}\) has a multiplicity of 2.
- Root \(\begin{aligned}&x = -1\end{aligned}\) also has a multiplicity of 2.
Other exercises in this chapter
Problem 8
State the degree of each polynomial equation. Find all of the real and imaginary roots to each equation. State the multiplicity of a root when it is greater tha
View solution Problem 8
Determine whether each given value of x is a zero of the given function. See Example 1. $$x=3, \quad P(x)=x-3$$
View solution Problem 9
Determine whether each given value of x is a zero of the given function. See Example 1. $$x=7, \quad P(x)=x+7$$
View solution Problem 9
Let \(f(x)=x^{4}-1, g(x)=x^{3}-3 x^{2}+5,\) and \(h(x)=4 x^{4}-\) \(3 x^{2}+3 x-1 .\) Find the following function values by using two different methods. See Exa
View solution