Problem 32
Question
Factor the polynomial completely, and find all its zeros. State the multiplicity of each zero. $$P(x)=x^{5}+7 x^{3}$$
Step-by-Step Solution
Verified Answer
Zeros are \(x = 0\) (multiplicity 3), \(x = i\sqrt{7}\) and \(x = -i\sqrt{7}\).
1Step 1: Identify Common Factors
Examine the terms in the polynomial \(P(x) = x^5 + 7x^3\). Both terms share a factor of \(x^3\).
2Step 2: Factor out the Greatest Common Factor (GCF)
Factor \(x^3\) from each term in the polynomial: \(P(x) = x^3(x^2 + 7)\).
3Step 3: Identify and Factor Further if Possible
Analyze \(x^2 + 7\). Notice it's a sum of squares which doesn't factor further over the real numbers. So, \(x^3(x^2 + 7)\) is fully factored.
4Step 4: Determine the Zeros from the Factors
Set each factor equal to zero to find the zeros:- From \(x^3 = 0\), we get \(x = 0\), which has a multiplicity of 3.- From \(x^2 + 7 = 0\), solve for \(x\): \[ x^2 = -7 \quad \Rightarrow \quad x = \pm i\sqrt{7} \].These roots, \(x = i\sqrt{7}\) and \(x = -i\sqrt{7}\), each have a multiplicity of 1.
Key Concepts
Greatest Common Factor (GCF)Multiplicity of ZerosComplex Roots
Greatest Common Factor (GCF)
When tackling problems in polynomial factorization, one effective strategy is to start by identifying the greatest common factor, or GCF. The GCF is the largest expression that divides all terms in a polynomial without leaving a remainder. In simple terms, it's like finding the biggest piece of cake you can cut from each term evenly.
In this case, the polynomial we are trying to factor is \(P(x) = x^5 + 7x^3\). Looking at each term, we notice both contain a factor of \(x^3\). Extracting this common factor allows us to rewrite the polynomial as \( x^3(x^2 + 7) \). This extraction simplifies the polynomial and sets up the next steps for finding zeros and understanding their multiplicities.
When you factor out the GCF, it makes the polynomial simpler and often reveals more about its structure. For instance, if further factorization is to be done, it'll be much easier with a reduced-degree polynomial, as in our case resulting from the factorization.
In this case, the polynomial we are trying to factor is \(P(x) = x^5 + 7x^3\). Looking at each term, we notice both contain a factor of \(x^3\). Extracting this common factor allows us to rewrite the polynomial as \( x^3(x^2 + 7) \). This extraction simplifies the polynomial and sets up the next steps for finding zeros and understanding their multiplicities.
When you factor out the GCF, it makes the polynomial simpler and often reveals more about its structure. For instance, if further factorization is to be done, it'll be much easier with a reduced-degree polynomial, as in our case resulting from the factorization.
Multiplicity of Zeros
Multiplicity in the context of polynomials refers to the number of times a particular zero appears. It highlights how a zero affects the graph of a polynomial, such as how the graph touches or crosses the x-axis.
In our original polynomial, after factoring out the GCF, we have \(x^3(x^2 + 7)\). Solving \(x^3 = 0\), we find that \(x = 0\) is a root. This root, however, has a multiplicity of 3 because it corresponds to the factor \(x^3\). Multiplicity of 3 means the graph will "bounce" off the x-axis at this point. It represents a zero with the same value repeated three times.
For the other factor, \(x^2 + 7 = 0\), which doesn't factor into real numbers, solving gives us \(x = \pm i\sqrt{7}\). These complex roots have a multiplicity of 1 each, which means each occurs just once in the factorization process.
In our original polynomial, after factoring out the GCF, we have \(x^3(x^2 + 7)\). Solving \(x^3 = 0\), we find that \(x = 0\) is a root. This root, however, has a multiplicity of 3 because it corresponds to the factor \(x^3\). Multiplicity of 3 means the graph will "bounce" off the x-axis at this point. It represents a zero with the same value repeated three times.
For the other factor, \(x^2 + 7 = 0\), which doesn't factor into real numbers, solving gives us \(x = \pm i\sqrt{7}\). These complex roots have a multiplicity of 1 each, which means each occurs just once in the factorization process.
Complex Roots
Complex roots emerge when dealing with non-real solutions to a polynomial equation. Such roots occur in conjugate pairs when coefficients of the polynomial are real numbers. Complex numbers are those that include the imaginary unit \(i\) (\(\sqrt{-1}\)).
For \(P(x) = x^5 + 7x^3\), after removing the GCF, we examine \(x^2 + 7 = 0\). Setting this equation to zero and solving, we find \(x = \pm i\sqrt{7}\). These solutions are complex roots, as they involve the square root of a negative number, specifically \(i\sqrt{7}\) and \(-i\sqrt{7}\).
It's important to note that because of their imaginary nature, complex roots don't appear on the regular cartesian coordinate plane in the ways real roots do. However, they are essential for the full understanding of a polynomial's behavior, especially in systems or studies involving complex analysis or electrical engineering.
For \(P(x) = x^5 + 7x^3\), after removing the GCF, we examine \(x^2 + 7 = 0\). Setting this equation to zero and solving, we find \(x = \pm i\sqrt{7}\). These solutions are complex roots, as they involve the square root of a negative number, specifically \(i\sqrt{7}\) and \(-i\sqrt{7}\).
It's important to note that because of their imaginary nature, complex roots don't appear on the regular cartesian coordinate plane in the ways real roots do. However, they are essential for the full understanding of a polynomial's behavior, especially in systems or studies involving complex analysis or electrical engineering.
Other exercises in this chapter
Problem 31
Find the quotient and remainder using synthetic division. $$\frac{x^{3}-8 x+2}{x+3}$$
View solution Problem 32
Find all horizontal and vertical asymptotes (if any). $$r(x)=\frac{x^{3}+3 x^{2}}{x^{2}-4}$$
View solution Problem 32
Evaluate the expression and write the result in the form \(a+b i\) $$(2 i)^{4}$$
View solution Problem 32
Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=2 x^{3}+7 x^{2}+4 x-4$$
View solution