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Magazine Chapter 3: Problem 20
Factor the polynomial completely and find all its zeros. State the multiplicity of each zero. $$Q(x)=x^{4}-625$$
Short Answer
Expert verified
Zeros: 5, -5, 5i, -5i (each with multiplicity 1).
Step by step solution
01
Recognize the Polynomial Form
The given polynomial is \[ Q(x) = x^4 - 625 \]Recognize that this is a difference of squares. Recall the identity \( a^2 - b^2 = (a - b)(a + b) \). Here, \( a^2 = x^4 \) and \( b^2 = 625 \), i.e., \( b = 25 \). Next, let's express \( x^4 \) as \( (x^2)^2 \).
02
Apply the Difference of Squares Formula
Using the identity from Step 1, express the polynomial as a product of two binomials:\[ Q(x) = (x^2)^2 - 25^2 = (x^2 - 25)(x^2 + 25) \]
03
Further Factor the Difference of Squares
The term \( x^2 - 25 \) can also be further factored using the same difference of squares identity:\[ x^2 - 25 = (x - 5)(x + 5) \]
04
Handle the Sum of Squares
The term \( x^2 + 25 \) is a sum of squares, and thus cannot be factored over the real numbers. Therefore, it remains as is.
05
Combine All Factors
Combine the factored components from the earlier steps, obtaining:\[ Q(x) = (x - 5)(x + 5)(x^2 + 25) \]
06
Solve for Zeros
Set each factor equal to zero to find the roots:1. \( x - 5 = 0 \Rightarrow x = 5 \)2. \( x + 5 = 0 \Rightarrow x = -5 \)3. \( x^2 + 25 = 0 \Rightarrow x^2 = -25 \Rightarrow x = \pm 5i \)
07
Determine Multiplicity of Zeros
Each zero of the polynomial appears once in the factored form. Therefore, each zero has a multiplicity of 1.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Difference of Squares
In polynomial factorization, a common method is the difference of squares. This is a simple yet powerful tool.
The difference of squares formula states:
For instance, in the polynomial \( Q(x) = x^4 - 625 \), we identify two perfect squares:
Moreover, we see that \( x^2 - 25 \) is itself another difference of squares, which can be further factored:
The difference of squares formula states:
- \( a^2 - b^2 = (a-b)(a+b) \)
For instance, in the polynomial \( Q(x) = x^4 - 625 \), we identify two perfect squares:
- \(a^2 = x^4 \), which can be rewritten as \((x^2)^2 \)
- \(b^2 = 625 \), where \(b = 25 \)
- \( (x^2)^2 - 25^2 = (x^2 - 25)(x^2 + 25) \)
Moreover, we see that \( x^2 - 25 \) is itself another difference of squares, which can be further factored:
- \( x^2 - 25 = (x-5)(x+5) \)
Sum of Squares
The sum of squares, although similar sounding, behaves differently from the difference of squares.
The formula \( a^2 + b^2 \) does not factor into real numbers easily. Instead, it appears as a complex expression. Unlike the difference of squares, the sum of squares remains prime over the real number system.
In our example, the term \( x^2 + 25 \) is a sum of squares.
To factor this term requires complex numbers, which introduces the concept of imaginary numbers:
The formula \( a^2 + b^2 \) does not factor into real numbers easily. Instead, it appears as a complex expression. Unlike the difference of squares, the sum of squares remains prime over the real number system.
In our example, the term \( x^2 + 25 \) is a sum of squares.
- This cannot be factored over the real numbers.
To factor this term requires complex numbers, which introduces the concept of imaginary numbers:
- Equating \( x^2 + 25 = 0 \) solves to \( x^2 = -25 \).
- \( x = \pm 5i \)
Multiplicity of Zeros
Finding the zeroes of a polynomial means identifying the values of \( x \) that make the polynomial equal zero.
Once a polynomial is factored, its roots (or zeros) become clear by setting the factors equal to zero.
In our factorization of \( Q(x) = (x-5)(x+5)(x^2 + 25) \), we find the zeros:
Multiplicity helps us understand the behavior of graphs at these zeros:
Once a polynomial is factored, its roots (or zeros) become clear by setting the factors equal to zero.
- Each solution is a zero of the polynomial.
In our factorization of \( Q(x) = (x-5)(x+5)(x^2 + 25) \), we find the zeros:
- \( x = 5 \)
- \( x = -5 \)
- \( x = 5i \) and \( x = -5i \)
Multiplicity helps us understand the behavior of graphs at these zeros:
- A multiplicity of 1 means the graph crosses the x-axis at that point.
- A higher multiplicity would indicate the graph touches but does not cross the x-axis.
