Problem 2
Question
(a) If a is a zero of the polynomial P, then ________ must be a factor of \(P(x)\). (b) If \(a\) is a zero of multiplicity \(m\) of the polynomial \(P,\) then ________ must be a factor of \(P(x)\) when we factor \(P\) completely
Step-by-Step Solution
Verified Answer
(a) \(x-a\); (b) \((x-a)^m\).
1Step 1: Understanding Polynomial Zeros
When we say that \(a\) is a zero of a polynomial \(P(x)\), it means that when we substitute \(a\) into \(P(x)\), it results in zero, i.e. \(P(a) = 0\). This concept is critical in determining factors of the polynomial.
2Step 2: Applying Factor Theorem (Part a)
According to the Factor Theorem, if \(a\) is a zero of \(P(x)\), then \(x - a\) must be a factor of the polynomial \(P(x)\). Essentially, \(P(x)\) can be expressed as \(P(x) = (x-a)Q(x)\) for some polynomial \(Q(x)\).
3Step 3: Incorporating Multiplicity (Part b)
If \(a\) is a zero of multiplicity \(m\), it means that \(a\) is a repeated zero, appearing \(m\) times. Consequently, \((x-a)^m\) must be a factor of \(P(x)\) in its completely factored form. This suggests \(P(x) = (x-a)^m R(x)\), where \(R(x)\) is some polynomial with \(R(a) eq 0\).
Key Concepts
Polynomial FactorizationFactor TheoremMultiplicity of Zeros
Polynomial Factorization
Polynomial factorization is a method of expressing a polynomial as a product of its factors. These factors are usually polynomials of lower degrees. When a polynomial is factored completely, it is broken down into the simplest polynomials that multiply together to form the original polynomial.
One primary application of polynomial factorization is finding the zeros of the polynomial. When you write a polynomial as a product of its factors, you can easily identify the values of the variable that make each factor zero. For example, if you have a polynomial P(x) and can express it as \(P(x) = (x - a)(x - b)(x - c)\), you know that the zeros of the polynomial are \(a, b, \) and \(c\).
Here are some key points about polynomial factorization:
One primary application of polynomial factorization is finding the zeros of the polynomial. When you write a polynomial as a product of its factors, you can easily identify the values of the variable that make each factor zero. For example, if you have a polynomial P(x) and can express it as \(P(x) = (x - a)(x - b)(x - c)\), you know that the zeros of the polynomial are \(a, b, \) and \(c\).
Here are some key points about polynomial factorization:
- Factorization simplifies complex polynomials, making them easier to work with and solve.
- When a polynomial is completely factored, all of its zeros are revealed.
- This approach is essential in calculus, especially for finding critical points, among others.
Factor Theorem
The Factor Theorem is a critical concept in polynomial algebra which connects the zeros of a polynomial with its factors. According to the Factor Theorem, if \(a\) is a zero of polynomial \(P(x)\), then \(x - a\) must be a factor of \(P(x)\).
This theorem essentially allows us to write the polynomial \(P(x)\) in a factored form, where \(P(x) = (x - a)Q(x)\). Here, \(Q(x)\) is another polynomial which, when multiplied by \(x - a\), gives us back the original polynomial \(P(x)\).
The Factor Theorem is especially useful when:
This theorem essentially allows us to write the polynomial \(P(x)\) in a factored form, where \(P(x) = (x - a)Q(x)\). Here, \(Q(x)\) is another polynomial which, when multiplied by \(x - a\), gives us back the original polynomial \(P(x)\).
The Factor Theorem is especially useful when:
- Checking if a given value is a zero of a polynomial.
- Factoring polynomials by using known zeros.
- Understanding the relationships between zeros and linear factors.
Multiplicity of Zeros
In polynomial expressions, the multiplicity of zeros refers to the number of times a particular zero appears as a root of the polynomial. If a zero has a high multiplicity, it impacts the graph of the polynomial at that zero's location, typically appearing as a flat point where the graph touches or crosses the x-axis.
If \(a\) is a zero of multiplicity \(m\) of a polynomial, it means that \(a\) shows up \(m\) times as a root. Consequently, the factor \((x-a)\) repeats \(m\) times in the complete factorization of the polynomial, expressed as \(P(x) = (x-a)^m R(x)\). Here, \(R(x)\) is a polynomial in which \(a\) is not a root.
Understanding multiplicity is useful because it:
If \(a\) is a zero of multiplicity \(m\) of a polynomial, it means that \(a\) shows up \(m\) times as a root. Consequently, the factor \((x-a)\) repeats \(m\) times in the complete factorization of the polynomial, expressed as \(P(x) = (x-a)^m R(x)\). Here, \(R(x)\) is a polynomial in which \(a\) is not a root.
Understanding multiplicity is useful because it:
- Helps in sketching the graph of the polynomial by identifying the behavior around its zeros.
- Shows the count of root appearances which can help in solving polynomial equations.
- Aids in deducing the degree of the polynomial based on its factors.
Other exercises in this chapter
Problem 2
Using Descartes' Rule of Signs, we can tell that the polynomial \(P(x)=x^{5}-3 x^{4}+2 x^{3}-x^{2}+8 x-8\) has _________, _________, or __________positive real
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If the rational function \(y=r(x)\) has the horizontal asymptote \(y=2,\) then \(y \rightarrow\) ________ as \(X \rightarrow \pm \infty\)
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(a) If we divide the polynomial \(P(x)\) by the factor \(x-c\) and we obtain a remainder of 0 , then we know that \(c\) is a ________of \(P\)
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Every polynomial has one of the following behaviors: $$\begin{array}{l}{\text { (i) } y \rightarrow \infty \text { as } x \rightarrow \infty \text { and } y \ri
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