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Questions and Answers
What is the multiplicity of the zero at 1 for the polynomial P(x)?
What is the multiplicity of the zero at 1 for the polynomial P(x)?
Which of the following is a possible zero of the polynomial P(x) = x4 - x3 + 7x2 - 9x - 18?
Which of the following is a possible zero of the polynomial P(x) = x4 - x3 + 7x2 - 9x - 18?
After using synthetic division to find that 2 is a zero, what is the resulting factor of the polynomial P(x)?
After using synthetic division to find that 2 is a zero, what is the resulting factor of the polynomial P(x)?
What are the complex zeros of the polynomial P(x), as found in the provided content?
What are the complex zeros of the polynomial P(x), as found in the provided content?
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When factoring the expression x^3 + x^2 + 9x + 9 = 0, which factors should be considered?
When factoring the expression x^3 + x^2 + 9x + 9 = 0, which factors should be considered?
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According to Descartes's Rule of Signs, what can be inferred about the number of positive zeros of a polynomial function F(x)?
According to Descartes's Rule of Signs, what can be inferred about the number of positive zeros of a polynomial function F(x)?
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If a polynomial has 4 variations of sign in F(x), what are the possible numbers of positive zeros?
If a polynomial has 4 variations of sign in F(x), what are the possible numbers of positive zeros?
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What does the Conjugate Pairs Theorem state about the complex zeros of polynomials?
What does the Conjugate Pairs Theorem state about the complex zeros of polynomials?
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When applying Descarte’s Rule of Signs, how should a zero with a multiplicity of m be counted?
When applying Descarte’s Rule of Signs, how should a zero with a multiplicity of m be counted?
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What is a necessary condition for finding bounds on the real zeros of a polynomial function F(x)?
What is a necessary condition for finding bounds on the real zeros of a polynomial function F(x)?
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If a polynomial function has a leading term of $-5x^3$ and a leading coefficient of $-5$, what can be said about the behavior of its zeros?
If a polynomial function has a leading term of $-5x^3$ and a leading coefficient of $-5$, what can be said about the behavior of its zeros?
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If a polynomial's leading coefficient is positive and it has an increasing degree, what does this indicate about the upper and lower bounds of its real zeros?
If a polynomial's leading coefficient is positive and it has an increasing degree, what does this indicate about the upper and lower bounds of its real zeros?
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What signifies an upper bound on the zeros of F(x) when k is greater than 0?
What signifies an upper bound on the zeros of F(x) when k is greater than 0?
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What is the significance of identifying the number of variations in the sign of F(-x)?
What is the significance of identifying the number of variations in the sign of F(-x)?
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When k is less than 0, what condition must the numbers in the last row meet to confirm it is a lower bound?
When k is less than 0, what condition must the numbers in the last row meet to confirm it is a lower bound?
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How many roots does every nth-degree polynomial have according to the fundamental theorem of algebra?
How many roots does every nth-degree polynomial have according to the fundamental theorem of algebra?
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What should be done to find the upper bound using synthetic division?
What should be done to find the upper bound using synthetic division?
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In the synthetic division for determining bounds, what occurs first for a lower bound given k < 0?
In the synthetic division for determining bounds, what occurs first for a lower bound given k < 0?
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How can polynomials that have complex coefficients be categorized?
How can polynomials that have complex coefficients be categorized?
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Which statement is true regarding the final row of synthetic division when trying to find bounds?
Which statement is true regarding the final row of synthetic division when trying to find bounds?
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What can be said about the roots of a polynomial in the complex number system?
What can be said about the roots of a polynomial in the complex number system?
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Study Notes
Descartes’s Rule of Signs
- The number of positive zeros of a polynomial with real coefficients is either equal to the number of variations of sign in the polynomial or less than that number by an even integer.
- The number of negative zeros of a polynomial with real coefficients is either equal to the number of variations of sign in the polynomial with x replaced by -x or less than that number by an even integer.
- In Descartes’s Rule, a zero of multiplicity m should be counted as m zeros.
Rules for Bounds
- An upper bound for the zeros of a polynomial with real coefficients is a real number that is greater than or equal to the largest zero of the polynomial.
- A lower bound for the zeros of a polynomial with real coefficients is a real number that is less than or equal to the smallest zero of the polynomial.
Finding the Bounds
- To find an upper bound for the zeros of a polynomial, use synthetic division with a positive integer until the last row contains all positive numbers or zeros.
- To find a lower bound for the zeros of a polynomial, use synthetic division with a negative integer until the last row alternates in signs (zeros can be considered either positive or negative).
Fundamental Theorem of Algebra
- Every polynomial function with complex coefficients has at least one complex zero.
Complex Zeros
- Every complex polynomial can be factored into exactly n linear factors, where n is the degree of the polynomial.
- Every nth-degree complex polynomial equation has exactly n roots.
Conjugate Pairs Theorem
- If a complex number a + bi (where b ≠ 0) is a zero of a polynomial with real coefficients, then its conjugate a – bi is also a zero of the polynomial. This can be used to find the other zeros of the polynomial if one complex zero is known.
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