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Questions and Answers
The number of positive real roots of a polynomial equation is equal to the number of ______ changes in the coefficients of the equation.
The number of positive real roots of a polynomial equation is equal to the number of ______ changes in the coefficients of the equation.
sign
If the polynomial has no ______ changes, it has no positive real roots.
If the polynomial has no ______ changes, it has no positive real roots.
sign
The number of negative real roots of a polynomial equation is equal to the number of ______ changes in the coefficients of the equation with alternating signs.
The number of negative real roots of a polynomial equation is equal to the number of ______ changes in the coefficients of the equation with alternating signs.
sign
To apply Descartes' Rule of Signs to find negative real roots, replace each term of the polynomial with its ______ (i.e., change the sign of each coefficient).
To apply Descartes' Rule of Signs to find negative real roots, replace each term of the polynomial with its ______ (i.e., change the sign of each coefficient).
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Complex roots of a polynomial equation always occur in ______ pairs.
Complex roots of a polynomial equation always occur in ______ pairs.
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The number of complex roots can be inferred by subtracting the number of real roots from the ______ of the polynomial.
The number of complex roots can be inferred by subtracting the number of real roots from the ______ of the polynomial.
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What is the maximum number of positive real roots of the equation x^4 + 2x^3 - 3x^2 - 4x + 5 = 0?
What is the maximum number of positive real roots of the equation x^4 + 2x^3 - 3x^2 - 4x + 5 = 0?
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If a polynomial equation has 3 sign changes, what is the minimum number of positive real roots it can have?
If a polynomial equation has 3 sign changes, what is the minimum number of positive real roots it can have?
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What is the maximum number of negative real roots of the equation x^5 - 2x^4 - 3x^3 + 4x^2 + 5x - 6 = 0?
What is the maximum number of negative real roots of the equation x^5 - 2x^4 - 3x^3 + 4x^2 + 5x - 6 = 0?
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If a polynomial equation has real coefficients and a complex root 3 + 2i, what can be said about the equation?
If a polynomial equation has real coefficients and a complex root 3 + 2i, what can be said about the equation?
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What can be said about the number of complex roots of a polynomial equation with real coefficients?
What can be said about the number of complex roots of a polynomial equation with real coefficients?
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What is the minimum number of sign changes required for a polynomial equation to have at least one positive real root?
What is the minimum number of sign changes required for a polynomial equation to have at least one positive real root?
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If a polynomial equation has 2 sign changes, what can be said about the number of positive real roots?
If a polynomial equation has 2 sign changes, what can be said about the number of positive real roots?
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What is the purpose of alternating the signs of the coefficients in Descartes' Rule of Signs?
What is the purpose of alternating the signs of the coefficients in Descartes' Rule of Signs?
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Study Notes
Descartes' Rule of Signs
Positive Real Roots
- The number of positive real roots of a polynomial equation is equal to the number of sign changes in the coefficients of the equation, or less than that by an even number.
- If the polynomial has no sign changes, it has no positive real roots.
- If the polynomial has one sign change, it has exactly one positive real root.
Negative Real Roots
- The number of negative real roots of a polynomial equation is equal to the number of sign changes in the coefficients of the equation with alternating signs, or less than that by an even number.
- To apply Descartes' Rule of Signs to find negative real roots, replace each term of the polynomial with its opposite (i.e., change the sign of each coefficient), and then count the number of sign changes.
- If the polynomial has no sign changes after replacing each term with its opposite, it has no negative real roots.
Complex Roots
- Descartes' Rule of Signs does not provide information about complex roots.
- However, it is known that complex roots of a polynomial equation always occur in conjugate pairs (i.e., if
a + biis a root, thena - biis also a root). - The number of complex roots can be inferred by subtracting the number of real roots from the degree of the polynomial.
Descartes' Rule of Signs
Positive Real Roots
- The number of positive real roots of a polynomial equation is equal to the number of sign changes in the coefficients of the equation, or less than that by an even number.
- If a polynomial has no sign changes in its coefficients, it has no positive real roots.
- If a polynomial has one sign change in its coefficients, it has exactly one positive real root.
Negative Real Roots
- The number of negative real roots of a polynomial equation is equal to the number of sign changes in the coefficients of the equation with alternating signs, or less than that by an even number.
- To apply Descartes' Rule of Signs to find negative real roots, replace each term of the polynomial with its opposite (i.e., change the sign of each coefficient), and then count the number of sign changes.
- If a polynomial has no sign changes after replacing each term with its opposite, it has no negative real roots.
Complex Roots
- Descartes' Rule of Signs does not provide information about complex roots.
- Complex roots of a polynomial equation always occur in conjugate pairs (i.e., if
a + biis a root, thena - biis also a root). - The number of complex roots can be inferred by subtracting the number of real roots from the degree of the polynomial.
Descartes' Rule of Signs
Positive Real Roots
- The number of positive real roots of a polynomial equation is equal to the number of sign changes in the coefficients of the equation, or is less than that by an even number.
- A sign change occurs when the coefficient of a term is positive and the coefficient of the next term is negative, or vice versa.
- The Upper Bound Theorem provides an upper bound for the number of positive real roots.
Negative Real Roots
- The number of negative real roots of a polynomial equation is equal to the number of sign changes in the coefficients of the equation with alternating signs, or is less than that by an even number.
- To find the number of negative real roots, alternate the signs of the coefficients (i.e., change the signs of the coefficients of the even-powered terms).
- The Lower Bound Theorem provides a lower bound for the number of negative real roots.
Complex Roots
- If a polynomial equation has real coefficients, then any complex roots occur in conjugate pairs (i.e., if a + bi is a root, then a - bi is also a root).
- Complex conjugate roots always occur in pairs.
- Descartes' Rule of Signs does not provide information about the number of complex roots, only about the number of positive and negative real roots.
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Description
Learn about Descartes' Rule of Signs, which determines the number of positive and negative real roots of a polynomial equation based on the sign changes in its coefficients.
