Descartes' Rule of Signs

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 + bi is a root, then a - bi is 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 + bi is a root, then a - bi is 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.