Roots of equations

Roots of Equations

Complex Roots

• Complex roots always come in conjugate pairs. If a + bi is a root of a polynomial with real coefficients, then its conjugate a - bi is also a root.
• The sum of complex roots and the product of complex roots both are also real numbers.

Root theorem

• The root theorem (or the Fundamental Theorem of Algebra) states that a polynomial of degree n has exactly n roots.

Polynomial roots

• If p and q are two roots of a polynomial, then x = p and x = q are solutions of that polynomial equation.
• The sum of the roots of a quadratic ax² + bx + c = 0 is given by -b/a and the product of the roots is c/a.

Cubic roots

• The sum of the roots of a cubic equation is equal to the coefficient of x² divided by the leading coefficient.
• The product of the roots taken three at a time is equal to the constant term divided by the leading coefficient.

Quartic roots

• For a quartic equation the sum of the roots is the opposite of the coefficient of the cubic term, divided by the leading coefficient.

Synthetic division

• You can use synthetic division to test whether a given value is a root of a polynomial. If the remainder is zero, then that value is a root.

Rational root theorem

• The Rational root theorem can be used to find possible rational roots of a polynomial. The possible rational roots of a polynomial are all the factors of the constant term divided by all the factors of the leading coefficient.

Descartes’ Rule of Signs

• Descartes’ Rule of Signs is a useful apparent rule of thumb for determining the possible number of positive and negative real roots of a polynomial.

Use of Technology

• While many roots can be solved analytically, technology (calculators, computer algebra systems) can be extremely helpful in finding roots of more complicated polynomials.