A Level Further Mathematics CAIE

Roots of polynomial equations

Definitions and Basic Concepts

• A polynomial equation is an expression that can be defined as a sum of terms, where each term is a product of a number and a power greater than or equal to 0.
• The highest power in any term of the polynomial is known as the degree of the equation.
• A root of a polynomial equation is a solution to the equation, i.e., a number that when substituted into the equation, makes the equation true.

Fundamental Theorem of Algebra

• The Fundamental Theorem of Algebra states that a polynomial with degree n has exactly n roots, though not all of them may be distinct or real.
• Some roots could be complex or imaginary numbers, especially if the polynomial has odd degree.

Relationship between Roots and Coefficients

• The sum of the roots of a polynomial is equal to the opposite of the coefficient of the second highest degree term, divided by the coefficient of the highest degree term. This is often represented as Sum of roots = -b/a for a quadratic equation ax^2+bx+c=0.
• The product of the roots of a polynomial equation ax^2+bx+c=0 is equal to the constant term (c) divided by the coefficient of the highest degree term (a).

Finding the Roots

• The quadratic formula, x = [-b ± sqrt(b^2 - 4ac)] / 2a, can be used to find the roots of a quadratic polynomial.
• For polynomials of higher degree, various methods such as factorisation, completing the square, rational root theorem, synthetic division or Newton's method can be used.
• Although not always feasible for pencil-and-paper calculations, the Durand-Kerner method or Jenkins-Traub method might be used in software for finding the roots of high-degree polynomials.

Complex Roots

• If a polynomial has real coefficients and a complex number is a root, then its complex conjugate is also a root.
• Complex roots always come in conjugate pairs when all coefficients in the polynomial are real numbers. This is called the Complex Conjugate Root Theorem.
• The argument or angle of a pair of complex conjugate roots is either equal or supplementary, depending on whether the roots are inside or outside the unit circle respectively.

Graphical Interpretation

• For a polynomial equation y = f(x), the x-values at which the curve intersects the x-axis represent the real roots of the equation, i.e., the values of x for which y = 0.
• The behaviour of the curve at each intersection point depends on whether the corresponding root is a 'single root', or a root with a multiplicity (e.g. a root 'repeated' several times).

Polynomial Division

• Polynomial long division or synthetic division can be used to simplify a higher order polynomial into a lower order polynomial by using one or more known roots.
• This results in a new polynomial, the roots of which correspond to the remaining roots of the original polynomial.