Further Mathematics

Paper 1

# Roots of polynomial equations

🤓 Study

📖 Quiz

Play audio lesson

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.