Exam Questions - Finding roots

Finding Roots Revision Content

A root of a function is a number that, when substituted into the function, makes the function equal zero.
The process of finding roots is also known as solving equations or finding zeros of the function.
Quadratic functions can have two roots, one root or no real root, depending on the discriminant of the function.
Higher degree functions (ex. cubic, quartic functions) can have more roots.

To solve a quadratic equation like ax^2 + bx + c = 0:
- Use factoring method if the equation can be easily factored.
- Use the quadratic formula x = [-b ± sqrt(b^2 - 4ac)] / (2a), where a, b, and c are coefficients in the equation.
To solve higher degree equations:
- Use factoring when it’s possible.
- The Rational Root Theorem can provide possible rational roots.
- The Remainder Theorem allows us to evaluate a function at a given point.

The Bisection Method is used to find roots by repeatedly bisecting an interval and then selecting a subinterval in which a root must lie.
The Newton-Raphson Method is an iterative method that begins with a guess and then improves the estimate using the calculus of the function.
The Secant Method is a numerical method that uses linear approximation to find roots.

If discriminant of a quadratic equation is negative, then it has complex roots.
By definition, complex roots come in conjugate pairs. If a+bi is a root, then a-bi is also a root.
Euler’s Formula, e^(ix) = cos(x) + i sin(x), is useful to manipulate and understand complex roots.

The Fundamental Theorem of Algebra states every non-zero single-variable polynomial with complex coefficients has a root among complex numbers.
Consequently, a polynomial equation of degree n has exactly n roots in the complex numbers.
The Vieta’s formulas relate the coefficients of a polynomial to sums and products of its roots, useful for quick calculations.

Root Theorem, also known as Factor theorem, is a special case of the Remainder theorem.
It states that a number c is a root of the function f(x) if and only if (x - c) is a factor of f(x).
The theorem is often used for factoring polynomials and finding roots.