In this section, the focus is on the Factor Theorem and Remainder Theorem, crucial concepts in algebra for solving polynomial equations. These theorems enable us to analytically find factors of polynomials, evaluate unknown coefficients, and solve polynomial equations with precision.

The Remainder Theorem

The Remainder Theorem posits that when a polynomial $f(x)$ is divided by a linear term $x - a$, the remainder is $f(a)$. This theorem simplifies the process of evaluating polynomials at specific points, eliminating the need for complex division.

Example: Evaluating a Polynomial

Consider the polynomial $f(x) = x^3 + 4x^2 - 5x - 14$. To find the remainder when dividing by $x - 2$, simply evaluate $f(2)$:

Thus, the remainder is 0.

The Factor Theorem

The Factor Theorem extends the Remainder Theorem. It states that $x - a$ is a factor of a polynomial $f(x)$ if and only if $f(a) = 0$. Essentially, $a$ is a root of the polynomial.

Example: Factorising a Polynomial

Using the polynomial $f(x) = x^3 + 4x^2 - 5x - 14$, and since $f(2) = 0$, we deduce $(x - 2)$ is a factor. To find the complete factorisation, we divide $f(x)$ by $(x - 2)$:

Hence, $f(x) = (x - 2)(x^2 + 6x + 7)$.

Solving Polynomial Equations

Example:

Solve the polynomial equation $x^3 + 4x^2 - 5x - 14 = 0$, given that $x = 2$ is a root.

Solution:

1. Given Polynomial: $x^3 + 4x^2 - 5x - 14 = 0$

2. Root: $x = 2$

3. Factor Theorem: Using $x = 2$ to factorize, we get the quadratic equation $x^2 + 6x + 7$.

4. Quadratic Formula: To find the other roots, apply $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 1, b = 6, c = 7$.

5. Calculate Roots: Substitute values into the quadratic formula to find the remaining roots.