2.4 Factorisation of Polynomials

Description

Quick Overview

This section introduces the concept of factorisation of polynomials and the Factor Theorem, illustrating how to identify polynomial factors based on their roots.

Standard

In this section, we explore the factorisation of polynomials, particularly focusing on the Factor Theorem, which connects a polynomial's zeroes with its factors. Various examples illustrate how to apply these concepts to quadratic and cubic polynomials, highlighting the significance of finding roots.

Detailed

Factorisation of Polynomials

In this section, we delve into the process of factorising polynomials, a crucial aspect of algebra that simplifies understanding polynomial behavior. The Factor Theorem plays a central role, stating that if a polynomial p(x) of degree n has a real number a such that p(a) = 0, then (x - a) is a factor of p(x). Conversely, if (x - a) is a factor of p(x), then p(a) = 0. This theorem arises from the Remainder Theorem, which establishes that dividing a polynomial by (x - a) yields a remainder of zero when a is a root.

The section also demonstrates factorisation techniques for quadratic polynomials, suggesting methods such as splitting the middle term or using the Factor Theorem. For instance, to factor a polynomial like ax² + bx + c, we must express b as the sum of two numbers whose product equals ac.

Examples illustrate these processes, such as checking if x + 2 is a factor of polynomials or identifying roots. By employing these methods, students gain a stronger grasp of polynomial equations and their factorizable forms, paving the way for more advanced algebraic manipulation.

Example: Find the value of \( k \), if \( x - 2 \) is a factor of \( 5x^3 + 4x^2 - 3x - 2 + k \).
Solution: As \( x - 2 \) is a factor of \( p(x) = 5x^3 + 4x^2 - 3x - 2 + k \), \( p(2) = 5(2)^3 + 4(2)^2 - 3(2) - 2 + k = 0 \)
Now,
\[ 5(2)^3 + 4(2)^2 - 3(2) - 2 + k = 0 \]
So,
\[ 5 \cdot 8 + 4 \cdot 4 - 6 - 2 + k = 0 \]
i.e.,
\[ k = -30 \]

Key Concepts

  • Factor Theorem: A method to determine factors of a polynomial based on its roots.

  • Polynomial Degree: The highest exponent of the polynomial determines its degree.

  • Quadratic Factorisation: Factor a quadratic by rewriting its middle term.

  • Cubic Factorisation: Requires finding one root to reduce the polynomial.

Memory Aids

🎵 Rhymes Time

  • Roots may be sweet, factors are neat; find one when you split, it’s a factoring hit!

📖 Fascinating Stories

  • Once there was a polynomial that wanted to find its factors. It remembered the Factor Theorem and finally discovered that every root could point towards a potential factor, and it joyously danced with its newfound friends, the factors.

🧠 Other Memory Gems

  • R.F. = Roots lead to Factors.

🎯 Super Acronyms

F.A.C.T. - Factorization And Checking Techniques.

Examples

  • Example of factoring x² - 3x + 2 to show x-1 and x-2 as factors.

  • Example of applying the Factor Theorem to check if x + 2 is a factor of x³ + 3x² + 5x + 6.

Glossary of Terms

  • Term: Polynomial

    Definition:

    An algebraic expression composed of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents.

  • Term: Factor Theorem

    Definition:

    A theorem that states if a polynomial p(x) has a root at a, then (x - a) is a factor of p(x).

  • Term: Root

    Definition:

    A value of x that makes the polynomial p(x) equal to zero.

  • Term: Quadratic Polynomial

    Definition:

    A polynomial of degree two, expressed in the form ax² + bx + c.

  • Term: Cubic Polynomial

    Definition:

    A polynomial of degree three, expressed in the form ax³ + bx² + cx + d.

  • Term: Zero Polynomial

    Definition:

    A polynomial whose coefficients are all zero, often denoted as 0 and has no defined degree.