Enrol to start learning

Reading is open to everyone. Enrolling is free, and it is what unlocks the audio lessons, practice tests and progress tracking.

9. Factorisation of Polynomials

Interactive Audio Lesson

Sessions

Session 1: Introduction to Factorisation

Unlock the classroom podcast

The transcript is above and free to read. A free account plays the conversation back.

Create a free account

SarahInstructor

Today we're going to discuss factorisation of polynomials. Can anyone tell me what you think factorisation means in algebra?

Noah

Is it breaking down a polynomial into simpler parts?

SarahInstructor

Exactly, Student_1! Factorisation involves expressing a polynomial as the product of simpler polynomials. This allows us to solve equations more easily. Why do you think understanding this is important?

Isabella

It helps us in graphing and understanding polynomial behaviors better!

SarahInstructor

Right! Let's start with the first method: taking out a common factor.

Session 2: Taking Common Factor

Unlock the classroom podcast

The transcript is above and free to read. A free account plays the conversation back.

Create a free account

RobertInstructor

When we take a common factor from a polynomial, we’re looking for the greatest common divisor of the coefficients. For example, in the polynomial $6x^3 + 9x^2$ , what can we factor out?

Akash

We can factor out $3x^2$ !

RobertInstructor

Exactly! So, it factors to $3x^2(2x + 3)$ . Let’s apply this on a tricky example: $x^3 - 3x^2 + 4x$ . What can we factor out?

Ananya

We can factor out $x$ , which gives us $x(x^2 - 3x + 4)$ .

RobertInstructor

Good job, Student_4! Let’s remember to keep an eye out for common factors in future examples.

Session 3: Using Identities

Unlock the classroom podcast

The transcript is above and free to read. A free account plays the conversation back.

Create a free account

SarahInstructor

Now, let’s dive into identities. Who can remind us what the difference of squares identity is?

Noah

It’s $a^2 - b^2 = (a - b)(a + b)$ !

SarahInstructor

Great, Student_1! This identity allows us to factor expressions like $x^2 - 16$ . Can anyone tell me how we would apply this?

Isabella

We would factor it into $(x - 4)(x + 4)$ !

SarahInstructor

Exactly! Let’s also briefly explore the square of a binomial identity. Can anyone recall?

Akash

That one is $(a + b)^2 = a^2 + 2ab + b^2$ !

SarahInstructor

Perfect! Knowing these identities can significantly ease the factorisation process.

Session 4: Splitting the Middle Term

Unlock the classroom podcast

The transcript is above and free to read. A free account plays the conversation back.

Create a free account

RobertInstructor

A common method for quadratic polynomials is splitting the middle term. What does that mean?

Ananya

It means we rewrite the middle term so we can factor by grouping!

RobertInstructor

Exactly! For example, in $x^2 + 5x + 6$ , we want two numbers that multiply to 6 and add to 5. What would those be?

Noah

That’s $2$ and $3$ !

RobertInstructor

Perfect! So we can express it as $(x + 2)(x + 3)$ . Great understanding, everyone!

Session 5: Using Factor and Remainder Theorems

Unlock the classroom podcast

The transcript is above and free to read. A free account plays the conversation back.

Create a free account

SarahInstructor

Finally, let’s discuss the Factor and Remainder Theorems. Can someone explain what the Remainder Theorem states?

Isabella

If we divide a polynomial by $x-a$ , the remainder is equal to $P(a)$ !

SarahInstructor

Correct! This theorem helps us find roots of polynomials. If the remainder is 0, then $x-a$ is a factor. How would that help us in factorisation?

Akash

We can identify factors quickly by plugging in values!

SarahInstructor

Exactly! Very well said. Remember these theorems; they are powerful tools in factorisation.

Overview

Short Summary

Factorisation of polynomials involves breaking down polynomials into simpler components, making it easier to solve equations and understand polynomial behavior.

Medium Summary

In this section, students learn various methods of factorising polynomials including taking common factors, applying identities, splitting the middle term, and using the Factor and Remainder Theorems. Practical examples help clarify these concepts, making the process of factorisation an essential skill in algebra.

Detailed Summary

Factorisation of Polynomials

Factorisation of polynomials is a key concept in algebra that simplifies polynomials into products of polynomials with lower degrees. This section discusses different methods of factorisation such as:

Taking Common Factor: Identify and factor out the common elements in a polynomial expression.
Using Identities: Apply algebraic identities such as:
- Difference of squares: $a^2 - b^2 = (a - b)(a + b)$
- Square of a binomial: $(a ± b)^2 = a^2 ± 2ab + b^2$
Splitting the Middle Term: Particularly useful for quadratic polynomials to express them as products of two binomials by finding two numbers that multiply and add to the coefficients.
Using Factor and Remainder Theorems: A theoretical foundation where, if a polynomial is divided by $x - a$ , the remainder is $P(a)$ which aids in the identification of factors.

Example:

To factorize $x^2 - 5x + 6$ , the expression can be re-written as $(x - 2)(x - 3)$ . This illustrates the concept of finding two values that yield the correct factors for the polynomial.

Factorisation is crucial not only for solving polynomial equations but also facilitates understanding polynomial graphs and their real-life applications.

Audio Book

Voice:

Playlist

Methods of Factorisation

Unlock the audio lesson

The script is above and free to read. A free account plays it back, in the voice you pick.

Create a free account

Methods: • Taking common factor • Using identities: o 𝑎2 −𝑏2 = (𝑎 −𝑏)(𝑎 +𝑏) o (𝑎 ±𝑏)2 = 𝑎2 ±2𝑎𝑏+𝑏2 • Splitting the middle term (for quadratics) • Using factor and remainder theorems

Detailed Explanation

Factorisation is the process of breaking down a polynomial into simpler components (factors) that, when multiplied together, result in the original polynomial. There are several methods to achieve this:

Taking a Common Factor: Identify a term that is common to all terms in the polynomial and factor it out.
Using Identities: Mathematical identities help in factorization. For instance:
- The difference of squares identity states that 𝑎² − 𝑏² can be factored into (𝑎 − 𝑏)(𝑎 + 𝑏).
- The square of a binomial can be expressed as (𝑎 ± 𝑏)² = 𝑎² ± 2𝑎𝑏 + 𝑏².
Splitting the Middle Term: Specifically for quadratic polynomials, this method involves rewriting the middle term in such a way that allows the polynomial to be factored easily.
Using Factor and Remainder Theorems: These theorems aid in finding factors of polynomials when given specific values that make the polynomial equal to zero.

Examples & Analogies

Think of factorisation like organizing a messy box of toys. If you have different types of toys (like action figures, cars, and blocks) all mixed together, factorising them involves taking a step back and grouping them by type. Similarly, in polynomials, you're looking for common values or patterns to group terms together, making it easier to handle and understand each 'type' (or factor) in the polynomial.

Example of Factorisation

Unlock the audio lesson

The script is above and free to read. A free account plays it back, in the voice you pick.

Create a free account

Example: Factorize 𝑥2 −5𝑥 +6 = (𝑥 −2)(𝑥−3)

Detailed Explanation

Let's look at an example of factorising a quadratic polynomial:

Given the polynomial 𝑥² − 5𝑥 + 6, our goal is to find two binomials that multiply to equal this polynomial. We can do this through the method of splitting the middle term:

Identify the Coefficients: In the polynomial 𝑥² − 5𝑥 + 6, the coefficients are:
- Leading coefficient (for 𝑥²) = 1
- Coefficient of 𝑥 = -5
- Constant term = 6.
Find the Two Numbers: We need two numbers that multiply to 6 (the constant) and add up to -5 (the coefficient of 𝑥). The numbers -2 and -3 fit this case because:
- (-2) * (-3) = 6
- (-2) + (-3) = -5.
Write the Factorization: Now we can express the quadratic as the product of two binomials: (𝑥 - 2)(𝑥 - 3).

Examples & Analogies

Imagine you have a rectangular garden that has an area represented by the polynomial 𝑥² − 5𝑥 + 6. To find out its dimensions (length and width), you can think of that area as being represented by the two dimensions (𝑥 - 2) and (𝑥 - 3). So, factorising helps you break down the overall area into manageable parts—just like how you find the length and width of a garden from its area!

Key Concepts

Core takeaways and short definitions to help you quickly recall the key ideas from this section.

Factorisation: The process of breaking down a polynomial into simpler expressions.

Common Factor: A shared factor among terms in a polynomial that can be factored out.

Algebraic Identities: Truthful statements regarding polynomial expressions that simplify calculations.

Remainder Theorem: A method to determine the remainder of polynomial division.

Factor Theorem: A guideline indicating if a polynomial is divisible by $x - a$ based on the polynomial's value at $a$ .

Examples

Step-by-step examples to apply the section's ideas and test your understanding.

To factor $x^2 - 9$ , use the difference of squares: $(x - 3)(x + 3)$ .

For the polynomial $2x^3 + 4x^2$ , factor out the common factor: $2x^2(x + 2)$ .

Use the middle term splitting to factor $x^2 + 5x + 6$ as $(x + 2)(x + 3)$ .

Memory Aids

Interactive tools to help you remember key concepts

🎵

Rhymes

To factor quickly and delay, think of common terms in play.

📖

Stories

Imagine a gardener splitting flowers and plants into smaller bouquets, making the garden easier to manage—just like we split polynomials into factors.

🧠

Memory Tools

GREAT FACTORS help: Grouping, Remainder, Extract common factor, Apply identities, Try splitting the middle term.

🎯

Acronyms

F.A.C.T.O.R

Find common terms

Apply identities

Consider middle term

Take out factors

Observe roots.

Flash Cards

Term

What is factorisation?

Definition

The process of breaking down a polynomial into simpler components.

Term

What is a common factor?

Definition

A factor that is shared by two or more terms in a polynomial.

Term

State the Remainder Theorem.

Definition

The remainder of $P(x)$ divided by $x - a$ is $P(a)$ .

Term

Provide an example of an algebraic identity.

Definition

Difference of squares: $a^2 - b^2 = (a - b)(a + b)$ .

Glossary

Factorisation

The process of breaking down a polynomial into simpler components (factors).

Common Factor

A number or expression that divides two or more numbers or expressions evenly.

Algebraic Identities

Equations that are true for all values of the variables.

Quadratic Polynomial

A polynomial of degree two.

Remainder Theorem

The theorem stating that the remainder of the division of a polynomial $P(x)$ by $x-a$ is equal to $P(a)$ .

Factor Theorem

A theorem stating that $x-a$ is a factor of polynomial $P(x)$ if and only if $P(a) = 0$ .

Reference Links

Introduction to Factorization of Polynomialsweb

Study Polynomial Factorizationweb

Factoring Quadraticsweb

Polynomial Identitiesweb

Remainder Theorem Explainedweb

Learn Content