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Introduction:

Factorising is an important mathematical concept that involves breaking down a mathematical expression into smaller, simpler components. It is a process of finding the factors that when multiplied together, give the original expression. Factorisation is used in a wide range of mathematical disciplines including algebra, calculus, number theory and many others. In this article, we will provide a comprehensive guide to factorising, including detailed explanations, examples, and frequently asked questions.

Definitions:

Before we dive into the details of factorising, let us first define some key terms:

• Factors: A factor of a number or expression is a number or expression that can be multiplied with another number or expression to give the original number or expression. For example, 3 and 5 are factors of 15, and (x+2) and (x-3) are factors of x^2-x-6.
• Prime Factorisation: Prime factorisation is the process of breaking down a number into its prime factors. A prime number is a number that can only be divided by 1 and itself. For example, the prime factors of 24 are 2, 2, 2, and 3, since 24 can be written as 2 x 2 x 2 x 3.
• Quadratic Equation: A quadratic equation is a second-degree polynomial equation in one variable. It is of the form ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable.
• Completing the Square: Completing the square is a technique used to solve quadratic equations by manipulating the equation into a perfect square form. This involves adding and subtracting a constant to both sides of the equation.

Factorising involves breaking down an expression into its constituent factors. This is usually done to simplify the expression or to solve an equation. There are several methods for factorising, and we will discuss some of the most common ones below.

Factorising by Common Factor:

The first method of factorising is to find the common factor of all the terms in the expression. For example, consider the expression 6x^2 + 12x. The common factor of both terms is 6x, so we can write:

6x^2 + 12x = 6x(x + 2)

Here, we have factored out the common factor of 6x. The resulting expression is simpler and easier to work with.

Factorising Quadratic Expressions:

Quadratic expressions are expressions that involve a second-degree polynomial equation in one variable. They are of the form ax^2 + bx + c, where a, b, and c are constants. There are several methods for factorising quadratic expressions, including:

a. Factorising by Inspection:

One method for factorising quadratic expressions is to inspect the expression and identify two numbers that multiply to give the constant term (c) and add to give the coefficient of the middle term (b). For example, consider the expression x^2 + 5x + 6. We need to find two numbers that multiply to give 6 and add to give 5. The numbers are 2 and 3, so we can write:

x^2 + 5x + 6 = (x + 2)(x + 3)

Here, we have factored the quadratic expression by inspection.

b. Factorising by Completing the Square:

Another method for factorising quadratic expressions is to use the completing the square technique. This involves manipulating the equation into a perfect square form. For example, consider the quadratic equation x^2 + 6x + 5 = 0. We can complete the square by adding and subtracting a constant:

x^2 +

6x + 5 = 0

(x+3)^2 – 4 = 0

(x+3)^2 = 4

Taking the square root of both sides, we get:

x + 3 = ±2

x = -3 ± 2

x = -1 or x = -5

So, the solutions to the quadratic equation are x = -1 and x = -5. We can also write the quadratic expression as a product of two factors:

x^2 + 6x + 5 = (x+1)(x+5)

Here, we have factored the quadratic expression using the completing the square method.

c. Factorising by Quadratic Formula:

Another method for factorising quadratic expressions is to use the quadratic formula. The quadratic formula is given by:

x = (-b ± sqrt(b^2 – 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation. For example, consider the quadratic equation x^2 – 3x – 4 = 0. We can use the quadratic formula to find the solutions:

x = (3 ± sqrt(9 + 16)) / 2

x = (3 ± 5) / 2

x = -1 or x = 4

So, the solutions to the quadratic equation are x = -1 and x = 4. We can also write the quadratic expression as a product of two factors:

x^2 – 3x – 4 = (x+1)(x-4)

Here, we have factored the quadratic expression using the quadratic formula.

Factorising Trinomials:

Trinomials are expressions that involve a polynomial equation of degree three in one variable. They are of the form ax^3 + bx^2 + cx + d, where a, b, c, and d are constants. There are several methods for factorising trinomials, including:

a. Factorising by Inspection:

One method for factorising trinomials is to inspect the expression and identify two numbers that multiply to give the constant term (d) and add to give the coefficient of the middle term (c). For example, consider the trinomial x^2 + 5x + 6. We need to find two numbers that multiply to give 6 and add to give 5. The numbers are 2 and 3, so we can write:

x^2 + 5x + 6 = (x + 2)(x + 3)

Here, we have factored the trinomial by inspection.

b. Factorising by Grouping:

Another method for factorising trinomials is to group the terms and factor out the common factor. For example, consider the trinomial x^3 – x^2 – x + 1. We can group the terms as follows:

x^3 – x^2 – x + 1 = x^2(x-1) – 1(x-1)

= (x^2-1)(x-1)

= (x+1)(x-1)(x-1)

Here, we have factored the trinomial by grouping.

Examples:

Factorise the expression 12x^2 + 24x.

Solution:

We can factorise 12x^2 + 24x by finding the common factor of both terms, which is 12x:

12x^2 + 24x = 12x(x + 2)

Therefore, the factorised form of the expression is 12x(x + 2).

Factorise the expression x^2 + 6x + 9.

Solution

We can factorise x^2 + 6x + 9 by using the method of perfect square trinomials:

x^2 + 6x + 9 = (x + 3)^2

Therefore, the factorised form of the expression is (x + 3)^2.

Factorise the expression 3x^2 – 12x + 9.

Solution:

We can factorise 3x^2 – 12x + 9 by finding the common factor of the three terms, which is 3:

3x^2 – 12x + 9 = 3(x^2 – 4x + 3)

We can then factorise x^2 – 4x + 3 using the method of inspection:

x^2 – 4x + 3 = (x – 1)(x – 3)

Therefore, the factorised form of the expression is 3(x – 1)(x – 3).

Factorise the expression 4x^2 – 9.

Solution:

We can factorise 4x^2 – 9 by using the method of difference of squares:

4x^2 – 9 = (2x)^2 – 3^2

= (2x + 3)(2x – 3)

Therefore, the factorised form of the expression is (2x + 3)(2x – 3).

Factorise the expression x^3 – 27.

Solution:

We can factorise x^3 – 27 by using the method of difference of cubes:

x^3 – 27 = (x – 3)(x^2 + 3x + 9)

Therefore, the factorised form of the expression is (x – 3)(x^2 + 3x + 9).

Factorise the expression 2x^2 – 7x – 15.

Solution:

We can factorise 2x^2 – 7x – 15 by using the method of quadratic formula:

x = (7 ± sqrt(7^2 + 4(2)(15))) / (2(2))

x = (7 ± 11) / 4

x = 9/4 or x = -5/2

Therefore, the solutions to the quadratic equation are x = 9/4 and x = -5/2. We can then write the quadratic expression as a product of two factors:

2x^2 – 7x – 15 = 2(x – 9/4)(x + 5/2)

Therefore, the factorised form of the expression is 2(x – 9/4)(x + 5/2).

Factorise the expression x^3 + 6x^2 + 11x + 6.

Solution:

We can factorise x^3 + 6x^2 + 11x + 6 by using the method of grouping:

x^3 + 6x^2 + 11x + 6 = x^2(x + 6) + 11(x + 6)

= (x^2 + 11)(x + 6)

Therefore, the factorised form of the expression is (x^2 + 11)(x + 6).

Factorise the expression 4x^3 – 36x.

Solution:

We can factorise 4x^3 – 36x by finding the common factor of both terms, which is 4x:

4x^3 – 36x = 4x(x^2 – 9)

We can then factorise x^2 –

Factorise the expression x^4 – 16.

Solution:

We can factorise x^4 – 16 by using the method of difference of squares:

x^4 – 16 = (x^2 + 4)(x^2 – 4)

We can then factorise x^2 – 4 using the method of difference of squares:

x^2 – 4 = (x + 2)(x – 2)

Therefore, the factorised form of the expression is (x^2 + 4)(x + 2)(x – 2).

Factorise the expression x^3 – 3x^2 + 3x – 1.

Solution:

We can factorise x^3 – 3x^2 + 3x – 1 by using the method of grouping:

x^3 – 3x^2 + 3x – 1 = (x^3 – 1) – 3x(x – 1)

= (x – 1)(x^2 + x + 1) – 3x(x – 1)

= (x – 1)(x^2 – 2x + 1 + x^2 + x + 1) – 3x(x – 1)

= (x – 1)(2x^2 + 2x) – 3x(x – 1)

= (x – 1)(2x(x + 1) – 3x)

= (x – 1)(2x^2 – x)

Therefore, the factorised form of the expression is (x – 1)(2x^2 – x).

FAQs

• What is factorising in math?

Factorising in math is the process of finding the factors of a given expression. In algebra, an expression can be written as a product of its factors. The process of factorising is used to simplify expressions, solve equations, and find roots of equations.

• What are the methods of factorising?

The methods of factorising include:

• Finding the common factor
• Grouping
• Difference of squares
• Difference of cubes
• Sum of cubes
• Quadratic formula
• Perfect square trinomials

• Why is factorising important in math?

Factorising is important in math because it helps simplify expressions and solve equations. Factorising is also useful in finding roots of equations and identifying the relationship between different algebraic expressions.

• What is the difference between factorising and expanding?

Expanding is the process of multiplying out brackets in an algebraic expression. For example, (x + 2)(x – 3) can be expanded to give x^2 – x – 6. Factorising, on the other hand, is the process of writing an expression as a product of its factors. For example, x^2 – x – 6 can be factorised to give (x + 2)(x – 3).

• How can I check my factorisation?

You can check your factorisation by multiplying out the factors to see if they give the original expression. For example, if you factorise x^2 + 5x + 6 as (x + 2)(x + 3), you can check your factorisation by multiplying (x + 2)(x + 3) and verifying that it gives x^2 + 5x + 6.

Quiz

• Factorise the expression x^2 – 9. a. (x + 3)(x – 3) b. (x + 9)(x – 1) c. (x – 9)(x + 1) d.

a. (x + 3)(x – 3)

• Factorise the expression 4x^2 – 25y^2. a. (2x + 5y)(2x – 5y) b. (4x + 25y)(x – 1) c. (2x – 5y)(2x + 5y) d. (4x – 25y)(x + 1)

c. (2x – 5y)(2x + 5y)

• Factorise the expression x^3 + 8. a. (x + 2)(x^2 – 2x + 4) b. (x + 2)(x^2 + 2x + 4) c. (x – 2)(x^2 – 2x + 4) d. (x – 2)(x^2 + 2x + 4)

a. (x + 2)(x^2 – 2x + 4)

• Factorise the expression x^2 + 6x + 8. a. (x + 2)(x + 4) b. (x – 2)(x – 4) c. (x + 2)(x + 2) d. (x – 2)(x + 4)

a. (x + 2)(x + 4)

• Factorise the expression x^3 – 27. a. (x + 3)(x^2 – 3x + 9) b. (x – 3)(x^2 + 3x + 9) c. (x + 3)(x^2 + 3x + 9) d. (x – 3)(x^2 – 3x + 9)

a. (x + 3)(x^2 – 3x + 9)

• Factorise the expression x^2 – 7x + 10. a. (x – 5)(x – 2) b. (x + 5)(x + 2) c. (x – 5)(x + 2) d. (x + 5)(x – 2)

a. (x – 5)(x – 2)

• Factorise the expression 2x^2 + 4x – 6. a. 2(x + 3)(x – 1) b. 2(x + 1)(x – 3) c. 2(x – 3)(x + 1) d. 2(x – 1)(x + 3)

a. 2(x + 3)(x – 1)

• Factorise the expression x^3 + x^2 – x – 1. a. (x – 1)(x + 1)(x + 1) b. (x – 1)(x – 1)(x + 1) c. (x + 1)(x – 1)(x – 1) d. (x + 1)(x + 1)(x – 1)

b. (x – 1)(x – 1)(x + 1)

• Factorise the expression x^2 – 6x + 9. a. (x + 3)(x + 3) b. (x – 3)(x – 3) c. (x + 3)(x – 3) d. (x – 3)(x + 3)

a. (x – 3)(

• Factorise the expression 2x^2 – 7x + 3. a. (x – 1)(2x – 3) b. (x + 1)(2x – 3) c. (x – 1)(2x + 3) d. (x + 1)(2x + 3)

a. (x – 1)(2x – 3)