Factoring Formulas

Factoring Formulas

Here is a review of what factoring is which must be understood by students prior to learning about unique Factoring Formulas. An algebraic expression is factored in when it is expressed as the product of two or more expressions. Expressions can be factored in using a variety of techniques. One among them makes use of unique Factoring Formulas. Factorisation, commonly referred to as factoring, is the division of a large number into a number of smaller numbers. Students shall obtain the genuine or original number when they multiply these smaller numbers. The knowledge of factorisation is often introduced to pupils in Class 6. There are study materials available on the Extramarks website that can assist students with learning more about this topic.

One of the key techniques for reducing an algebraic or quadratic equation to a simple form is factorisation. Therefore, one should be familiar with Factoring Formulas in order to deconstruct complex equations. Students will receive all the knowledge they require pertaining to various factorisation formulas for Polynomials, Trigonometry, Algebra, and Quadratic Equations through the resources provided by Extramarks. The Factoring Formulas PDF is also made available to students and can be downloaded from the Extramarks online learning portal.

As special Factoring Formulas, one employs a few algebraic formulas. The LHS and RHS of the expression on both sides can be solved in order to confirm these algebraic identities. The following are a few Factoring Formulas that can be used:

• a2 – b2 = (a – b)(a + b)
• (a + b)2 = a2 + 2ab + b2
• (a – b)2 = a2 – 2ab + b2
• a3 – b3 = (a – b)(a2 + ab + b2)
• a3 + b3 = (a + b)(a2 – ab + b2)
• (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
• (a – b – c)2 = a2 + b2 + c2 – 2ab + 2bc – 2ca

The breaking or deconstruction of an entity (such as a number, a matrix, or a polynomial) into a product of another entity, or factors, whose multiplication results in the original number, matrix, etc., is known as factorisation or factoring in Mathematics. Students will mostly learn this idea in their junior secondary classes from grades 6 to 8.

What are Factoring Formulas?

A polynomial or integer is simply resolved into components that, when multiplied together, produce the original or starting polynomial or integer. The factorisation method allows us to simplify any algebraic or quadratic equation by representing the equations as the product of factors rather than by expanding the brackets. Any equation can have an integer, a variable, or the algebraic expression itself as a factor.

An algebraic expression is factorised when it is written as the product of its factors. These variables, factors, or algebraic expressions could be present.

A number is broken down for the factor into components that can be multiplied to produce the original number. For instance,

Numbers can also be factored into a variety of combinations. There are various approaches to determining a number’s factors. It is simple to determine the factors of an integer, but it is more difficult to determine the factors of algebraic equations. So students will learn how to determine a quadratic polynomial’s factors.

Factoring Formula 1: (a + b)2 = a2 + 2ab + b2

One of the Factoring Formulas is derived through a distinct process henceforth. The left-hand side of this formula is given here, leading to the right-hand side at the conclusion.

=(a + b)2 = (a + b) (a + b)

= a2 + ab + ba + b2 (Multiplied the binomials)

= a2 + 2ab + b2

The formula is so produced. By clicking here, students can learn this formula in depth.

Therefore, the factors of a2 + 2ab + b2 are (a+b) and (a+b).

Factoring Formula 2: (a – b)2 = a2 – 2ab + b2

Another one of the Factoring Formulas is provided below.

One should start with the left side of this equation and work our way to the right side at the conclusion.

(a – b)2 = (a – b) (a – b)

= a2 – ab – ba + b2 (Multiplied the binomials)

= a2 – 2ab + b2

Formulas are produced in this way. Students can learn more about this formula by clicking here.

Factoring Formula 3: (a + b) (a – b) = a2 – b2

Third one of the Factoring Formulas is given on the Extramarks website.

Students must start with the left side of this equation and work their way to the right side at the conclusion.

(a + b) ( a – b) = a2 – ab + ba + b2 (Multiplied the binomials)

= a2 – b2

This formula is produced in this way. Students can learn this formula in detail by clicking here.

Factoring Formula 4: (x + a) (x + b) = x2 + (a + b) x + ab

The fourth one of the Factoring Formulas 4 is derived in the following manner.,

One ought to start with the left side of this equation and solve further to acquire the right side at the conclusion.

(x + a) ( x + b) = x2 + xb + ax + b2 (Multiplied the binomials)

= x2 + (a + b) x + ab

The formula is so produced.

Factoring Formula 5: (a + b)3 = a3 + b3 + 3ab (a + b)

Fifth one of the Factoring Formulas 5 is derived using a particular mathematical process.

In order to get to the right side of this equation, one must start with the left side.

(a + b)

3 = (a + b)

2 (a + b)

= (a2 + 2ab + b2) (a + b)

= a3 + 2a2b + ab2 + a2b + 2ab2 + b3

= a3 + b3 + 3a2b + 3ab2 (or)

= a3 + b3 + 3ab (a + b)

This is how the formula is generated. This formula can be learned in depth by clicking here.

Factoring Formula 6: (a – b)3 = a3 – b3 – 3ab (a – b)

One of the Factoring Formulas 6 is derived by following the given steps of calculation.

It is important that students start with the left side of the equation and work their way towards the right side at the end.

(a – b)

3 = (a – b)

2 (a – b)

= (a2 – 2ab + b2) (a – b)

= a3 – 2a2b + ab2 – a2b + 2ab2 – b3

= a3 – b3 – 3a2b + 3ab2 (or)

= a3 – b3 – 3ab (a – b)

The formula is so produced. By clicking here, students may learn more about this formula in depth.

Factoring Formula 7: (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca

The derivation process of the seventh one of the Factoring Formulas is provided henceforth.,

Students must begin solving with the left side of this equation and work their way to the right side at the conclusion.

(a + b + c)

2 = (a + b + c) (a + b + c)

= a2 + ab + ac + ba + b2 + bc + ca + bc + c2

= a2 + b2 + c2 + 2ab + 2bc + 2ca

This formula is produced in this manner. This formula can be learned in greater detail by clicking here.

Factoring Formula 8: x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – xy – yz – xz)

The derivation of the eighth one of the Factoring Formulas is provided below.

In order to solve this equation, students must begin by solving the right side and work their way to the left at the end.

(x + y + z) (x2 + y2 + z2 – xy – yz – xz)

= (x3 + xy2 + xz2 – x2y – xyz – x2z) + (x2y + y3 + yz2 – xy2 – y2z – xyz) + (x2z + y2z + z3 – xyz – yz2 – xz2)

= x3 + y3 + z3 – 3xyz (All other terms are null and void)

The formula is produced in the aforementioned manner.

Factoring Formula 9: x3 + y3 = (x + y) (x2 – xy + y2)

The process of deriving the ninth one of the Factoring Formulas is provided on the Extramarks website.,

It is necessary for students to begin with the right side of the equation and work their way to the left at the conclusion.

(x + y) (x2 – xy + y2)

= x3 – x2y + xy2 + x2y – xy2 + y3

= x3 + y3

As a result, the formula has been produced. Click here to learn more about this formula.

Factoring Formula 10: x3 – y3 = (x – y) (x2 + xy + y2)

The final one of the Factoring Formulas is derived by adhering to the given process.

In order to derive this equation, students should begin with the right side of the equation and work their way to the left.

(x – y) (x2 + xy + y2)

= x3 + x2y + xy2 – x2y – xy2 – y3

= x3 – y3

Thus, the formula is produced. It is possible for students to gain a deeper understanding of this formula by clicking here.

Examples Using Factoring Formulas

1. Factorise the expression 8×3 + 27.

Factoring: 8×3 + 27.

To factorise this, students must utilise the a3 + b3 formula, one of the unique Factoring Formulas.

The given expression can be written as

8×3 + 27 = (2x)3 + 33.

The formula a3 + b3 will be changed to a = 2x and b = 3.

The equation is (a3 + b3 = (a + b) (a2 – ab + b2)

(2x)3 +33 = (2x+3) ((2x)2 − (2x)(3)+32)

=(2x+3) (4×2−6x+9)

Answer is 8×3 + 27 = (2x + 3) (4×2 – 6x + 9).

2. Factorize x2 + 4xy + 4y2.

Factoring:  x2 + 4xy + 4y2.

The given expression can be written as (x)2 + 2 (x) (2y) + (2y)2.

With the use of the (a + b)2 formula (one of the unique Factoring Formulas):

a2 + 2ab + b2 = (a + b)2

Replace a = x and b = 2y in the following formula:

(x)2 + 2 (x) (2y) + (2y)2 = (x + 2y)2

x2 + 4xy + 4y2 = (x + 2y)2 (or) (x + 2y) (x + 2y).