Factoring Formulas
Before knowing special factoring formulas, let us recall what is factoring. Factoring an algebraic expression is writing it as the product of two or more expressions. There are several methods to factorize expressions. One of them is using special factoring formulas.
What are Factoring Formulas?
We use some algebraic formulas as special factoring formulas. These algebraic identities can be verified by solving the LHS and RHS of the expression on both sides. Some factoring formulas that can be used are as given below,
Let us derive each of these formulas and more.
Factoring Formula 1: (a + b)2 = a2 + 2ab + b2
Let us start with the left-hand side of this formula and reach the right-hand side at the end.
(a + b)2 = (a + b) (a + b)
= a2 + ab + ba + b2 (Multiplied the binomials)
= a2 + 2ab + b2
Hence the formula is derived. You can learn this formula in detail by clicking here.
Factoring Formula 2: (a - b)2 = a2 - 2ab + b2
Let us start with the left-hand side of this formula and reach the right-hand side at the end.
(a - b)2 = (a - b) (a - b)
= a2 - ab - ba + b2 (Multiplied the binomials)
= a2 - 2ab + b2
Hence the formula is derived. You can learn this formula in detail by clicking here.
Factoring Formula 3: (a + b) (a - b) = a2 - b2
Let us start with the left-hand side of this formula and reach the right-hand side at the end.
(a + b) ( a - b) = a2 - ab + ba + b2 (Multiplied the binomials)
= a2 - b2
Hence the formula is derived. You can learn this formula in detail by clicking here.
Factoring Formula 4: (x + a) (x + b) = x2 + (a + b) x + ab
Let us start with the left-hand side of this formula and reach the right-hand side at the end.
(x + a) ( x + b) = x2 + xb + ax + b2 (Multiplied the binomials)
= x2 + (a + b) x + ab
Hence the formula is derived.
Factoring Formula 5: (a + b)3 = a3 + b3 + 3ab (a + b)
Let us start with the left-hand side of this formula and reach the right-hand 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 + 3a2 b + 3ab2 (or)
= a3 + b3 + 3ab (a + b)
Hence the formula is derived. You can learn this formula in detail by clicking here.
Factoring Formula 6: (a - b)3 = a3 - b3 - 3ab (a - b)
Let us start with the left-hand side of this formula and reach the right-hand 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 - 3a2 b + 3ab2 (or)
= a3 - b3 - 3ab (a - b)
Hence the formula is derived. You can learn this formula in detail by clicking here.
Factoring Formula 7: (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
Let us start with the left-hand side of this formula and reach the right-hand side at the end.
(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
Hence the formula is derived. You can learn this formula in detail by clicking here.
Factoring Formula 8: x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – xy – yz – xz)
Let us start with the right-hand side of this formula and reach the left-hand side 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 the other terms are canceled)
Hence the formula is derived.
Factoring Formula 9: x3 + y3 = (x + y) (x2 – xy + y2)
Let us start with the right-hand side of this formula and reach the left-hand side at the end.
(x + y) (x2 – xy + y2)
= x3 - x2y + xy2 + x2y - xy2 + y3
= x3 + y3
Hence the formula is derived. You can learn this formula in detail by clicking here.
Factoring Formula 10: x3 - y3 = (x - y) (x2 + xy + y2)
Let us start with the right-hand side of this formula and reach the left-hand side at the end.
(x - y) (x2 + xy + y2)
= x3 + x2y + xy2 - x2y - xy2 - y3
= x3 - y3
Hence the formula is derived. You can learn this formula in detail by clicking here.
Examples Using Factoring Formulas
Example 1: Factorize the expression 8x3 + 27.
Solution:
To factorize: 8x3 + 27.
We will use the a3 + b3 formula (one of the special factoring formulas) to factorize this.
We can write the given expression as
8x3 + 27 = (2x)3 + 33
We will substitute a = 2x and b = 3 in the formula of a3 + b3.
a3 + b3 = (a + b) (a2 - ab + b2)
\[ \begin{align} (2x)^3+3^3 &=(2x+3)((2x)^2-(2x)(3)+3^2)\\[0.2cm] &= (2x+3) (4x^2-6x+9)\end{align}\]
Answer: 8x3 + 27 = (2x + 3) (4x2 - 6x + 9).
Example 2: Factorize x2 + 4xy + 4y2.
Solution:
To factorize: x2 + 4xy + 4y2.
We can write the given expression as: (x)2 + 2 (x) (2y) + (2y)2.
Using (a + b)2 formula (one of the special factoring formulas):
a2 + 2ab + b2 = (a + b)2
Substitute a = x and b = 2y in this formula:
(x)2 + 2 (x) (2y) + (2y)2 = (x + 2y)2
Answer: x2 + 4xy + 4y2 = (x + 2y)2 (or) (x + 2y) (x + 2y).
Example 3: Factorize x2 - 6x + 9 using factoring formulas.
Solution:
We have, x2 - 6x + 9 = x2 - 2(3)(x) + 32
Using (a - b)2 factoring formula we have, (a - b)2 = a2 - 2ab + b2
x2 - 2(3)(x) + 32 = (x - 3)2
Answer: x2 - 6x + 9 = (x - 3)2
FAQs on Factoring Formulas
What Are Factoring Formulas?
Factoring formulas are used to write an algebraic expression as the product of two or more expressions. Some important factoring formulas are given as,
- (a + b)2 = a2 + 2ab + b2
- (a - b)2 = a2 - 2ab + b2
- (a + b) (a - b) = a2 - b2
- (x + a) (x + b) = x2 + (a + b) x + ab
How to Apply Factoring Formulas?
Factoring formulas are used to factorize expressions depending upon their forms. The terms in expression can be compared with a suitable factoring formula to factorize.
What Is the Factoring Formula For Difference of Cubes?
The factoring formula for difference of cubes is given as, x3 - y3 = (x - y) (x2 + xy + y2).
What Is the Factoring Formula For Sum of Cubes?
The factoring formula for sum of cubes is given as, x3 + y3 = (x + y) (x2 – xy + y2).