Factoring Formulas in Algebra

Factoring Formulas in Algebra

Factoring formulas in algebra are special algebraic identities used to break down complex polynomials into the product of two or more simpler expressions. Factoring is the exact opposite of expanding. Mastering these 10 special factoring formulas is vital for CBSE Class 8 to 10 board exams and competitive exams like JEE.

Class: 8, 9 & 10
Topic: Algebra & Polynomials
Exams: CBSE · ICSE · JEE Foundation

What are Special Factoring Formulas?

We use standard algebraic identities as special factoring formulas. Instead of expanding two brackets, factoring requires us to look at an expanded polynomial and shrink it back into its multiplied factors. These formulas can be verified by solving the Left-Hand Side (LHS) and Right-Hand Side (RHS) of the expression.

✓ Quick TipFactoring is simply the reverse of expanding! If you multiply your final factored brackets together and get the original algebraic expression back, your factoring is correct.

1. Basic Quadratic Factoring Formulas

These are the foundational identities taught in CBSE Class 8. They deal with powers of 2 (squares).

Perfect Square of a Sum: a2 + 2ab + b2 = (a + b)2
Perfect Square of a Difference: a2 − 2ab + b2 = (a − b)2
Difference of Two Squares: a2 − b2 = (a + b)(a − b)
Product of Two Binomials: x2 + (a + b)x + ab = (x + a)(x + b)

2. Cubic Factoring Formulas

Introduced in Class 9, these formulas handle polynomials with a power of 3. Students often confuse the perfect cube formulas with the sum/difference of cubes.

Sum & Difference of Cubes
a3 + b3 = (a + b)(a2 − ab + b2)
a3 − b3 = (a − b)(a2 + ab + b2)
Perfect Cube Formulas
a3 + b3 + 3ab(a + b) = (a + b)3
a3 − b3 − 3ab(a − b) = (a − b)3

3. 3-Variable Factoring Formulas (Class 9 CBSE)

For complex polynomials containing variables x, y, and z (or a, b, c), these standard identities are essential for advanced factorization.

Square of a Trinomial:
a2 + b2 + c2 + 2ab + 2bc + 2ca = (a + b + c)2
Three-Variable Cube Identity:
x3 + y3 + z3 − 3xyz = (x + y + z)(x2 + y2 + z2 − xy − yz − xz)Note: If x + y + z = 0, then x3 + y3 + z3 = 3xyz.

List of All 10 Special Factoring Formulas

Formula No. Algebraic Identity (Expanded to Factored)
Formula 1 a2 + 2ab + b2 = (a + b)2
Formula 2 a2 − 2ab + b2 = (a − b)2
Formula 3 a2 − b2 = (a + b)(a − b)
Formula 4 x2 + (a + b)x + ab = (x + a)(x + b)
Formula 5 a3 + b3 + 3ab(a + b) = (a + b)3
Formula 6 a3 − b3 − 3ab(a − b) = (a − b)3
Formula 7 a2 + b2 + c2 + 2ab + 2bc + 2ca = (a + b + c)2
Formula 8 x3 + y3 + z3 − 3xyz = (x + y + z)(x2 + y2 + z2 − xy − yz − xz)
Formula 9 x3 + y3 = (x + y)(x2 − xy + y2)
Formula 10 x3 − y3 = (x − y)(x2 + xy + y2)

Examples Using Factoring Formulas

Example 1: Factoring Sum of Cubes

Factorize the expression: 8x3 + 27

Solution:
We can rewrite the given expression as perfect cubes: (2x)3 + (3)3.
Now substitute a = 2x and b = 3 into the formula a3 + b3 = (a + b)(a2 − ab + b2).= (2x + 3) [ (2x)2 − (2x)(3) + (3)2 ]
= (2x + 3)(4x2 − 6x + 9)
Answer: 8x3 + 27 = (2x + 3)(4x2 − 6x + 9)

Example 2: Perfect Square Trinomial

Factorize the algebraic expression: x2 + 4xy + 4y2

Solution:
We can write the given expression as: (x)2 + 2(x)(2y) + (2y)2.
Using the special formula a2 + 2ab + b2 = (a + b)2.
Substitute a = x and b = 2y into this formula:(x)2 + 2(x)(2y) + (2y)2 = (x + 2y)2
Answer: x2 + 4xy + 4y2 = (x + 2y)2

Example 3: Difference Perfect Square

Factorize: x2 − 6x + 9

Solution:
We have, x2 − 6x + 9 = x2 − 2(3)(x) + 32.
Using the factoring formula a2 − 2ab + b2 = (a − b)2, where a = x and b = 3: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 standard algebraic identities used to write an expanded algebraic expression as the product of two or more simpler expressions. For example, a² - b² factors perfectly into (a - b)(a + b).
How to apply factoring formulas?
To apply them, observe the form of your expression. Look for perfect squares, perfect cubes, or common terms. Once you match the expression to the Left-Hand Side of a standard identity, you simply substitute the values into the Right-Hand Side.
What is the factoring formula for difference of cubes?
The factoring formula for the 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 the sum of cubes is given as: x³ + y³ = (x + y)(x² - xy + y²).
Can we factor the sum of two squares (a² + b²)?
No, a² + b² is a prime polynomial over real numbers. It cannot be factored using real numbers, unlike the difference of squares (a² - b²).