Algebraic Identities and Factorisation Notes for SSC Exams
Rahul Kumar
SSC Exam Expert & Content Editor
In this note
- 1.Identities of Degree Two
- 2.Identities of Degree Three
- 3.Using the Identities for Quick Calculation
- 4.Finding the Value from Given Data
- 5.The x + 1/x Family
- 6.Factorisation Methods
- 7.Quick Tips
- 8.Quick Revision Points
- 9.Practice Questions With Answers
- 10.How to Practise This Chapter
- 11.Frequently Asked Questions
PDF Notes: Algebraic Identities and Factorisation Notes for SSC Exams
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Algebraic identities are equations that are true for all values of the variables, and they are the most useful tool in the algebra part of SSC maths. A student who knows the identities can solve questions on squares, cubes and the value of x plus 1 by x in one or two lines. These algebraic identities and factorisation notes for SSC give all the important identities in tables with solved examples and practice questions.
Identities of Degree Two
| Identity | Form |
|---|---|
| Square of a sum | (a + b)2 = a2 + 2ab + b2 |
| Square of a difference | (a − b)2 = a2 − 2ab + b2 |
| Difference of squares | a2 − b2 = (a + b)(a − b) |
| Product of two binomials | (x + a)(x + b) = x2 + (a + b)x + ab |
| Sum and difference | (a + b)2 + (a − b)2 = 2 (a2 + b2); (a + b)2 − (a − b)2 = 4ab |
| Three variables | (a + b + c)2 = a2 + b2 + c2 + 2 (ab + bc + ca) |
Identities of Degree Three
| Identity | Form |
|---|---|
| Cube of a sum | (a + b)3 = a3 + b3 + 3ab (a + b) |
| Cube of a difference | (a − b)3 = a3 − b3 − 3ab (a − b) |
| Sum of cubes | a3 + b3 = (a + b)(a2 − ab + b2) |
| Difference of cubes | a3 − b3 = (a − b)(a2 + ab + b2) |
| Three cubes | a3 + b3 + c3 − 3abc = (a + b + c)(a2 + b2 + c2 − ab − bc − ca) |
| Special case | If a + b + c = 0, then a3 + b3 + c3 = 3abc |
Using the Identities for Quick Calculation
Example 1. Find 1012.
(100 + 1)2 = 10000 + 200 + 1 = 10201.
Example 2. Find 98 × 102.
(100 − 2)(100 + 2) = 1002 − 22 = 9996.
Example 3. Find 472 − 432.
(47 + 43)(47 − 43) = 90 × 4 = 360.
Finding the Value from Given Data
Example 4. If a + b = 7 and ab = 10, find a2 + b2.
a2 + b2 = (a + b)2 − 2ab = 49 − 20 = 29.
Example 5. If a − b = 2 and ab = 15, find a2 + b2.
a2 + b2 = (a − b)2 + 2ab = 4 + 30 = 34.
Example 6. If a + b = 5 and ab = 6, find a3 + b3.
a3 + b3 = (a + b)3 − 3ab (a + b) = 125 − 90 = 35.
The x + 1/x Family
| Given | Find | Formula |
|---|---|---|
| x + 1/x = p | x2 + 1/x2 | p2 − 2 |
| x + 1/x = p | x3 + 1/x3 | p3 − 3p |
| x − 1/x = q | x2 + 1/x2 | q2 + 2 |
| x − 1/x = q | x3 − 1/x3 | q3 + 3q |
Example 7. If x + 1/x = 3, then x2 + 1/x2 = 9 − 2 = 7 and x3 + 1/x3 = 27 − 9 = 18.
Factorisation Methods
| Method | Example |
|---|---|
| Taking out the common factor | 6x + 9 = 3 (2x + 3) |
| Grouping | ax + ay + bx + by = (a + b)(x + y) |
| Splitting the middle term | x2 + 5x + 6 = x2 + 2x + 3x + 6 = (x + 2)(x + 3) |
| Using identities | x2 − 16 = (x + 4)(x − 4) |
| Using the cube identities | x3 + 8 = (x + 2)(x2 − 2x + 4) |
Quick Tips
- Memorise the nine main identities with their signs, because a wrong sign is the most common error.
- Use the identity of the difference of squares for quick multiplication.
- For x + 1/x problems, use the formulas and avoid finding x.
- Look for a + b + c = 0 in cube problems.
Quick Revision Points
- (a + b)2 = a2 + 2ab + b2.
- a2 − b2 = (a + b)(a − b).
- a3 + b3 = (a + b)3 − 3ab (a + b).
- If x + 1/x = p, then x2 + 1/x2 = p2 − 2.
- If a + b + c = 0, then a3 + b3 + c3 = 3abc.
Practice Questions With Answers
- Find 992.
Answer: 9801 - Find 103 × 97.
Answer: 9991 - If a + b = 9 and ab = 20, find a2 + b2.
Answer: 41 - If x + 1/x = 4, find x2 + 1/x2.
Answer: 14 - If x − 1/x = 2, find x2 + 1/x2.
Answer: 6 - Factorise x2 + 7x + 12.
Answer: (x + 3)(x + 4) - Factorise x2 − 25.
Answer: (x + 5)(x − 5) - If a + b + c = 0, find (a3 + b3 + c3) / abc.
Answer: 3 - If a − b = 1 and a2 + b2 = 25, find ab.
Answer: 12 - Find 522 − 482.
Answer: 400
How to Practise This Chapter
Learn three identities a day and solve five questions with each. Then attempt timed sets from our practice papers and the previous year papers page. Read the previous chapter, Linear Equations in Two Variables, and the next chapter, Lines and Angles. See all topics on the Mathematics notes page, and also read Algebra Formulas.
Frequently Asked Questions
What is an algebraic identity?
An algebraic identity is an equation that is true for all values of its variables, such as (a plus b) squared equals a squared plus 2ab plus b squared.
What is the formula of a squared minus b squared?
It is (a plus b) times (a minus b).
What is the formula of a plus b whole cube?
It is a cube plus b cube plus 3ab times (a plus b).
What is the formula of a cube plus b cube?
It is (a plus b) times (a squared minus ab plus b squared).
What is the identity of a cube plus b cube plus c cube minus 3abc?
It equals (a plus b plus c) times (a squared plus b squared plus c squared minus ab minus bc minus ca).
What happens if a plus b plus c is zero?
Then a cube plus b cube plus c cube equals 3abc.
How do I find x squared plus 1 by x squared when x plus 1 by x is given?
Square the given value and subtract 2.
What is factorisation?
Factorisation is writing an expression as a product of simpler expressions.
What are the methods of factorisation?
The methods are taking out the common factor, grouping, splitting the middle term and using identities.
How can identities help in quick calculation?
They turn a hard multiplication into an easy one, such as 98 times 102 into 100 squared minus 2 squared.
Are identities important for SSC?
Yes. Many algebra questions in SSC CGL and CHSL are solved with identities.
How many questions come from identities in SSC exams?
Usually two or three questions come from algebra and identities, but the number changes from year to year.
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Linear Equations in Two Variables Notes for SSC Exams
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Lines and Angles Notes for SSC Exams
Rahul Kumar
SSC Exam Expert & Content Editor
Rahul is a senior SSC exam strategist with 8+ years of experience helping aspirants crack CGL, CHSL and MTS exams. He writes in-depth notification breakdowns, exam pattern guides and preparation strategies.