Concept of Algebra:

Algebra is the branch of Maths that uses alphabetical letters to find unknown numbers. Basically these letters are called as variables. The values which are known in the given expression such as numbers are called as constants.

Algebra includes several forms of mathematical representations, such as real numbers, complex numbers, vectors, matrices, and so on. Equations form a crucial part of algebraic applications. 

Terms related to basic algebra skills are mentioned below.

 •  Exponent     •  Expression     •  Polynomial (Monomial, binomial and trinomial)

 •  Like terms and Unlike terms     •  Constants

1.An equation is a statement which implies two same identities separated by the “=” sign. 

2.An expression is a group of different terms separated by ‘+’ or ‘-‘ sign. 

3.Like terms are those terms whose variables and their exponents are the same. 

Properties of Algebra:

The ‘commutative law’ 

a + b = b + a

a * b = b * a

The ‘associative law’ 

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

The ‘distributive law’

a (b + c) = a * b + a * c

While solving questions in algebra follow the below steps:

 •  Identify the relationship between expressions. 

 •  If an expression has a known value, write the other expression in terms of the expression with a known value. 

 •  Substitute the known value for the expression.

 •  Evaluate the expression.

Some Important Algebraic Formulae:

1. (a + b)² = a² + 2ab + b²

2. (a – b)² = a² – 2ab + b²

3. (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

4. (a + b)³ = a³ + b³ + 3ab (a + b)

5. (a – b)³ = a³ – b³ – 3ab (a – b)

6. (a + b + c)³ = a³ + b³ + c³ + 3 (a + b)(b + c)(c + a)

7. a² – b² = (a + b)(a – b)

8. (a + b)² – (a – b)² = 4ab

9. a³ + b³ = (a + b)(a² – ab + b²)

10. a³ – b³ = (a – b)(a² + ab + b²)

11. a³ + b³ + c³ – 3abc = (a + b + c)(a² + b² + c² – ab – bc – ca)

If x + (1 / x) = 3, Find the value of x² + (1 / x²).

a) 5                              b) 7                              c) 9                              d) 11

Solution:

Given that, x + (1 / x) = 3

Squaring both sides,

(x + 1 / x)² = (3)²

=> x² + (1 / x²) + 2 × x × (1 / x) = 9

=> x² + (1 / x²) + 2 = 9

=> x² + (1 / x²) = 9 – 2

=> x² + (1 / x²) = 7.

Answer: b

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