Algebraic expressions are mathematical expressions that consists of numbers, variables and mathematical operators.
Terms, factors and coefficients
 Factors: If algebraic expressions are expressed as the product of numbers, variables or expressions, then each of these numbers and expressions is called the factor of algebraic expressions.

Monomials, Binomials, & Polynomials
Like & Unlike terms
Addition & Subtraction of Algebraic Expressions



Multiplication of algebraic expressions

 Product of two monomials = (product of their numerical coefficients) × (product of their variable parts)
Example 1: 1ab and 2a²b³
SOLUTION: (1ab) × (2a²b³)
= {1 × (2)} × {ab × a²b³}
= 2 x a^{1+2} x b^{1+3}= 2a³b⁴.
 Multiplying a polynomial by a polynomial:
Suppose (x + y) and (p + q) are two polynomials. By using the distributive property of multiplication over addition twice, we may find their product as given below.
EXAMPLE 3: Multiply (1a³ – 2a² – a + 3) by (4 – 5a + 6a^{2})
SOLUTION: Arranging the terms of the given polynomials in descending power of a and then multiplying:
1a³ – 2a² – a + 3
× (4 – 5a + 6a²)
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6a⁵  12a⁴ – 6a³ + 18a² ⇐ multiplication by 6a^{2}.
 5a⁴ + 10a³ + 5a² – 15a ⇐ multiplication by 5a.
+ 4a³ – 8a²  4a + 12 ⇐ multiplication by 4.
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6a⁵ – 17a⁴ + 8a³ + 15a² – 19a + 12 ⇐ multiplication by (4 – 5a + 6a²)
_________________________________Identities and its applications
 Some standard identities are:
EXAMPLE 1: Expand (a – 2b)^{3} using standard algebraic identities.
SOLUTION: (a– 2b)^{3 }is of the form (ab)^{3} where a = a and b = 2b. So we have,
(a – 2b)^{3} = (a)^{3} – (2b)^{3}– 3(a)(2b)(a – 2b) = a^{3} – 8b^{3} – 6a^{2}b + 12ab^{2 } [ANS]
EXAMPLE 2: Find . Given = 12 and ab=3.
SOLUTION:
Practice these questions
Q1) Multiply the following:
a) 11x^{2}y and 3 b)12 and 33 c)4 and 6Q2) Find the value of when
Q3) Simplify the following:
a) (x+2y)(3x+4y) – (5x+6y)(7x+8y) b)(9a+10b)(2x+3y) – (4a+5b)(6c+7d)Q4) Find the product of the following using identities:
a)(8x+9y)(8x9y) b)(10x+11y+23z)^{2} c)(45a6b)^{3} d)(7a+8b)^{3 }Q5) Simplify using identities:
a)121 x 121 – 1001^{2 } b)999^{2 }c)101^{2 }d)Q6) If , find the value of .
Q7) Find the value of x if 12x=
Q8) Tick the pair of like terms:
Q9) If , find the value of .
Recap
 Algebraic expressions are mathematical expressions that consists of numbers, variables and mathematical operators.
 If algebraic expressions are expressed as the product of numbers, variables or expressions, then each of these numbers and expressions is called the factor of algebraic expressions.
 Algebraic expressions which contain one nonzero term only are called monomials.
Algebraic expressions which contain two nonzero terms are called binomials.
 Product of two monomials = (product of their numerical coefficients) × (product of their variable parts)



