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.

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  Monomials, Binomials, & Polynomials

  

  Like & Unlike terms

  

  Addition & Subtraction of Algebraic Expressions

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        Multiplication of algebraic expressions

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          • 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 a¹⁺² x b¹⁺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 + 6a²)

          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 6a².
          
          - 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)
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          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 (a-b)³ where a = a and b = 2b. So we have,

          (a – 2b)³ = (a)³ – (2b)³– 3(a)(2b)(a – 2b) = a³ – 8b³ – 6a²b + 12ab² [ANS]
          

          EXAMPLE 2: Find . Given = 12 and ab=3.

          SOLUTION:

          Practice these questions

          Q1) Multiply the following:
          a) 11x²y and 3 b)12 and 33 c)4 and 6

          Q2) 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)(8x-9y) b)(10x+11y+23z)² c)(45a-6b)³ d)(7a+8b)³

          Q5) Simplify using identities:
          a)121 x 121 – 1001² b)999² c)101² 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 non-zero term only are called monomials.

          Algebraic expressions which contain two non-zero terms are called binomials.