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# Algebraic Expressions

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.

• # 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 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.

## Quiz for Algebraic Expressions

 Q.1 3x/y, 7xy⁻¹, 19 x/y, all these expressions can be termed as ____________. a) Unlike terms b) Polynomials c) Like terms d) None of the above
 Q.2 A man travel 2/9th of his journey by car, 4/6th of his journey by bus and for the rest of his journey he opted to walk. If he covered a total distance of y km, what distance did he travel on foot? a) y/18 b) 2y/9 c) y/3 d) y/9
 Q.3 Add the expressions: (i) 5xy + 6x - 24y² (ii) -x +7 xy - 4y² (iii) 5y² + 9x - 4 a) 12xy + 14x - 28y² -4 b) 12xy + 16x -28y² -4 c) 12xy + 16x -20y² -4 d) none of the above
 Q.4 Take away 5x - y² from 8y - 6y² + xy a) 8y - 7y² -xy b) 8y - 5y² +xy - 5x c) 8y - 7y² -xy + 5x d) 8y + 5y² +xy - 5x
 Q.5 What is 35% of 4y/7? a) y/5 b) 4y/7 c) 7y/20 d) none of the above
 Q.6 Find the sum: (i) 7a² - 5a + 2 (ii) 3a² - 7 (iii) 2a + 9 (iv) 1 + 2a - 5a² a) a² - 5a + 1 b) a² - 5a + 5 c) 5a² - 5a + 1 d) 5a² - a + 5
 Q.7 Find the sum: 2.5a + 4.7a - 3.2a - 5.8a + a a) 1.8a b) -1.8a c) -0.8a d) 0.8a
 Q.8 Simplify: 3 5/6 ab - 1 2/3 ab - 2ab + 3 1/2 ab a) 11/5 ab b) 11/3 ab c) 11/6 ab d) 3/11 ab
 Q.9 Simplify: 5x⁴ - 7x² + 8x - 1 + 3x³ - 9x² + 7 - 3x⁴ + 11x - 2 + 8x² a) 2x⁴ + 3x³ - 8x² + 19x + 4 b) 2x⁴ + 3x³ - 8x² - 19x + 4 c) 2x⁴ - x³ - 8x² + 19x + 4 d) none of the above
 Q.10 Subtract x³ - y³ +2 from x³ + y³ + 3xy + 1 a) 2y³ - x³ +3 xy - 1 b) 2y³ + x³ - 1 c) 2y³ - x³ +3 xy + 1 d) 2y³ + 3 xy - 1