Algebraic expressions are mathematical expressions that consists of numbers, variables and mathematical operators.

# Terms, factors and coefficients

- Term: A term can be a number, a variable, or a constant multiplied by a variable or variables. Each term in an algebraic expression is separated by a positive sign or negative sign.

- 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.

- Coefficients: When a term of an algebraic expression is made up of a constant multiplied by a variable or variables, then that constant is called a coefficient of the algebraic expression.

# Monomials, Binomials, & Polynomials

# Like & Unlike terms

# Addition & Subtraction of Algebraic Expressions

- Horizontal method of addition:

Step1: Suppose we have to add 3 expression 1a+2b+3c, 4a+5b+6c, and 7a+8b+9c

Step2: write the expression in a horizontal line and then add all the like terms:

- Column Method of addition:

Write the 3 expressions in 3 rows with like terms below each other and then add the like terms column-wise.

- Subtraction of algebraic expression:

Step1: Suppose we have to subtract 10a+23b+45c from 67a+89b+20c

Step2: Write the above expressions in two rows such that the like terms occur one below the other. Keep the expression to be subtracted in the second row.

# Multiplication of algebraic expressions

- In multiplication of algebraic expressions:

(i) The product of two factors having like signs is positive, and the product of two factors having unlike signs is negative.

(ii) if a is a variable and p, q are positive integers, then

- Multiplication of Two Monomials:

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 monomial:

Multiply each term of the polynomial by the monomial, using the distributive property p × (q + r) = p × q + p × r.

**EXAMPLE 2: **1x²y² × (2x² - 3xy + 4y²)

**SOLUTION: **1x²y² × (2x² - 3xy + 4y²)

= (1x²y²) × (2x²) + (1x²y²) × (-3xy) + (1x²y²) × (4y²)

= 2x⁴y² - 3x³y³ + 4x²y⁴

- 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)

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# Identities and its applications

- If an equation is true for all values of the variables in it, then that equation is called an identity.

- 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) = 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 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 – 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.
- A term can be a number, a variable, or a constant multiplied by a variable or variables. Each term in an algebraic expression is separated by a positive sign or negative sign.
- 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.
- When a term of an algebraic expression is made up of a constant multiplied by a variable or variables, then that constant is called a coefficient of the algebraic expression.
- Algebraic expressions which contain one non-zero term only are called monomials.

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