Simplify cos⁴theta – sin⁴theta

Algebraic expressions are those equations that are obtained when operations like addition, subtraction, multiplication, division, etc. are operated upon by any variable. Algebraic expressions contain variables, constants, operators, exponents, etc. Let’s try and understand algebraic expressions in more detail,

Types of Algebraic expression

There are 3 main types of algebraic expressions which are monomial algebraic expressions, binomial algebraic expressions, and polynomial algebraic expressions. Let’s take a look at their definitions,

• Monomial Expression: An algebraic expression which is having only one term is known as a monomial.

Examples: 3x⁴, 3xy, 3x, 8y, etc.

• Binomial Expression: A binomial expression is an algebraic expression that contains two terms, which are unlike, that is, are different from each other.

Examples: 5xy + 8xyz, 9x – 7xy etc.

• Polynomial Expression: A polynomial expression is defined as an expression with more than one term with non-negative integral exponents of a variable.

Examples: ax + by + ca, x³ + 2x + 3, etc.

Terms in algebraic expressions

There are different terms used in algebraic expressions like variable, constant, term, coefficients, degree, etc. Below are the proper definitions of various terms,

1. Variable: In mathematics, a symbol that does not have a particular value is called a variable.
2. Constant: A symbol that has a fixed numerical value is called to be a constant. All numbers are constants.
3. Term: A term can be a variable alone (or) a constant alone (or) it can be a mixture of variables and constants by the operation of multiplication or division.
4. Coefficients: The fixed (or constant) number part along with the sign (positive or negative) associated with each algebraic term is called it’s coefficient.
5. Degree: The degree of the polynomial is that the highest integral power of the variable(s) of its terms when the polynomial is expressed in its standard form. It is the sum of exponents of the variables within the term if has quite more than one variable. 

Classification on the basis of the degree

Based on the exponents present in algebraic expressions, there are different types, for instance, if the degree is 1, then the algebraic expression has a first degree. If the degree is 2, then the algebraic expression has a second degree, and so on.

• First Degree: It is an algebraic expression whose degree is 1. Example: 5x, x, y,…etc.
• Second Degree: It is an algebraic expression whose degree is 2. Example: 5x², x² + 3xy + 12y² + 3x – 8y + 9,…etc.
• Third Degree: It is an algebraic expression whose degree is 3. Example: 5x³, x³ + 3xy + 12y², y² + 3x – 8y³ + 9,…etc.

Classification on the basis of variables

Based on the number of terms present in the algebraic expression, the expression is defined. For instance, if only one term is present in the algebraic expression, and so on.

• With One Variable: It is an algebraic expression that contains one variable only. Example: 5x, x + 2, y – 9,…etc.
• With Two Variables: It is an algebraic expression that contains two variables only. Example: 7xy, 5x² + z, x² + 3xy + 12y², y² + 3x – 8y + 9,…etc.
• With Three Variables: It is an algebraic expression that contains three variables only. Example: 6xyz, 5x³ + 3y + z, x³ + 3xy + 12y2z, y² + 3xz – 8y³ + 9,…

Simplify cos⁴ θ – sin⁴ θ

Solution:

cos⁴θ – sin⁴θ

= (cos²θ)² – (sin²θ)²

= (cos²θ + sin²θ)(cos²θ – sin²θ)

= 1 × (cos²θ – sin²θ)

= cos²θ – (1 – cos²θ)

= 2cos²θ – 1

Sample Problems

Question 1: Subtract (2x² – 5x + 7) from (3x² + 4x – 6)

Solution:

 (3x² + 4x – 6) – (2x² – 5x + 7)

= 3x² + 4x – 6 – 2x² + 5x – 7

= x² + 9x – 13.

Question 2: Simplify the expression: 12m² – 9m + 5m – 4m² – 7m + 10.

Solution:

Rearranging the terms, 

= (12 – 4)m² + (5 – 9 – 7)m + 10

= 8m² + (-4 – 7)m + 10

= 8m² + (-11)m + 10