Simplify y = sin 4x cos4x

An algebraic expression in mathematics can be said to be an expression that is made up of variables and constants, along with algebraic operations like addition, subtraction, etc. Expressions are made up of terms. Algebraic expressions are represented with the help of unknown variables, constants, and coefficients. The combination of these three (as terms) is called to be an expression. It is to be mentioned that, unlike the algebraic equation, an algebraic expression has no sides or is equal to a sign.

Types of Algebraic expression

There are 3 main types of algebraic expressions which are monomial algebraic expressions, binomial algebraic expressions, and polynomial algebraic expressions. Lets 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.

Expression of other types

Other than monomial, binomial, and polynomial types of expressions, and algebraic expression can also be classified into two additional types which are:

• Numeric Expression: It consists of numbers and operations, but never includes any variable. Examples are 10 + 5, 15 ÷ 2, etc.
• Variable Expression: It is an expression that contains variables along with numbers and operations to define an expression. Examples include 4x + y, 5ab + 33, etc.

General algebraic formulas

1. (a + b)² = a² + 2ab + b²
2. (a – b)² = a² – 2ab + b²
3. a² – b² = (a – b)(a + b)
4. (a + b)³ = a³ + b³ + 3ab(a + b)
5. (a – b)³ = a³ – b³ – 3ab(a – b)  
6. a³ – b³ = (a – b)(a² + ab + b²)
7. a³ + b³ = (a + b)(a² – ab + b²)

Simplify: y = sin 4x × cos4x

Solution:

y = 1/2 sin(8 x)

y = 1/4 i e^{(-8 i x)} – 1/4 i e^{(8 i x)}

y = 4 sin(x) cos⁷(x) – 28 sin³(x) cos⁵(x) + 28 sin⁵(x) cos³(x) – 4 sin⁷(x) cos(x)

y = 16 sin(π/4 – 2 x) sin(π/4 – x) sin(x) sin(x + π/4) sin(2x + π/4) cos(x)

Sample Problems

Question 1: Are the algebraic expressions polynomials?

Answer:

No, not all the algebraic expressions are polynomials. But we can say that all polynomials are algebraic expressions. The only difference is polynomials include only variables and coefficients with mathematical operations(+, -, ×) but algebraic expressions include irrational numbers in the powers as well.

Also, is there one factor, polynomials are continuous function (eg: x² + 2x + 1) but algebraic expression may not be continuous sometimes (eg: 1/x² – 1 is not continuous at 1).

Question 2: What are like and unlike terms?

Answer:

1. Like Term: The terms having the same algebraic factors are known as like terms.
2. Unlike Term: The terms having different algebraic factors are known as, unlike terms.

Question 3: Simplify: 12m² – 9m + 5m – 4m² – 7m + 10.

Solution:

12m² – 9m + 5m – 4m² – 7m + 10

= (12 – 4)m² + (5 – 9 – 7)m + 10

= 8m² + (-4 – 7)m + 10

= 8m² + (-11)m + 10

= 8m² – 11m + 10.

Question 4: Find the degree of the monomial 7?

Solution: 

The degree of any constant term is zero (0). So, the degree of the monomial 7 is 0.