Algebraic Expressions and Identities

Algebraic expressions and identities are fundamental concepts in mathematics. In this chapter, we will explore what algebraic expressions and identities are, how to work with them and their significance in mathematics and beyond. They offer a powerful toolkit for simplifying expressions, solving equations and understanding the relationships between variables and constants. By mastering these concepts, one can enhance their problem-solving skills and analytical abilities which make them better equipped to tackle the challenges of the modern world.

Algebraic Expression

An algebraic expression is a combination of variables and constants connected by addition (+), subtraction (−), multiplication (×) and division (÷). 

An algebraic expression 5a − 17 has one constant, one variable and two terms.

An algebraic expression 7a² − 3a + 9 has one constant, two variables and three terms.

Like and Unlike Terms

In algebraic expressions, terms that have the same variable and exponent are called "like terms".

Examples:

1) 2a, −5a and 3a are like terms.
2) 5x and −3x/2 are like terms.
3) 7p² and 11p² are like terms.

Terms with different variables or exponents are called "unlike terms".

Examples:

1) 2a and −3b are unlike terms.
2) 5x, 7y and −3z are unlike terms.
3) 7y² and 11z² are unlike terms.

Types of Algebraic Expression

1. Monomial: An algebraic expression with just one term is called a monomial.
   Examples of monomials are −7a, 2ab, 3x⁵, −2ab/5, −0.5p, −51, etc.
2. Binomial: An algebraic expression with two terms is called a binomial.
   Examples of binomials are (3a − 2b/5), (4x + 9y), (3x² − 8x³), etc.
3. Trinomial: An algebraic expression with three terms is called a trinomial.
   Examples of trinomials are (a − 5b + 3c), (3x + 2y − z/3), (a² − 2b² − c³), etc.
4. Quadrinomial: An algebraic expression that has four terms is called a quadrinomial.
   Examples of quadrinomials are 2x − 3x² + y² − 2z, 5a − 3b + 7c² + 11d, etc.
5. Polynomial: Any algebraic expression with one or more terms that have non-negative whole number exponents is called a polynomial.
   An example of a polynomial is 3x² − 2x + 5.

Degree of a Polynomial

The degree of a polynomial is determined by the highest exponent of the variable in that polynomial.

Examples:

1. In the polynomial 3x − 5, the highest exponent is 1.
   So, the degree is 1.
2. In the polynomial x2 + 3x − 5, the highest exponent is 2.
   So, the degree is 2.
3. c. In the polynomial 5 − 2x³ + 3x², the highest exponent is 3.
   So, the degree is 3.

Types of Polynomials by Degree:

→ If the degree of a polynomial is 1, it is called a linear polynomial.
→ If the degree of a polynomial is 2, it is called a quadratic polynomial.
→ If the degree of a polynomial is 3, it is called a cubic polynomial.
→ If the degree of a polynomial is 4, it is called a bi-quadratic polynomial.

Addition and Subtraction of Polynomials

Adding and subtracting polynomials is a fundamental operation.
Follow these steps for addition and subtraction of polynomials:

Step 1: Identify Like Terms.

First, identify the like terms in the polynomials.

Example: In the polynomials 3x² + 4y − 11 and 5x² − y + 5, the like terms are:

I) 3x² and 5x² (both have x² terms)
II) 4y and −y (both have y terms)
III) −11 and 5 (both are constants)

Step 2: Add or Subtract Like Terms.

Now, add or subtract the like terms separately.

In the example above, you would add the like terms as follows:

I) 3x² + 5x² = 8x² (Add)
II) 4y − y = 3y (Subtract)
III) −11 + 5 = −6 (Add)

So, the sum of the two polynomials is 8x² + 3y − 6

Step 3: Combine Unlike Terms.

If there are any unlike terms that cannot be added or subtracted further, leave them as they are in the final expression.

In this example, there are no more, unlike terms in the polynomial 8x² + 3y − 6. So, the answer is 8x² + 3y − 6.

Multiplication of Polynomials

Use the distributive property to multiply two polynomials which states that for any real numbers a, b, and c: a × (b + c) = (a × b) + (a × c)

Follow the steps to multiplying two polynomials:

Step 1: Multiply each term in the first polynomial by each term in the second polynomial.

For example, multiply the polynomials (2x + 3) and (2x − 1).

In this case, you would do the following:

(2x) × (2x) = 4x²
(2x) × (−1) = −2x
(3) × (2x) = 6x
(3) × (−1) = −3

Step 2: Combine like terms.

After multiplying each term, combine like terms (terms with the same variable and exponent).

In this case: 4x2 − 2x + 6x − 3

Step 3: Simplify the result.

Combine the like terms by adding or subtracting polynomials.

4x² −2x + 6x − 3

= 4x² + (−2 + 6)x − 3
= 4x² + (4)x − 3
= 4x² + 4x − 3

So, the product of (2x + 3) and (2x − 1) is 4x² + 4x −3.

Identity

An identity is an algebraic equation which is true for every value of the variables in them.

Standard Identities

Some of the most common standard identities are as follows: