Hello, and welcome to your mathematics lesson today. We are going to explore fundamental concepts in algebra. By the end of this lesson, you will understand what algebra is, how to work with algebraic expressions, and how to add, subtract, multiply, and divide them.
Let us begin with the very foundation: what is algebra? Algebra is simply a generalized form of arithmetic. In arithmetic, you work with numbers that have definite values, like 5, or minus 8, or 0.64. But in algebra, we use letters alongside numbers. These letters are called variables, or literal numbers, or simply literals. Unlike numbers, variables do not have fixed values. For example, expressions like 7x, 3x – 2, 5a + b, and x + 2y – 7z are all algebraic expressions.
Now, let us look at the signs and symbols we use in algebra. The basic operations — plus, minus, multiply, and divide — work exactly as they do in arithmetic. But algebra introduces some special symbols. The equals sign means "is equal to." The symbol ≠ means "is not equal to." The symbol < means "is less than." The symbol > means "is greater than." We also have ≮ meaning "is not less than." And ≯ meaning "is not greater than." The symbol ∴ means "therefore." The symbol ∵ means "because" or "since." The symbol ~ means "difference between." The symbol ⇒ means "implies that."
Next, we need to distinguish between constants and variables. A constant is a symbol with a fixed numerical value in every situation. Numbers like 5, 30, 256, –7, or fractions like 5/3 are all constants. A variable, on the other hand, is a symbol whose value changes depending on the situation.
Letters like x, y, p, q, or expressions like 5x are variables.
Here is an important insight. In the expression 3x, 3 is a constant and x is a variable. But 3x itself is also a variable. Why? Because as the value of x changes, the value of 3x changes too. Similarly, 3 + x, x – 3, and x ÷ 3 are all variables.
Remember this: any combination of a constant and a variable is always a variable.
Now let us understand what a term is. A term is a constant, or a variable, or a product or quotient of constants and variables. For instance, x is a term that is just a variable. 4x is a term that is the product of a constant and a variable.
And 3/y is a term that is the quotient of a constant and a variable.
A constant term is a term that contains no variables at all.
So 3, –20, 5/7, or –4/9 are all constant terms.
Terms can be classified as like terms or unlike terms. Like terms are terms that have the same literal coefficients. They may differ only in their numerical coefficients. For example, xy, 5xy, and –4xy are all like terms.
Similarly, –8x²y, 7x²y, and 1.5x²y are like terms.
Unlike terms do not have the same literal coefficients. For example, 6a, 6ab, and 6ac are unlike terms.
Also, 2xy, 2x²y, and 2xy² are unlike terms.
Let us move on to algebraic expressions. An algebraic expression is a collection of one or more terms, separated by plus or minus signs. For example, 5x has one term. 3x + 8z has two terms: 3x and 8z. 4x – y + 7 has three terms: 4x, y, and 7.
In this last expression, 7 is the constant term.
Based on the number of terms, we classify algebraic expressions. A monomial has exactly one term. Examples include –8, z, xy, 2x, or 2x/5y.
Remember, terms are separated by plus and minus signs only, not by multiplication or division signs.
A binomial has two unlike terms. Examples are 5x + 2y, 7 – x, or 2a + b/2.
Remember, the terms must be unlike — you cannot have 5x + 3x as a binomial because these are like terms that combine to 8x.
A trinomial has three unlike terms.
Examples include ax² + bx + c, 2x² – 7x + 4, or xy – x + y².
A multinomial has two or more terms. So every binomial and trinomial is also a multinomial. A polynomial is an algebraic expression with one or more unlike terms, where the power of each variable must be a whole number.
This means every monomial, binomial, trinomial, and multinomial is also a polynomial.
But expressions like 1/x, 3/(x+5), or x^(2/3) are not polynomials because variables must have whole number powers.
But expressions like 1/x, 3/(x+5), or x^(2/3) are not polynomials because variables must have whole number powers.
Also, expressions with variables under roots, like the square root of x, are not polynomials.
Let us understand products and factors. A product is the result of multiplying two or more constants or variables together. For example, 5xy is the product of 5, x, and y.
Each of these — 5, x, and y — is called a factor of the product.
A coefficient is any factor or group of factors of a product.
In 5axy, 5 is the coefficient of axy, 5x is the coefficient of ay, and xy is the coefficient of 5a.
If the coefficient is a number, we call it a numeral coefficient. If it involves letters, we call it a literal coefficient. In 3xy, 3 is the numeral coefficient of xy, while x is a literal coefficient of 3y. When the coefficient is 1, we usually do not write it.
So we write b instead of 1b.
Now we come to the degree of polynomials.
For a polynomial in one variable, the degree is the greatest exponent of its various terms.
Consider 4x² – 3x⁵ + 8x⁶. The exponent of 4x² is 2. The exponent of 3x⁵ is 5. The exponent of 8x⁶ is 6.
The greatest exponent is 6, so the degree of this polynomial is 6.
For polynomials with two or more variables, we find the sum of powers in each term, and the greatest sum gives us the degree. Take 3x + xy² – 8yz. In 3x, the sum is 1. In xy², the sum is 1 plus 2, which equals 3. In 8yz, the sum is 1 plus 1, which equals 2.
The greatest sum is 3, so the degree is 3.
A constant like 8 can be written as 8x⁰, since x to the power 0 equals 1.
Therefore, the degree of any constant is zero.
Let us now learn how to add algebraic expressions. When adding like terms, we add their coefficients and keep the same literal part. For example, 3x + 8x equals 11x. 8x²y – 5x²y equals 3x²y.
2xy + 3xy + 5xy equals 10xy.
When adding unlike terms, we cannot combine them into a single term. We simply write them connected by plus signs. For example, 2ab + 4bc stays as it is.
Similarly, 5x² + 8xy cannot be simplified further.
Subtraction follows similar rules. To subtract like terms, change the sign of the term being subtracted, then add.
Remember, the result of subtracting two like terms is also a like term. Subtract 4x from –8x: we get –8x – 4x, which equals –12x.
Subtract –3x from –7x: we get –7x + 3x, which equals –4x.
For unlike terms, subtraction also leaves us with an expression that cannot be simplified to a single term.
4bc – 2ab remains as written.
When adding or subtracting polynomials, we can use two methods. The row method: write all polynomials in a row, remove brackets, group like terms, and simplify.
Or the column method: arrange like terms in vertical columns and add column-wise.
Let me show you the column method with an example.
Add 4a + 2b, 3a – 3b + c, and –2a + 4b + 2c.
Write them with like terms aligned: 4a + 2b 3a – 3b + c
–2a + 4b + 2c
Adding the a terms: 4a plus 3a minus 2a gives 5a. Adding the b terms: 2b minus 3b plus 4b gives 3b. Adding the c terms: 0 plus c plus 2c gives 3c.
The answer is 5a + 3b + 3c.
For subtraction using the column method, first write the expressions with like terms aligned. Then change the sign of every term in the lower expression, and add.
Subtract 3a – 4b + 5c from 4a – b + 6c.
Write: 4a – b + 6c
3a – 4b + 5c
Change signs of the lower row: plus 3a becomes minus 3a, minus 4b becomes plus 4b, and plus 5c becomes minus 5c. Now add: 4a minus 3a gives a. Minus b plus 4b gives 3b. 6c minus 5c gives c.
The answer is a + 3b + c.
Now for multiplication. When multiplying monomials, multiply the coefficients and apply the product law for exponents. First, multiply the numerical coefficients together.
The product law states: when multiplying powers with the same base, add the exponents.
In symbols, aᵐ × aⁿ = aᵐ⁺ⁿ.
Multiply 8x²y³, 6y²z⁵, and 3xz². Coefficients: 8 times 6 times 3 equals 144. For x: x² × x = x³. For y: y³ × y² = y⁵. For z: z⁵ × z² = z⁷.
The product is 144x³y⁵z⁷.
To multiply a polynomial by a monomial, multiply each term of the polynomial by the monomial. Multiply 4x²y – 3xy² + 4xy by 2xy. 2xy × 4x²y = 8x³y². 2xy × (–3xy²) = –6x²y³. 2xy × 4xy = 8x²y².
The result is 8x³y² – 6x²y³ + 8x²y².
To multiply two binomials, multiply each term of the first by each term of the second, then combine like terms. Multiply x + 3 by x + 5. x times x gives x². x times 5 gives 5x. 3 times x gives 3x. 3 times 5 gives 15.
Combine: x² + 5x + 3x + 15 equals x² + 8x + 15.
Finally, let us look at division. When dividing monomials, we use the quotient law.
The quotient law states: when dividing powers with the same base, subtract the exponents. In symbols, aᵐ/aⁿ = aᵐ⁻ⁿ when m is greater than n.
If n is greater than m, we get 1 over a to the power n minus m.
Divide 12m⁵ by 4m³. Coefficients: 12 divided by 4 equals 3. For m: m⁵⁻³ = m².
The quotient is 3m².
Divide 35a³b⁵ by 5a⁶b². Coefficients: 35 divided by 5 equals 7. For b: b⁵⁻² = b³. For a: since 6 is greater than 3, we write this as 1/a³ in the denominator.
The result is 7b³/a³.
To divide a polynomial by a monomial, divide each term separately. Divide 12x⁵ – 9x³ by 3x². 12x⁵/3x² = 4x³. –9x³/3x² = –3x.
The answer is 4x³ – 3x.
Let me conclude with some beautiful generalizations about numbers. The sum of the first n even natural numbers equals n times n plus 1, or n(n + 1).
For example, the sum of the first 5 even numbers — 2, 4, 6, 8, 10 — is 30, which equals 5 times 6.
The sum of the first n odd natural numbers equals n squared, or n².
For example, the sum of the first 5 odd numbers — 1, 3, 5, 7, 9 — is 25, which is 5 squared.
These patterns show the elegant structure hidden in numbers.
Let us quickly recap the key takeaways from today's lesson.
First, algebra uses letters called variables alongside numbers, and these variables can take different values.
Second, like terms have identical literal coefficients and can be combined by adding or subtracting their numerical coefficients.
Third, algebraic expressions are classified by their number of terms: monomial, binomial, trinomial, or multinomial.
Fourth, the degree of a polynomial is the highest power of its variable, or for multiple variables, the highest sum of powers in any term.
Fifth, when adding or subtracting polynomials, always group like terms together.
Sixth, for multiplication and division, apply the laws of exponents and work with coefficients and variables systematically.
You have now built a solid foundation in algebraic concepts. Practice these operations until they feel natural, and you will find algebra becoming easier with every problem you solve.
Keep exploring, keep practicing, and enjoy your mathematical journey. Until next time, goodbye and happy learning!