Algebraic Expressions: Complete Guide with Formulas, Examples & Worksheets
Introduction to Algebraic Expressions
Algebraic expressions form the foundation of modern mathematics, bridging the gap between arithmetic and advanced mathematical concepts. An algebraic expression is a mathematical phrase that combines constants, variables (literals), and operators (+, -, ×, ÷) to represent quantities and relationships. Unlike equations, expressions don't contain an equality sign they simply represent a value that can vary based on the variables involved.
For students working with algebraic expressions grade 7 through high school, understanding these fundamental building blocks is essential for success in mathematics. Whether you're searching for an algebraic expressions worksheet or need clarity on algebraic expressions and equations, this comprehensive guide provides expert insights grounded in mathematical pedagogy.
The concept of algebraic expressions dates back centuries, with contributions from mathematicians across civilizations. Today, they're essential tools in fields ranging from physics and engineering to economics and computer science. A firm grasp of algebraic expressions enables students to model real-world situations mathematically, solve complex problems, and develop critical analytical thinking skills.
Components of Algebraic Expressions
Variables and Constants
Variables (or literals) are symbols typically letters like x, y, a, b that represent unknown or changeable numerical values. For example, in the expression for the perimeter of a square (P = 4a), the variable a represents the side length, while P is also a variable representing the perimeter.
Constants are fixed numerical values that don't change. These include numbers like 7, -2, π, √3, and fractions such as 3/8. In the expression 5x + 7, the number 7 is a constant term, while 5 is the coefficient of the variable x.
Coefficients and Terms
A coefficient is the numerical factor of a term containing variables. In the term 7xy, the coefficient is 7, while x and y are the literal factors. Understanding coefficients is crucial when working with algebraic expressions examples and performing operations like addition and multiplication.
Terms are the individual parts of an algebraic expression separated by addition or subtraction signs. The expression 3x² + 5x - 8 contains three terms: 3x², 5x, and -8. Terms with identical literal factors are called like terms and can be combined through addition or subtraction.
Classification of Algebraic Expressions
Based on Number of Terms
Algebraic expressions are classified by the number of terms they contain:
- Monomial: Contains exactly one term (e.g., 5x, -9, 3x²yz)
- Binomial: Contains exactly two terms (e.g., 5x + 9y, a² - 3b²)
- Trinomial: Contains exactly three terms (e.g., x + 2y - 3z, z² - 2xy + 5)
- Multinomial/Polynomial: Contains more than three terms
This classification is particularly important when working with algebraic expressions worksheet pdf materials, as different types of expressions require different manipulation strategies.
Polynomials and Their Degrees
A polynomial is an algebraic expression where all variables have non-negative integer exponents. The degree of a polynomial is determined by the highest power of the variable(s) present.
For single-variable polynomials:
- Linear polynomial (degree 1): 3x + 5, 6 - 2y
- Quadratic polynomial (degree 2): 6x² - 5x + 4
- Cubic polynomial (degree 3): 6x³ - 5x² + 2x + 1
- Constant polynomial (degree 0): 6 (a single constant)
For multi-variable polynomials, the degree is the highest sum of exponents in any single term. For example, in 6x³ - 2x²y + xy² - 3x²y², the term -3x²y² has degree 4 (2+2), making this a polynomial of degree 4.
Operations on Algebraic Expressions
Addition and Subtraction
Addition of algebraic expressions follows the principle of combining like terms. When adding expressions, identify terms with identical literal factors and combine their coefficients:
Example: Add (7x + 3y - 2z) and (-2x + 4y + 3z)
- Group like terms: (7x - 2x) + (3y + 4y) + (-2z + 3z)
- Result: 5x + 7y + z
Subtraction works similarly, but requires changing the signs of all terms in the expression being subtracted before combining like terms. This is a common topic in algebraic expressions and equations curricula.
Multiplication
Multiplying monomials involves multiplying coefficients and applying the product rule of exponents (aᵐ × aⁿ = aᵐ⁺ⁿ):
Example: (2x³y) × (4x²y³) = (2 × 4)(x³⁺²)(y¹⁺³) = 8x⁵y⁴
Multiplying a polynomial by a monomial uses the distributive property:
Example: 2x²(4x² + 6x + 7) = 8x⁴ + 12x³ + 14x²
Multiplying two binomials requires applying the distributive property twice (often remembered as FOIL):
Example: (x + 8)(x - 5) = x² - 5x + 8x - 40 = x² + 3x - 40
Division
Division of algebraic expressions applies the quotient rule of exponents (aᵐ ÷ aⁿ = aᵐ⁻ⁿ):
Dividing monomials: Example: (24a²bc³) ÷ (-6abc²) = (24 ÷ -6)(a²⁻¹)(b¹⁻¹)(c³⁻²) = -4ac
Dividing a polynomial by a monomial requires dividing each term separately:
Example: (4x⁵ - 14x⁴ + 6x³ - 2x²) ÷ 2x² = 2x³ - 7x² + 3x - 1
Essential Algebraic Identities
Algebraic identities are equations that hold true for all values of the variables involved. Mastering these identities significantly simplifies complex calculations and is fundamental for success with algebraic expressions examples at higher levels.
Standard Identities Table
| Identity Name | Formula | Explanation |
|---|---|---|
| Square of Sum | (a + b)² = a² + 2ab + b² | Expanding the square of a binomial sum |
| Square of Difference | (a - b)² = a² - 2ab + b² | Expanding the square of a binomial difference |
| Difference of Squares | (a + b)(a - b) = a² - b² | Product of sum and difference equals difference of squares |
| Binomial Product | (x + a)(x + b) = x² + (a + b)x + ab | General form for binomial multiplication |
| Square of Trinomial | (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca | Expanding three-term square |
| Cube of Sum | (a + b)³ = a³ + b³ + 3ab(a + b) | Alternative: a³ + 3a²b + 3ab² + b³ |
| Cube of Difference | (a - b)³ = a³ - b³ - 3ab(a - b) | Alternative: a³ - 3a²b + 3ab² - b³ |
| Sum of Cubes | a³ + b³ = (a + b)(a² - ab + b²) | Factorization of sum of cubes |
| Difference of Cubes | a³ - b³ = (a - b)(a² + ab + b²) | Factorization of difference of cubes |
| Sum of Cubes (Three Terms) | a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca) | When a + b + c = 0, then a³ + b³ + c³ = 3abc |
Practical Applications of Identities
These identities enable rapid mental calculation and elegant problem-solving approaches. For instance, calculating 103² becomes simple using (100 + 3)² = 10000 + 600 + 9 = 10609.
Similarly, finding 52² - 48² can be solved instantly using the difference of squares: (52 + 48)(52 - 48) = 100 × 4 = 400.
Working with Algebraic Expressions: Advanced Techniques
Evaluation and Substitution
Evaluating an algebraic expression means finding its numerical value when specific values are assigned to the variables. This skill is essential when working through algebraic expressions worksheet problems.
Example: Evaluate 3x(4x - 5) + 3 when x = 3
- Substitute: 3(3)[4(3) - 5] + 3
- Calculate: 9(12 - 5) + 3 = 9(7) + 3 = 63 + 3 = 66
Simplification Strategies
Simplifying algebraic expressions involves:
- Removing parentheses using distributive property
- Combining like terms
- Applying algebraic identities where appropriate
- Reducing to the simplest form
Example: Simplify 3x(x² - x) - 3x²(x² + 2x) - 2(x³ - 3x)
- Expand: 3x³ - 3x² - 3x⁴ - 6x³ - 2x³ + 6x
- Combine: -3x⁴ + (3x³ - 6x³ - 2x³) - 3x² + 6x
- Result: -3x⁴ - 5x³ - 3x² + 6x
Practice Problems and Worksheets
Sample Problems for Self-Assessment
Problem 1: If x + 1/x = 4, find the value of x² + 1/x²
Solution: Square both sides: (x + 1/x)² = 16 Expand: x² + 2(x)(1/x) + 1/x² = 16 Simplify: x² + 2 + 1/x² = 16 Therefore: x² + 1/x² = 14
Problem 2: If x + y = 12 and xy = 14, find x² + y²
Solution: Use identity (x + y)² = x² + y² + 2xy (12)² = x² + y² + 2(14) 144 = x² + y² + 28 Therefore: x² + y² = 116
Resources for Further Practice
Students seeking additional practice should look for:
- Algebraic expressions worksheet pdf materials from reputable educational websites
- Algebraic expressions grade 7 curriculum-aligned practice sets
- Interactive online platforms offering immediate feedback
- Textbook exercises with progressive difficulty levels
Common Questions About Algebraic Expressions
What is the difference between algebraic expressions and equations?
An algebraic expression is a mathematical phrase containing variables, constants, and operators (e.g., 3x + 7), while an equation includes an equals sign showing that two expressions have the same value (e.g., 3x + 7 = 22). Expressions are evaluated for different values; equations are solved to find specific values.
How do you identify like terms?
Like terms have identical literal (variable) parts, including the same variables raised to the same powers. For example, 3xy and -7xy are like terms, but 3xy and 3x²y are not, since the power of x differs.
Why are algebraic identities important?
Algebraic identities provide shortcuts for calculations, enable factorization, and reveal underlying mathematical relationships. They're essential tools for simplifying complex expressions and solving equations efficiently—critical skills for algebraic expressions and equations mastery.
Conclusion
Mastering algebraic expressions is a foundational skill that opens doors to advanced mathematics and numerous practical applications. From basic operations to complex identities, the concepts covered in this guide provide a comprehensive framework for understanding and manipulating algebraic expressions.
Whether you're a student working through algebraic expressions examples, a teacher preparing an algebraic expressions worksheet, or a parent supporting a child in algebraic expressions grade 7 studies, consistent practice with varied problem types builds confidence and competence.
The journey from simple monomials to complex polynomial manipulations develops not just mathematical skill, but also logical thinking, pattern recognition, and problem-solving abilities that extend far beyond the classroom. With the identities, strategies, and examples provided here, you're well-equipped to tackle algebraic expressions at any level of complexity.
How do you simplify algebraic expressions with like terms
Simplifying algebraic expressions with like terms is a foundational algebra skill that helps make longer expressions easier to understand and solve. Like terms are terms that have the same variable raised to the same power, though their coefficients may differ. For example, in 3x + 4x, both terms are like terms because they have the same variable x. To simplify, you just add or subtract their coefficients. Therefore, 3x + 4x becomes 7x.
To simplify expressions with multiple variables, identify and group similar terms first. Consider 5x + 2y + 7x – 3y. Combine x terms (5x + 7x = 12x) and y terms (2y – 3y = -y), resulting in 12x – y. This approach keeps your expressions clean and concise.
A common mistake learners make is trying to combine terms that are not alike, such as adding x² and x. These cannot be added because the powers of x are different. Always focus on matching both the variable and its exponent. Simplifying algebraic expressions helps make later steps like solving linear equations more efficient and less prone to error. Practicing regularly with increasingly complex expressions improves clarity and accuracy in mathematical problem-solving, preparing students for equations and real-world applications in physics, engineering, and economics.
Examples of evaluating expressions Step-by-Step
Evaluating algebraic expressions involves substituting values for variables and simplifying the result step by step. This process helps learners understand how variables and constants interact. For example, take the expression 3x + 2y when x = 2 and y = 5. Substitute these values into the expression: 3(2) + 2(5). Then perform multiplication first (following the order of operations), giving 6 + 10, and finally, add to get 16.
Let’s consider a slightly more complex example: 4a² – 3a + 2 when a = 3. Substitute first to get 4(3)² – 3(3) + 2. Evaluate the exponent (3² = 9), then multiply to get 36 – 9 + 2. Finally, simplify the arithmetic: 36 – 9 + 2 = 29. Step-by-step evaluation ensures accuracy, especially when dealing with negative numbers or parentheses.
Students should also pay attention to the order of operations, known as BODMAS (Brackets, Orders, Division/Multiplication, Addition/Subtraction). When expressions include parentheses like 2(3x + 4), start by simplifying the part inside the brackets. Regular practice strengthens understanding and builds confidence for tackling more advanced algebraic topics such as simultaneous linear equations and inequalities, making this skill essential for all math learners.
How to add and subtract polynomial expressions
Adding and subtracting polynomial expressions follows the same principle as combining like terms. Polynomials are multi-term expressions involving variables raised to whole number powers, such as 2x² + 3x + 4. To add or subtract polynomials, align like terms and combine their coefficients. For example, to add (3x² + 4x + 5) and (2x² + x – 3), add coefficients of similar powers: (3x² + 2x²) + (4x + x) + (5 – 3). The simplified answer is 5x² + 5x + 2.
For subtraction, distribute the negative sign carefully before simplifying. If we subtract (x² + 2x – 1) from (4x² – 3x + 6), rewrite it as (4x² – 3x + 6) – (x² + 2x – 1), then change the signs of the second polynomial’s terms: 4x² – 3x + 6 – x² – 2x + 1. Combine like terms: (4x² – x²) + (–3x – 2x) + (6 + 1) = 3x² – 5x + 7.
Students should always structure their work so that like terms line up neatly; this minimizes sign mistakes and simplifies verification. Mastery in polynomial addition and subtraction lays the groundwork for learning polynomial multiplication, factoring, and graphing skills used extensively in calculus and data modeling.
How to multiply binomials using FOIL
The FOIL method is a systematic way to multiply two binomials, ensuring that every term multiplies correctly. FOIL stands for First, Outer, Inner, Last, describing the order in which you multiply terms. Consider the example (x + 3)(x + 2). Multiply the First terms (x × x = x²), Outer terms (x × 2 = 2x), Inner terms (3 × x = 3x), and Last terms (3 × 2 = 6). Combine like terms to get x² + 5x + 6.
When handling expressions with negative signs, carefully manage the signs during multiplication. For example, (x – 4)(x + 3) becomes x² + 3x – 4x – 12, which simplifies to x² – x – 12. The FOIL method ensures no term is missed and is particularly useful for beginning algebra students.
Real-world applications include area and volume problems where dimensions are expressed as binomials. For instance, if the length and width of a rectangle are (x + 2) and (x + 3), the area is found by multiplying them—perfectly illustrating the FOIL method in practice. Consistent practice helps build fluency, enabling swift recognition of patterns when expanding quadratic expressions for graphing or factoring purposes.
Practice problems with solutions on algebraic expressions
Practicing algebraic expressions helps strengthen conceptual clarity and speed. Here are a few examples with step-by-step solutions:
1. Simplify: 3x + 2x – 5 + 4
Solution: Combine like terms for x: 3x + 2x = 5x. Combine constants: –5 + 4 = –1. Final expression: 5x – 1.
2. Simplify: 2(3x + 4) – (x + 5)
Solution: Distribute first: 6x + 8 – x – 5 = (6x – x) + (8 – 5) = 5x + 3.
3. Evaluate: 4y – 3 when y = 2
Solution: Substitute: 4(2) – 3 = 8 – 3 = 5.
4. Simplify: (x² + 2x + 1) + (3x² – x + 4)
Solution: Combine like terms: x² + 3x² = 4x²; 2x – x = x; 1 + 4 = 5. The simplified expression is 4x² + x + 5.
Show step-by-step examples of combining like terms
Combining like terms makes complex algebraic expressions easier to solve and understand. The process starts by identifying terms that share the same variables and exponents; only those can be added or subtracted. For example, with the expression 2x + 3y – x + 5y, first, group like terms: (2x – x) and (3y + 5y). Subtract to get x for the x terms, and add to get 8y for the y terms, yielding x + 8y. Rearranging terms can sometimes help, but the critical point is to preserve the sign in front of each term. When a term is written as –x, that negative follows the term when combining.
Here’s another step-by-step example: 4a + 3b – 2a + 6 – b.
-
Group a terms: 4a – 2a = 2a
-
Group b terms: 3b – b = 2b
-
Constant 6 stands alone
The final simplified expression: 2a + 2b + 6.
For longer expressions or real-life scenarios—such as totaling item costs in a shopping list you follow the same process: group quantities and types first, then do the arithmetic. Practicing these steps regularly builds strong algebraic thinking, enabling easy progress to solving equations and word problems down the line.
How to handle negative signs when combining terms
Managing negative signs is crucial for accurate simplification of algebraic expressions. Every term’s sign, whether positive or negative, must be observed and consistently carried through every step. When combining terms, always look at the operation preceding each variable or constant. For example, in 3x – 5x, the – (minus) before 5x means you subtract its coefficient from the 3x, resulting in –2x.
A frequent source of confusion is distributing negative signs with parentheses. For the expression –(4a + 2b), distribute the negative to both terms: –1 × 4a = –4a and –1 × 2b = –2b, producing –4a – 2b. If you have –(2x – 7), the negative outside reverses the sign of both terms, yielding –2x + 7. The key is to treat every negative sign outside parentheses as multiplication by –1.
Whenever simplifying lengthy expressions, pause at each negative sign and confirm its effect on succeeding terms. Carefully handling these ensures no accidental sign errors, which could otherwise lead to incorrect answers. Developing the habit of rewriting distributed expressions before combining like terms can minimize mistakes and boost confidence in solving even the trickiest algebraic problems.
Practice problems with solutions on algebraic expressions
Practicing algebraic expressions helps strengthen conceptual clarity and speed. Here are a few examples with step-by-step solutions:
1. Simplify: 3x + 2x – 5 + 4
Solution: Combine like terms for x: 3x + 2x = 5x. Combine constants: –5 + 4 = –1. Final expression: 5x – 1.
2. Simplify: 2(3x + 4) – (x + 5)
Solution: Distribute first: 6x + 8 – x – 5 = (6x – x) + (8 – 5) = 5x + 3.
3. Evaluate: 4y – 3 when y = 2
Solution: Substitute: 4(2) – 3 = 8 – 3 = 5.
4. Simplify: (x² + 2x + 1) + (3x² – x + 4)
Solution: Combine like terms: x² + 3x² = 4x²; 2x – x = x; 1 + 4 = 5. The simplified expression is 4x² + x + 5.
Algebraic Expression Theory - PDF
SOME IMPORTANT TERMS
VARIABLES OR LITERALS:
A symbol which takes on various numerical values is known as a variable or a literal.
We know that the perimeter of a square of side a is given by the formula, P = 4a.
Here 4 is a constant, while a and P are variables.
Constants:
A symbol having a fixed numerical value is called a constant.
e.g. 7, - 2, 3/8, root 3, pi etc. are all constants.
FUNCTION:
Any expression involving a letter is called a function of that letter.
2a 5b + 6c – 7c + 8xd
e.g. x2 + 3x + 2 is a function of x.
ALGEBRAIC EXPRESSIONS :
A combination of constants and variables, connected by +, - , and is known as an algebraic expression.
e.g. (i) 6a + 7b is an expression containing 2 terms, namely 6a and 7b.
(ii) 7 + 5x – 8xy is an expression containing 3 terms, namely 7, 5x and – 8xy.
TYPES OF ALGEBRAIC EXPRESSIONS
MONOMIALS:
An algebraic expression containing one term only, is called a monomial.
e.g. 5x, 6y2, 3x2 yz, - 9 are all monomials.
BINOMIALS:
An algebraic expression containing 2 terms is called a binomial.
e.g. (i) 5x + 9y is a binomial having 2 terms, namely 5x and 9y.
(ii) 2xy – 3 is a binomial having 2 terms, namely, 2xy and – 3.
(iii) a2b – 3b2c is a binomial having 2 terms, namely, a2b and -3b2c.
TRINOMIALS:
An algebraic expression containing 3 terms is called a trinomial.
e.g. (i) x + 2y – 3z is a trinomial having 3 terms, namely x, 2y and – 3.
(ii) z2 – 2xy + 5 is a trinomial having 3 terms, namely, z2, - 2xy and 5.
MULTINOMIAL:
An algebraic expression containing more than 3 terms, is called a multinomial.
e.g. a3b3 + 3ab - 5/a + 7a/b2 is a multinomial, having 4 terms, namely, a3b3, -5/a and 71/b2
POLYNOMIALS:
An algebraic expression in which the variables involved have only nonnegative integral powers, is called a polynomial.
Degree of a Polynomial in One Variable
The highest power of the variable in a polynomial of one variable is called the degree of the polynomial.
(i) 5x3 − 3x2 + 4x − 8 is a polynomial of degree 3.
(ii) 6 + 5y2 − 7y4 is a polynomial of degree 4.
Linear Polynomial
A polynomial of degree 1 is called a linear polynomial.
(a) 3 + 5x (b) 6 − 2y (c) 3/2 + 7z are all linear polynomials.
Quadratic Polynomial
A polynomial of degree 2 is called a quadratic polynomial.
(a) 6x2 − 5x + 4 (b) 2y2 + 3y − 1 are both quadratic polynomials.
Cubic Polynomial
A polynomial of degree 3 is called a cubic polynomial.
(a) 6x3 − 5x2 + 2x + 1 (b) z3 − 3z + 7 are both cubic polynomials.
CONSTANT POLYNOMIAL:
A polynomial having one term consisting of a constant only is a constant polynomial.
The degree of a constant polynomial is 0.
Thus, 6 is a constant polynomial.
DEGREE OF A POLYNOMIAL IN TWO OR MORE VARIABLES:
If a polynomial involves two or more variables, then the sum of the powers of all the variables in each term is taken up and the highest sum so obtained is the degree of the polynomial.
e.g. (i) 6x3 – 2x2y + xy2 – 3x2y2 is a polynomial in x and y, of degree 4.
(ii) 2x2y3 – 3xy2 + 5x3y3 is a polynomial in x and y of degree 6.
(iii) 3ab2 – 4a root b + 5b3 is an expression but not a polynomial, as it contains a term in which the sum of the powers of the variables is 3/2, which is not a non-negative integer.
FACTORS AND COEFFICIENTS
FACTORS:
Each term in an algebraic expression is a product of one or more number(s) and / or literal number(s). These number(s) and / or literal number(s) are known as the factors of that term. For example:
The monomial 7x is the product of number 7 and literal x. So, 7 and x are factors of the monomial 7x.
In the binomial and are two terms. In the tem for instance, 3, x and y are its factors. Clearly, number 3 is the numerical factor, and x and y are literal factors.
In the term - the numerical factor is – 4 whereas x, y and z are literal factors.
In the binomial expression the term has -1 as the numerical factor while x and y are literal factors. The term 3 has only numerical factor. It has no literal factor.
In the algebraic expression the term ab has numerical factor as 1 and literal factors are a and b. The term has numerical factor as -1 and literal factors are c and The third term -7 has no literal factor.
COEFFICIENT:
In a term of an algebraic expression, any of the factors with the sign of the term is called the coefficient of the product of the other factors.
CONSTANT TERM:
A term of the expression having no literal factor is called a constant term. For example:
In the binomial expression 5x + 7, the constant term is 7.
In the trinomial expression a2 + b2 - 3/4, the constant term is -3/4
Like terms and unlike terms:
The terms having the same literal factors are called the like terms other wise they are known as unlike terms.
In the expressions 2xy + 3x – 7xy + 5x, (2xy, – 7xy) and (3x, 5x) are like terms
[have same literals factors]
3x, – 7xy are unlike terms [donot have same literal factors]
OPERATIONS OF ALGEBRAIC EXPRESSIONS
(1) The plus sign (+) indicates what number is to be added to what precedes it
e.g.: p + q
(2) The minus sign (–) when placed before a number indicates that the number is to be subtracted from what precedes it
e.g.: x – y
(3) The sign plus or minus (±) is read plus or minus and when placed before a number indicates that the number is to be either added to or subtracted from what precedes it.
(4) The sign ~ when placed between the two numbers indicates that the less of the two is to be subtracted from the greater
e.g.: If a = 5, b = 8, then a ~ b means 8 – 5 = 3
(5) An intelligible collection of letters and signs of operation is called an algebraic expression.
The parts of an algebraic expression that are connected by the + or – are called its terms.
E.g.: 7a + ab + c × d – 2c × f × g is an algebraic expression.
7a, ab + c × d, –2c × f × g are called terms.
(6) Expressions are either simple or compound.
WORKING RULES FOR ADDITION OF ALGEBRAIC EXPRESSIONS:
Rule 1: When 2 positive quantities are added the sum is a positive quantity
(+a) + (+b) = + (a + b)
e.g.: (+7a) + (+5a) = + (7a + 5a) = +12a
Rule 2: When 2 negative quantities are added together the sum is a negative quantity.
(–a) + (–b) = – (a + b)
e.g.: (– 8x) + (– 6x) = – (8 + 6)x = – 14x
Rule 3: When a negative quantity is added to a positive quantity, the sign of the result is positive or negative according as the absolute value is less or greater than that of the positive quantity and the absolute value of the result is always equal to the difference between the absolute values of the quantities.
If a > b (– a) + (+ b) = – (a – b) Eg: (– 7a) + (+ 3a) = – 4a
If a < b (– a) + (+ b) = + (b – a) E.g.: (– 2a) + ( + 7a) = + 5a
When any number of quantities are added together the result will be the same in whatever order the quantities may be taken.
E.g.: (– a) + (+ b) = (+b) + (– a), (– 8p) + (+ 3p) = (+ 3p) + (– 8p)
When any number of quantities are added together they can be divided into groups and the result expressed as the sum of the these groups.
– 7y + 6y + 2y – 8y + 9y + 3y = (– 7y + 6y) + (2y – 8y) + (9y + 3y)
= (– 1y) + (– 6y) + (12y) = (– 7y) + (12y) = + 5y
When any number of quantities are to be added some of which are positive and other negative, then collect the positive terms in one group and the negative terms in another and express the result as the sum of there groups.
4m – 3m + 2m – 9m + 8m – 6m = (4m + 2m + 8m) – (+3m + 9m + 6m)
= (+ 14m) – (+ 18m) = – 4m
Column method for addition of Algebraic expressions:
Write each expression in a separate row such that their like terms are arranged one below the other in a column. Then the terms are added column wise.
Add 7a2 + 3a – 8 and – 2a2 + 4a – 3
7a2 + 3a – 8
– 2a2 + 4a – 3
_____________
+ 5a2 + 7a – 11
Horizontal method for addition of algebraic expressions:
We add two or more expressions by collecting the like terms together and then simplifying.
E.g.: 7x + 3y – 2z is added to – 2x + 4y + 3z
= 7x – 2x + 3y + 4y – 2z + 3z
= 7x + (– 2x) + (+3y) + (+4y) +(– 2z) + (+3z) = 5x + 7y + 1z
E.g. is added to =
L.C.M of 2 and 3 is 6. L.C.M of 3 and 6 is 6
WORKING RULE FOR SUBTRACTION OF ALGEBRAIC EXPRESSION:
Any quantity b is said to be subtracted from any other quantities a, when a third quantity c is found such that the sum of b and c is equal to a. In other words c = a – b. When c is such that a = b + c, a is called minuend, b is called subtrahend and c is called the remainder.
Subtraction of like terms can be performed in a manner exactly similarly to that used in subtraction of integers. For any 2 integers a and b we have a – b = a + (additive inverse of b). Change the sign of term to be subtracted and add the new monomial to the one from which subtraction is to be made.
E.g.: (+7a) – (+ 3a) = + 7a + (– 3a) = + 4a
(– 8a) – (– 2a) = (– 8a) + (+ 2a) = – 6a
(– 5a) – (+ 3a) = (– 5a) + (– 3a) = – 8a
(+ 8a) – (– 2a) = (+ 8a) + (+ 2a) = + 10a
(– 6a) – (– 9a) = (– 6a) + (+ 9a) = + 3a
Column method for subtraction of Algebraic expressions:
Write the expression to be subtracted below the other expression such that like terms of the two expressions are in the same column. Now change the sign of each term of lower expression and add term wise
e.g. subtract 3p – 8q + 5r from 7q + 10p – 3r
7q + 10p – 3r
– 8q + 3p + 5r
+ – –
_______________
15q + 7p – 8r
Horizontal method for subtraction of Algebraic expressions:
Change the sign of each term of the expression to be subtracted and then add
E.g.: 7a3 + 3a2 – 8a + 6) – (3a3 – 2a2 + 3a – 7)
= (7a3 +3a2 – 8a + 6) + (– 3a3 + 2a2 – 3a + 7)
= (7a3) + (– 3a3) + (3a2) + (2a)2 + (– 8a) + (– 3a) + (+ 6) + (+ 7) = 4a3 + 5a2 – 11a + 13
E.g.: =
What should be added to 2x – 3y + 5z to get 7x – 3y + 2z
7x – 3y + 2z
2x – 3y + 5z
– + –
__________
5x – 3z should be added
What should be subtracted from 2x2– 9x + 6 to get 3x2 –2x + 7
2x2 – 9x + 6
3x2 – 2x + 7
– + –
______________
– x2 – 7x – 1 should be subtracted from 2x2 – 9x + 6
By how much 7x – 3y + 2z exceeds 2x + 7y – 8z
7x – 3y + 2z
2x + 7y – 8z
– – +
_______________
5x – 10y + 10z
From the sum of 2x + 3y – 5z, 4x – 2y + 7z, subtract 5x – 9y – 2z
2x + 3y – 5z
+ 4x – 2y + 7z
________________
6x + y + 2z
+ 5x – 9y – 2z
– + +
______________
x + 10y + 4z
Multiplication of Monomials
Product of monomials = (Product of their numerical coefficients) (Product of their variable parts)
The product law of exponents i.e. am x an x = am+n finds great use in the multiplication of algebraic expressions.
Multiplication of Two Polynomials
Multiply term of the multiplicand by each term of the multiplier and take the algebraic sum of these products.
- Multiply (3x – 5 + 2x2) by (5x + 3).
Solution. Arranging the multiplicand in descending order of powers of x and then multiplying, we get:
2x2 + 3x − 5 5x + 3 ────────────── 10x3 + 15x2 − 25x (Multiplying by 5x) + 6x2 + 9x − 15 (Multiplying by 3) ────────────── 10x3 + 21x2 − 16x − 15
Division of Polynomials
The quotient law of exponents i.e. am ÷ an = am−n finds great use in the division of algebraic expressions.
Division of a Monomial by a Monomial
Quotient of two monomials = (Quotient of their numerical coefficients) × (Quotient of their variable parts)
DIVISION OF A POLYNOMIAL BY A MONOMIAL:
For dividing a polynomial by a monomial, we divide each term of the polynomial be the monomial.
DIVISION OF A POLYNOMIAL BY A POLYNOMIAL:
Working Rule
Fundamentals propositions
1) a b × b = a
2) a b c = a bc
3) a b =
4) a b × c = a × c b
Sign convertions
If a b = c, then
1) (+ a) (+ b) = + c
2) (– a) (– b) = + c
3) ( – a) (b) = – c
4) (+ a) (– b) = – c
Division of a monomial by a monomial
Quotient can be found by subtracting smaller power of a letter from greater power of the same letter.
E.g.: 16a5 b4 8a2 b3
To divide a multinomial by a monomial we have to divide each term of the dividend and take the sum of those partial quotients for the complete quotient
Ex: Divide 18x8 + 24x6 + 12x4 6x2
= 3x6 + 4x4 + 2x2
Division of a multinomial by another multinomial
- The dividend and the divisor both stand arranged according to descending powers of a common letter, namely, a.
- Divide the first term of the dividend by the first term of the divisor and write down the result as the first term of the quotient. Multiply the divisor by the quantity thus found and subtract the product from the dividend.
- Regard the remainder as a new dividend and see if it is arranged according to the descending powers of the common letter. Divide its first terms by the first term of the divisor and write down the result as the next term of the quotient. Multiply the divisor by this term and subtract the product from the new dividend. Then go similarly with the successive remainders until there is no remainder.
Example:
x4 - 4x2 + 12x - 9 ÷ (x2 - 2x + 3)
Dividend = x4 - 4x2 + 12x - 9
Divisor = x2 - 2x + 3
Quotient = x2 + 2x - 3
Remainder = 0
You will observe that
Dividend = Divisor × Quotient + Remainder
IDENTITY
An identity is an equality which is true for all values of the variable(s).
Standard identities
- (a + b)2 = a2 + 2ab + b2
- (a − b)2 = a2 − 2ab + b2
- (a + b)(a − b) = a2 − b2
- (x + a)(x + b) = x2 + (a + b)x + ab
- (a + b + c)2 = (a2 + b2 + c2) + 2(ab + bc + ca)
- (a + b − c)2 = (a2 + b2 + c2) + 2(ab − bc − ca)
- (a + b)3 = a3 + b3 + 3ab(a + b)
- (a − b)3 = a3 − b3 − 3ab(a − b)
Formulae
- (a + b)2 = a2 + 2ab + b2
- (a − b)2 = a2 − 2ab + b2
- (a − b)2 = (a + b)2 − 4ab
- (a + b)2 = (a − b)2 + 4ab
- (a + b)2 − (a − b)2 = 4ab
- (a + b)2 + (a − b)2 = 2(a2 + b2)
- a4 + a2b2 + b4 = (a2 + ab + b2)(a2 − ab + b2)
- (a + b)3 = a3 + b3 + 3ab(a + b) or a3 + b3 + 3a2b + 3ab2
- (a − b)3 = a3 − b3 − 3ab(a − b) or a3 − b3 − 3a2b + 3ab2
- (a + b)3 + (a − b)3 = 2a3 + 6ab2
- (a + b)3 − (a − b)3 = 2b3 + 6a2b
- (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
- a3 + b3 + c3 − 3abc = (a + b + c)(a2 + b2 + c2 − ab − bc − ca)
- If a + b + c = 0 then a3 + b3 + c3 = 3abc
- a3 + b3 = (a + b)(a2 − ab + b2)
- a3 − b3 = (a − b)(a2 + ab + b2)
- (x + a)(x + b)(x + c) = x3 + x2(a + b + c) + x(ab + bc + ca) + abc
- (x + a)(x + b) = x2 + x(a + b) + ab