An algebraic expression is a combination of constants and variables connected by four fundamental operations such as addition, subtraction, multiplication and division. Algebraic expressions are used to represent real-life situations concisely and clearly, allowing us to analyze and predict outcomes.
Example: 2x + 3 is an algebraic expression,
Where;
x is a variable
2 and 3 are constants
These expressions can be evaluated by substituting a specific value for the variable and performing the operations in the correct order. Algebraic expressions are often used in solving equations, graphing functions and simplifying expressions.
The addition sign (+) indicates that the two values should be added together. If we were to substitute a specific value for x (such as x = 5), we could simplify the expression to
2x + 3
= 2(5) + 3
= 10 + 3
= 13
Algebraic expression with a single term.
For example: 5x, -7xy, 7/5
Algebraic expression with two terms.
For example: 7x - 3y, 6x^{2} - 4xy
Algebraic expression with three terms.
For example: -6x + 7y -4xy, 5p - 106p^{2}q + 2
An algebraic expression with one or more terms, each with a non-zero coefficient and variables with only positive exponents.
For example: 3x^{2} + 2x + 1 or 4y^{3} - 2y^{2} + y + 6
Variables: In algebraic expressions, variables are used to represent values that are not yet known or can vary based on the context of the problem. These variables are typically represented by letters such as x, y or z.
For example, in the expression 7x - 3, x is the variable and its value can change depending on the context.
Constants: Constants, on the other hand, are values that do not change and are represented by numbers that have a specific value.
For example, in the expression 7x - 3 the number 3 is constant. They have specific values and do not change.
Terms: Terms are the parts of an algebraic expression that are separated by addition or subtraction.
For example, in the expression 7x - 3, the term 7x represents the product of 7 and x and the term 3 represents the constant value.
Like and Unlike Terms: Terms having the same variables with the same exponents are called like terms, otherwise unlike terms.
Coefficient: The coefficient of an algebraic expression is the numerical value that is multiplied by the variable.
For example, in the expression 7x - 3, the coefficient of the variable x is 7. This means that x is multiplied by 7 in the term 7x.
Degree: Highest power of the variable in that expression is known as a degree.
For example, in the expression 7x^{3} + 2x^{2} - 1, the degree is 3, because that is the highest power of x that appears in the expression.
All of the above are important concepts in algebraic expressions. Variables represent unknown values, constants represent specific values, terms are the parts of an expression separated by addition or subtraction and the coefficient is the numerical value that is multiplied by the variable. Understanding these concepts is crucial for solving and manipulating algebraic expressions.
Algebraic identities are mathematical statements that are always true for specific values of the variables involved.
Algebraic expressions serve the purpose of simplifying and manipulating equations, and they have a crucial role in solving equations and dealing with problems in algebra and other mathematical disciplines. Additionally, they are used in the factorization of polynomials. To find algebraic expressions and solve various polynomials, algebraic identities are used.
The standard Algebraic Identities are formed from the Binomial Theorem.
Some Standard Algebraic Identities list are listed below:
Identity I: (a + b)^{2} = a^{2} + 2ab + b^{2}
Identity II: (a – b)^{2} = a^{2} – 2ab + b^{2}
Identity III: a^{2} – b^{2}= (a + b) (a – b)
Identity IV: (x + a) (x + b) = x^{2} + (a + b) x + ab
Identity V: (a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ca
Identity VI: (a + b)^{3} = a^{3} + b^{3} + 3ab (a + b)
Identity VII: (a – b)^{3} = a^{3} – b^{3} – 3ab (a – b)
Identity VIII: a^{3} + b^{3} = (a + b) (a^{2} – ab + b^{2})
Identity IX: a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2})
Identity X: a^{3} + b^{3} + c^{3} – 3abc = (a + b + c) (a^{2} + b^{2} + c^{2} – ab – bc – ca)
To multiply the monomials put the constants together and multiply the like variables together.
Therefore, 3z^{2} x 5z^{3}y^{2} = (3 x 5) (z^{2} x z^{3}y^{2})
= 15 z^{5}y^{2}
To multiply the binomials (x + 2) and (x - 3) first take term of (x + 2) as x with (x - 3) than 2 with (x - 3) and add them.
Therefore, (x + 2) (x - 3) = x (x - 3) + 2(x - 3)
= x^{2} - 3x + 2x - 6
= x^{2} - x – 6
NOTE: It's important to remember the order of operations (PEMDAS) when multiplying algebraic expressions.
CREST Olympiads has launched this initiative to provide free reading and practice material. In order to make this content more useful, we solicit your feedback.
Do share improvements at info@crestolympiads.com. Please mention the URL of the page and topic name with improvements needed. You may include screenshots, URLs of other sites, etc. which can help our Subject Experts to understand your suggestions easily.