Algebraic expressions are fundamental mathematical constructs that play a crucial role in representing and solving a wide range of mathematical and real-world problems. They are composed of variables, constants, and mathematical operations, such as addition, subtraction, multiplication, and division. Algebraic expressions are a bridge between the abstract world of mathematics and the practical world of problem-solving.
Key components of an algebraic expression:
Variables: These are symbols (usually letters) that represent unknown values or quantities. Common variables include "x," "y," and "z." Variables allow us to generalize mathematical relationships and solve problems with unknowns.
Constants: These are fixed numerical values that do not change within the expression. Examples include numbers like 2, 5, π (pi), or any other specific constant value.
Mathematical Operations: Algebraic expressions include operations like addition (+), subtraction (-), multiplication (*), division (/), and exponentiation (^ or **). These operations define how the variables and constants interact within the expression.
Coefficients: Coefficients are the numerical values that multiply variables. For example, in the expression 3x, 3 is the coefficient of the variable x.
Algebraic expressions can take various forms, from simple linear expressions like 2x + 3 to more complex ones like (x^2 - 4)(x + 1). They are used in a wide range of mathematical contexts, including equations, inequalities, and functions.
Expansion of Algebraic Expressions:
Expanding an algebraic expression involves simplifying it by removing parentheses and combining like terms. This process is essential for solving equations, simplifying complex expressions, and gaining a better understanding of the underlying mathematical relationships.
Here's how to expand algebraic expressions:
Distribute: When an expression contains parentheses, you distribute each term within the parentheses to every term outside the parentheses using the appropriate mathematical operation (usually multiplication or addition).
Example: To expand 2(x + 3), you distribute the 2 to both terms inside the parentheses: 2x + 6.
Combine Like Terms: After distributing and simplifying, you look for like terms (terms with the same variable(s) and exponent(s)) and combine them.
Example: In the expression 3x + 2x, you combine the like terms to get 5x.
Remove Parentheses: If there are nested parentheses, continue to distribute and simplify until no parentheses remain.
Expanding algebraic expressions is a crucial step in solving equations and simplifying complex expressions. It allows mathematicians and scientists to manipulate and analyze mathematical relationships efficiently, making it an essential tool in various fields, including physics, engineering, and computer science.
3. Learning Outcomes
“A star does not compete with other stars around it; it just shines.”
Algebraic
Expressions
Operations Expansions
4.
5.
6. You are familiar with the
following type of numerical
expressions:
12 + 6
3 (12)
6 (3 + 2)
15 -4 (6)
7. Variable
In the expression
12 + B,
the letter “B” is a variable.
A variable is a letter or symbol that represents an
unknown value.
8. Constants
In the expression:
5x + 7y + 2
the constant is 2
In the expression: x -3
the constant is -3
A constant is a number that cannot change its value.
11. MOTIVATION
Algebra is a fascinating and essential part of mathematics. It provides the
written language in which mathematical ideas are described.
Many parts of mathematics are initiated by finding patterns and relating to
different quantities. Before the introduction and development of algebra, these
patterns and relationships had to be expressed in words. As these patterns and
relationships became more complicated, their verbal descriptions became
harder and harder to understand. Our modern algebraic notation greatly
simplifies this task.
12. MOTIVATION
A well-known formula, due to Einstein, states that
E = mc2 .This remarkable formula gives the relationship
between energy, represented by the letter E, and mass,
represented by letter m. The letter c represents the speed
of light, a constant, which is about 300 000 000 metres
per second.
E = mc2.
13. MOTIVATION
The simple algebraic statement E = mc2 states that some
matter is converted into energy (such as happens in a
nuclear reaction),
then the amount of energy produced is equal to the mass
of the matter multiplied by the square of the speed of
light. You can see how compact the formula is
compared with the verbal description.
E = mc2.
14. Al-Khwārizmī's contributions
to mathematics, geography,
astronomy.
He established the basis for
innovation in algebra
and trigonometry.
His systematic approach to
solving linear and quadratic
equations led to algebra.
Muhammad ibn Musa al-Khwarizmi
15. A number x is multiplied by itself and then doubled.
x × x × 2 = x2 × 2 = 2x2.
A number x is squared and then multiplied by the
square of a second number y.
x2 × y2 = x2y2.
A number x is multiplied by a number y and the result
is squared.
(x × y)2 = (xy)2 which is equal to x2y2
EXAMPLES (concise algebraic notation.)
20. Algebraic
Expressions Definition: Like and Unlike Terms.
If you have 3 pencil case with the same number x
of pencils in each,
you have 3x pencils altogether.
x pencils x pencils x pencils
21. Algebraic
Expressions Definition: Like Terms.
If there are 2 more pencil cases with x pencils in
each,
then you have 3x + 2x = 5x pencils altogether.
This can be done as the number of pencils in
each case is the same. The terms 3x and 2x are
said to be like terms.
x pencils x pencils x pencils
x pencils x pencils
25. Algebraic
Expressions Addition and Subtraction
Working Rule:
In adding algebraic expressions, we collect
different groups of like terms and find the
sum/difference of like terms.
26. 1) 2x + 3x + 5x
2) 3xy + 2xy
3) 4x2 – 3x2
4) 2x2 + 3x + 4x
5) 4x2y – 3x2y + 3xy2
EXAMPLES (Simplify each of the following by adding or subtracting like terms)
2x + 3x + 5x = 10x
3xy + 2xy = 5xy
4x2 – 3x2 = x2
2x2 + 3x + 4x
= 2x2 + 7x
4x2y – 3 x2 y +3xy2
= x2 y + 3x y2
31. This process of rewriting
an expression to remove
brackets is usually referred
to as expanding brackets.
EXPANDING
BRACKETS AND
COLLECTING LIKE
TERMS
Expansions of Algebraic Expressions
35. »Descartes (La Geometrie, 1637)
»Wallis (1693)
»Egyptian scribe Ahmes (1650 BC)
»Scottish antiquarian, Henry Rhind (1858 )
»Euclid (circa 300 BC)
»Diophantus (circa 275 AD )
»An Indian mathematician Baudhayana (800 BC)
»The Indian mathematician Brahmagupta (680 AD )
»al-Khwarizmi,(circa 825 AD)
»Fibonacci European writer (circa 1225 AD)
»Cardano, Tartaglia (16th century),
(algebra has a very long history)
Name of mathematician and Countries
36. » A History of Mathematics: An
Introduction, 3rd Edition, Victor
J. Katz, Addison-Wesley, (2008)
REFERENCES
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