Terms used in Algebraic expression:
- An algebraic expression consists of one or more terms.
- A term in an algebraic expression is a combination of variables, constants, coefficients, and exponents, usually separated by mathematical operators. Some examples of algebraic terms are:
- Constant Term:
- Definition: A term that contains only a numerical constant.
- Example: 5, -7, 0.25
- Linear Term:
- Definition: A term with a variable raised to the power of 1 (exponent of the variable is 1).
- Example: 3x, -2y, 0.5a
- Quadratic Term:
- Definition: A term with a variable raised to the power of 2 (exponent of the variable is 2).
- Example: 2x^2, -4y^2, 1.5a^2
- Cubic Term:
- Definition: A term with a variable raised to the power of 3 (exponent of the variable is 3).
- Example: 4x^3, -8y^3, 2a^3
- Coefficient Term:
- Definition: The numerical factor that multiplies a variable in a term.
- Example: In the term 3x, the coefficient is 3. In -2y^2, the coefficient is -2.
- Exponent Term:
- Definition: The exponent represents the power to which a variable is raised in a term.
- Example: In 2x^2, the exponent is 2. In a^3, the exponent is 3.
- Like Terms:
- Definition: Terms that have the same variable(s) with the same exponent(s) and can be combined.
- Example: 3x and 2x are like terms because they both have the variable x raised to the power of 1.
- Unlike Terms:
- Definition: Terms that have different variables or different exponents on the same variable and cannot be combined.
- Example: 3x^2 and 2y are unlike terms because they involve different variables (x and y) or have different exponents.
- For better understanding consider below example:
P(a) = 5 — 3a + 2a2 + 4a3
- Here
- 5 is a constant term because it contains only a numerical constant.
- —3a is a linear term because it has the variable a raised to the power of 1.
- 2a2 is a quadratic term because it has the variable a raised to the power of 2.
- 4a3 is a cubic term because it has the variable a raised to the power of 3.
- —3a This term have the coefficient is —3.
- 2a2 and 4a3 Here the terms 2a2 and 4a3 have exponents are 2 and 3, respectively.
- —3a and 4a3 Here the terms have the same variable a but with different exponents. They are unlike terms and cannot be combined.
Some Key components of an algebraic expression:
- Variables:
- Symbols or letters that represent unknown or unspecified values are known as variables. Common variables include x, y, a, b, etc. Variables can take on different values, and the expression may involve manipulating or solving for these variables.
- Constants:
- Constants are Numerical values that do not change. Common constants include numbers like 1, 2, π, e, and so on.
- Mathematical Operator:
- Algebraic expressions include mathematical Operator such as addition (+), subtraction (-), multiplication (x), division (/), and exponentiation (^) to combine variables and constants. Some Mathematical Operator and their use are:
- Addition (+):
- The addition operator is used to combine two or more numbers or expressions to find their sum. For example, 3 + 4 means adding 3 and 4 to get 7.
Subtraction (—):
The subtraction operator is used to find the difference between two numbers or expressions. For example, 7—2 means subtracting 2 from 7 to get 5.
- Multiplication (x):
- The multiplication operator is used to multiply two or more numbers or expressions. For example, 3 x 4 means multiplying 3 and 4 to get 12.
- Division (/):
- The division operator is used to divide one number by another. For example, 10/2 means dividing 10 by 2 to get 5.
- Exponentiation (^ or**):
- The exponentiation operator is used to raise a number or expression to a certain power. For example, 23 means 2 raised to the power of 3, which is 8.
- Equality or Equal (=):
- The equality operator is used to indicate that two expressions or values are equal. For example, 3 + 4 = 7 means that the sum of 3 and 4 is equal to 7.
- Inequality Operators (<, >, ≤, ≥):
- These operators are used to compare two values. For example, 5<8 means that 5 is less than 8.
- Modulus (%):
- The modulus operator is used to find the remainder when one number is divided by another. For example, 10 means finding the remainder when 10 is divided by 3, which is 1.
- Parentheses (()):
- Parentheses are used to group numbers and operations, indicating the order of operations. For example, 2 x (3 + 4) ensures that the addition inside the parentheses is performed first.
- Square Root (√):
- The square root symbol is used to find the square root of a number. For example, √9 means finding the square root of 9, which is 3.
BODMAS /PEMDAS
The order of mathematical operations (BODMAS /PEMDAS) is often used to determine the sequence in which these operators are applied when solving complex mathematical expressions. Meaning and sequence are :
B: Brackets or (P) Parentheses
Perform calculations inside brackets (parentheses) first. If there are nested brackets, start with the innermost ones and work outward.
O: Orders or (E)Exponents
Evaluate expressions involving exponents (powers and roots). This includes square roots, cube roots, and any other operations with exponents.
D: Division
Perform division from left to right. Division has equal priority with multiplication.
M: Multiplication
Perform multiplication from left to right. Multiplication has equal priority with division.
A: Addition
Perform addition from left to right. Addition has equal priority with subtraction.
S: Subtraction
Perform subtraction from left to right. Subtraction has equal priority with addition.
Linear
polynomial:
- A linear polynomial is a polynomial of degree 1, which means it has only one term involving a variable raised to the first power. The general form of a linear polynomial is:
ax + b
In this form, a and b are constants, with a being the coefficient of the variable x. The variable x is raised to the first power, and there are no other terms involving x raised to higher powers.
- Example 1.
3x + 2
In this linear polynomial, 3x is the term with the variable x raised to the first power (degree 1), and 2 is the constant term. The coefficient of x is 3.
- Example 2.
— 2x + 7
In this linear polynomial, —2x is the term with the variable x raised to the first power, and 7 is the constant term. The coefficient of x is —2.
- Linear polynomials represent straight-line equations in a Cartesian coordinate system.
- They have a constant slope (the coefficient of x), and their graph is a straight line.
- The value of x determines the position on the line, and the value of the polynomial at a specific x gives the corresponding y-coordinate on the line.
Quadratic Polynomial:
- A quadratic polynomial is a polynomial of degree 2, which means it contains terms involving a variable raised to the second power (quadratic term) and lower powers (linear and constant terms). The general form of a quadratic polynomial is:
ax2 + bx + c
In this form, a, b and c are constants, with a being the coefficient of the quadratic term (x2), b being the coefficient of the linear term (x) and c being the constant term.
- Quadratic polynomials can be graphically represented as parabolas when plotted on a Cartesian coordinate system.
- The shape of the parabola depends on the sign and value of the coefficient a:
- If a>0, the parabola opens upward, and it has a minimum point.
- If a<0, the parabola opens downward, and it has a maximum point.
Example 1.
2x2 — 3x + 1
In this quadratic polynomial, 2x2 is the quadratic term, —3x is the linear term, and 1 is the constant term.
Example 2.
—x2 + 4x — 5
In this quadratic polynomial, —x2 is the quadratic term, 4x is the linear term, and —5 is the constant term.
Example 3.
3x2
This is a quadratic polynomial with no linear or constant terms, meaning it's a pure quadratic expression.
- Quadratic polynomials are commonly used in various mathematical and scientific operations, including physics, engineering and optimization problems.
- Their graphical representations are used to analyze real-world scenarios, such as motion of objects, projectile motion and finding the maximum or minimum values of functions.
Cubic Polynomial:
- A cubic polynomial is a polynomial of degree 3, meaning it contains terms involving a variable raised to the third power (cubic term), second power (quadratic term), first power (linear term), and a constant term. The general form of a cubic polynomial is:
ax3 + bx2 + cx + d
In this form, a, b, c and d are constants, with a being the coefficient of the cubic term (x3 ), b being the coefficient of the quadratic term (x2 ), c being the coefficient of the linear term (x) and d being the constant term.
- Cubic polynomials can take various shapes when graphed on a Cartesian coordinate system.
- Its appearance depends on the values of the coefficients.
- They may have one or two local extrema (maxima or minima) and may intersect the x-axis one, two, or three times.
Example 1.
x3 + 2x2 — 3x + 4
In this cubic polynomial, x3 is the cubic term, 2x2 is the quadratic term, —3x is the linear term and 4 is the constant term.
Example 2.
—2x3 + x2 — 5x + 1
In this cubic polynomial, —2x3 is the cubic term, x2 is the quadratic term, —5x is the linear term and 1 is the constant term.
Example 3.
3x3 — 6x2 + 2x — 7
In this cubic polynomial, 3x3 is the cubic term, 6x2 is the quadratic term, 2x is the linear term and —7 is the constant term.
- Cubic polynomials have uses in various fields’ like- physics, engineering, economics, and computer graphics.
- They are used to model and analyze complex relationships in real-world problems, such as modeling the behavior of certain physical systems, predicting trends in data, and designing curves in computer-generated graphics.
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