Topics
Number Systems
Number Systems
Polynomials
Algebra
Algebraic Expressions
Algebraic Identities
Coordinate Geometry
Linear Equations in Two Variables
Coordinate Geometry
Geometry
Area
Constructions
- Introduction of Constructions
- Geometric Constructions
- Some Constructions of Triangles
Introduction to Euclid’S Geometry
Mensuration
Statistics and Probability
Lines and Angles
- Introduction to Lines and Angles
- Basic Terms and Definitions
- Intersecting Lines and Non-intersecting Lines
- Parallel Lines
- Concept of Pairs of Angles
- Concept of Transversal Lines
- Basic Properties of a Triangle
Probability
Triangles
Quadrilaterals
- Properties of Quadrilateral
- Another Condition for a Quadrilateral to Be a Parallelogram
- Theorem of Midpoints of Two Sides of a Triangle
- Property: The Opposite Sides of a Parallelogram Are of Equal Length.
- Theorem: A Diagonal of a Parallelogram Divides It into Two Congruent Triangles.
- Theorem : If Each Pair of Opposite Sides of a Quadrilateral is Equal, Then It is a Parallelogram.
- Property: The Opposite Angles of a Parallelogram Are of Equal Measure.
- Theorem: If in a Quadrilateral, Each Pair of Opposite Angles is Equal, Then It is a Parallelogram.
- Property: The diagonals of a parallelogram bisect each other. (at the point of their intersection)
- Theorem : If the Diagonals of a Quadrilateral Bisect Each Other, Then It is a Parallelogram
Circles
Areas - Heron’S Formula
- Area of a Triangle by Heron's Formula
- Application of Heron’s Formula in Finding Areas of Quadrilaterals
- Geometric Interpretation of the Area of a Triangle
Surface Areas and Volumes
Statistics
- Introduction
- Definition: Polynomial in One Variable
- Definition: Degree
- Definition: Polynomials of Two or More Variables
- Real-Life Application
- Key Points Summary
Introduction
Imagine you're building with mathematical blocks. Each block has three parts: a number (coefficient), a letter (variable), and a small number above it (exponent).
When you add or subtract such terms (with whole-number exponents), you get a polynomial.
Definition: Polynomial
A polynomial is an algebraic expression made up of terms in which the variables have non‑negative whole-number exponents.
Definition: Polynomial in One Variable
When an algebraic expression is made of only one variable, it is called a polynomial in one variable.
Examples of Polynomials in One Variable:
| Polynomial | Variable | Why it’s a polynomial |
|---|---|---|
| 3 + 5x − 7x2 | x | All exponents (0, 1, 2) are whole numbers |
| 9y3 − 5y2 + 8 | y | All exponents (3, 2, 0) are whole numbers |
| z4 + z - 1 | z | All exponents (4, 1, 0) are whole numbers |
Definition: Degree
The degree of a polynomial is simply the highest exponent (power) in the expression.
Example 1: 4x² - 3x⁵ + 8x⁶
- Term 1: 4x² → exponent = 2
- Term 2: -3x⁵ → exponent = 5
- Term 3: 8x⁶ → exponent = 6
- Degree = 6 (highest exponent)
Example 2: 25 - x⁴
- Term 1: 25 → exponent = 0 (since 25 = 25x⁰)
- Term 2: -x⁴ → exponent = 4
- Degree = 4
Definition: Polynomials of Two or More Variables
A polynomial made of two or more variables is called a polynomial in two or more variables.
Examples of Polynomials in Two/More Variables:
- Term 1: 3x → exponent sum: 1
- Term 2: xy2 → exponent sum: 1 + 2 = 3
- Term 3: −8yz → exponent sum: 1 + 1 = 2
Degree = 3
Real-Life Application
Physics: Motion of Objects
In physics, polynomials are used to describe the motion of objects. For instance, the equation for an object’s position at any time can often be expressed as a polynomial.
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Example:
The equation s(t) = 4t2 + 3t + 2 describes the position of an object over time, where s(t) is the position and t is time. -
Here, the degree of the polynomial is 2, indicating that the object is undergoing accelerated motion (i.e., its velocity is changing over time).
Key Points Summary
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Polynomials are algebraic expressions with whole number exponents only.
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Degree = highest exponent in the expression.
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For multiple variables: add exponents in each term, take the highest sum.
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Polynomials are everywhere: sports, business, technology, and science.
