Grade 9
  1. Number Systems  1.1. Real Numbers  1.2. Rational Numbers  1.3. Irrational Numbers  1.4. Properties of Real Numbers  1.5. Laws of Exponents and Surds  1.6. Representation on the Number Line  1.7. Decimal Representation of Real Numbers
  2. Polynomials  2.1. Definition and Types of Polynomials  2.2. Degree of a Polynomial  2.3. Zeroes of a Polynomial  2.4. Remainder Theorem  2.5. Factorization of Polynomials  2.6. Algebraic Identities  2.7. Polynomial Equations and Roots
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Plotting Points on the Plane  3.3. Distance Formula  3.4. Section Formula  3.5. Area of a Triangle  3.6. Coordinates of Midpoint and Centroid
  4. Linear Equations in Two Variables  4.1. Solutions of a Linear Equation  4.2. Graphical Method  4.3. Algebraic Method  4.4. Applications of Linear Equations  4.5. Linear Inequalities
  5. Introduction to Euclidean Geometry  5.1. Basic Definitions and Terms  5.2. Axioms and Postulates  5.3. Theorems on Angles and Lines  5.4. Properties of Parallel Lines  5.5. Construction of Triangles  5.6. Congruence Axioms
  6. Lines and Angles  6.1. Angle Pair Relationships  6.2. Parallel Lines and Transversal  6.3. Properties of Angles Formed by Parallel Lines  6.4. Angle Sum Property of a Triangle  6.5. Types of Angles
  7. Triangles  7.1. Congruence of Triangles  7.2. Criteria for Congruence  7.3. Properties of Triangles  7.4. Inequalities in a Triangle  7.5. Similarity of Triangles  7.6. Pythagorean Theorem
  8. Quadrilaterals  8.1. Properties of Quadrilaterals  8.2. Types of Quadrilaterals  8.3. Midpoint Theorem  8.4. Properties of Parallelograms  8.5. Trapezium and Kite Properties  8.6. Diagonals and their Properties
  9. Areas of Parallelograms and Triangles  9.1. Area of a Parallelogram  9.2. Area of a Triangle  9.3. Proofs of Area Theorems  9.4. Applications of Area Theorems
  10. Circles  10.1. Definitions and Terms  10.2. Properties of a Circle  10.3. Chord Properties  10.4. Tangents to a Circle  10.5. Cyclic Quadrilaterals  10.6. Arcs and Angles Subtended
  11. Constructions  11.1. Basic Constructions  11.2. Constructing Bisectors  11.3. Constructing Triangles  11.4. Constructing Tangents to a Circle  11.5. Constructing Angles of Given Measure
  12. Heron's Formula  12.1. Area of a Triangle Using Heron’s Formula  12.2. Application to Different Triangles and Quadrilaterals  12.3. Proof of Heron’s Formula
  13. Surface Areas and Volumes  13.1. Surface Area of a Cube Cuboid and Cylinder  13.2. Volume of a Cube Cuboid and Cylinder  13.3. Surface Area and Volume of a Cone  13.4. Surface Area and Volume of a Sphere and Hemisphere  13.5. Conversion of Units  13.6. Volume of a Prism
  14. Statistics  14.1. Collection of Data  14.2. Presentation of Data  14.3. Measures of Central Tendency  14.4. Mean Median and Mode  14.5. Graphical Representation of Data  14.6. Range and Measures of Dispersion
  15. Probability  15.1. Introduction to Probability  15.2. Experimental Probability  15.3. Theoretical Probability  15.4. Simple Problems on Probability

Grade 9 → Polynomials ↓


Degree of a Polynomial


In mathematics, a polynomial is an expression that can contain constants, variables, and exponents, which are combined using addition, subtraction, multiplication, and non-negative whole number exponents of the variables. Understanding polynomials is a fundamental part of algebra, a major field of study in mathematics.

What is a polynomial?

A polynomial is a sum of terms, where each term includes a coefficient (a number), a variable (such as x) and an exponent (a non-negative integer that tells you how many times to multiply the variable by itself). For example, consider the following polynomial:

4x^3 + 3x^2 - 2x + 1

This polynomial has four terms: 4x^3, 3x^2, -2x, and 1 Each term can be described as follows:

  • 4x^3: The coefficient is 4, the variable is x, and the exponent is 3.
  • 3x^2: The coefficient is 3, the variable is x, and the exponent is 2.
  • -2x: The coefficient is -2, the variable is x, and the exponent is 1.
  • 1: This is a constant term with coefficient 1, and it has no variables.

Understanding the degree of a polynomial

The degree of a polynomial is the highest exponent of the variable within the polynomial. It tells us about the highest power of the variable present in any term of the polynomial when it is expressed in its standard form. The standard form of a polynomial arranges the terms in the descending order of their exponents.

For the polynomial 4x^3 + 3x^2 - 2x + 1, the highest exponent of x is 3. Therefore, the degree of this polynomial is 3.

Examples of finding a degree

Example 1: Consider the polynomial: 
7x^5 + 6x^3 + 4x^2 + x + 9
The degrees of the posts are as follows:
  • 7x^5: degree is 5
  • 6x^3: degree is 3
  • 4x^2: degree is 2
  • x: degree is 1
  • 9: degree is 0 (constant term)
The highest power is 5, so the degree of this polynomial is 5.
Example 2: Consider the polynomial: 
10 - 4x^2 + x^8 - 3x^7
Rearranging the exponents in descending order, we get: 
x^8 - 3x^7 - 4x^2 + 10
The highest power is 8, so the degree of this polynomial is 8.

Illustrating polynomials and their degrees

Polynomials can be visualized graphically, where the degree of the polynomial often gives a hint about the shape and nature of its graph. For example, let's explore the difference in the plot for different polynomial degrees.

Degree 1: Linear polynomials

A polynomial of degree 1 is called a linear polynomial. It is in the form of ax + b and represents a straight line on the graph.

(x = 0, y = b) y = ax + b

Degree 2: Quadratic polynomials

A polynomial of degree 2 is known as a quadratic polynomial and generally has the form ax^2 + bx + c. Its graph is a parabola.

y = ax² + bx + c

Degree 3: Cubic polynomials

A polynomial of degree 3 is called a cubic polynomial. It has the form ax^3 + bx^2 + cx + d and can form an S-shaped curve.

y = ax³ + bx² + cx + d

Degree and polynomial expression

Different polynomial expressions can have the same degree, even though they look different. The degree helps to understand the behavior of polynomial functions, including their graphs and intersection points with axes. Here's how different configurations of the coefficients do not change the underlying degree of the polynomial, as long as the highest power remains unchanged.

Example 3: Consider the following polynomials:
  • 8x^4 + 7x^3 - x^2 + 5
  • -3x^4 + 2x^2 + x - 12
  • x^4 - x^3 + 3x^2 + 2
All these polynomials have the same degree, which is 4. Despite having different coefficients, the highest power of x remains the same.

Special cases of polynomials

Stable polynomials

A constant polynomial refers to a polynomial that has only one constant term and no variable terms, such as 7 or -4. These have degree 0 because the variable term is absent, which can be considered as x^0.

Zero polynomial

The zero polynomial is a special case, expressed simply in terms of 0 The degree of the zero polynomial is undefined, although sometimes it is considered to be negative infinity for some computational reasons.

Why is a degree important?

The degree of a polynomial provides important information about the properties and behavior of the polynomial, such as:

  • Number of roots: A polynomial of degree n can have at most n roots. These roots can be real or complex numbers.
  • Behavior as x approaches infinity: The degree helps in predicting the final behavior of the polynomial, when x approaches infinity or negative infinity.
  • Slope and curvature: For polynomials represented graphically, the degree affects the number of inflection points of the curve.

Conclusion

Understanding the degree of a polynomial is important in both algebraic expressions and graphical interpretations. By identifying the highest exponent present in a polynomial, you gain information about the shape of the graph, the number of approximate solutions, and the overall behavior of the expression. Whether dealing with simple linear polynomials or complex cubic equations, recognizing and working with the degree of polynomials is a fundamental skill in mathematics that lays the groundwork for advanced topics.


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Definition and Types of Polynomials
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Degree of a Polynomial

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