Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 → Algebra → Polynomials ↓


Degree of a Polynomial


In algebra, one of the important concepts students encounter is the concept of a polynomial. Polynomials are expressions made up of variables and coefficients. They involve operations such as addition, subtraction, multiplication, and sometimes division. Understanding the structure of a polynomial helps in solving algebraic equations. An important aspect of understanding polynomials is learning about the degree of a polynomial.

What is a polynomial?

A polynomial is an algebraic expression consisting of terms. Each term is composed of a coefficient (a constant) and a variable raised to a non-negative integer exponent. The general form of a polynomial in one variable (usually represented as x) is given as:

P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0

Here, each a_i is a coefficient and n is a non-negative integer indicating the degree of the polynomial. The variable x is undefined or a placeholder in the expression.

What is the degree of a polynomial?

The degree of a polynomial is the largest exponent of the variable in the polynomial. It tells us the highest power of the variable x in the expression. For example, in the polynomial:

P(x) = 4x^3 + 3x^2 + 5x + 7

The degree is 3 because the highest exponent of x in this polynomial is 3 The degree gives us an idea of the "size" or "complexity" of the polynomial. It also helps to determine the shape of the graph of the polynomial function.

Visual example of a polynomial of degree 2

y = 2x 2 + 3x + 1

Examples of finding the degree of a polynomial

Example 1

Consider the polynomial:

P(x) = 5x^4 + 2x^3 + x^2 + 7x + 9

Here, the power is 4 because it is the highest power of x.

Example 2

For polynomials:

Q(x) = 3x^5 - 4x^3 + 2x + 8

The power is 5 because 5 is the highest exponent of x.

Example 3

Consider the polynomial:

R(x) = 7x + 12

This polynomial is a linear polynomial (degree 1) because the highest power of x is 1.

Example 4

For a constant polynomial:

S(x) = 6

The degree of a polynomial is 0 because it has no variable with an exponent. It is just a constant.

Common misconceptions

Sometimes students get confused between the degree of a polynomial and the number of its terms. The degree deals only with the highest exponent of the variable, not with the number of terms, coefficients, or specific values of the coefficients.

Example of different positions vs. degrees

Polynomials:

P(x) = 6x^4 + 2x + 5

has three terms, but the exponent is still 4 because of the term 6x^4.

Applications of understanding polynomial degree

Knowing the degree of a polynomial is fundamental in algebra and calculus. It plays an important role in the following:

  1. Solving Polynomial Equations: The degree of the polynomial determines the maximum number of solutions or roots of the equation.
  2. Graphing a Polynomial: This degree helps in predicting the behaviour of the polynomial graph, especially to understand the end-behaviour and number of turning points.

Additional example: Graphical representation

Below is the representation of a second degree (quadratic) polynomial:

y = 5x²

Conclusion

It is important to understand the degree of a polynomial as it is directly related to the properties and behavior of a polynomial function. This concept serves as a bridge to more advanced topics in algebra, providing a foundational understanding needed to deal with equations, functions, and eventually calculus concepts. In short, the degree does not only tell us about the highest power of a polynomial but also guides us in solving equations and analyzing graphs effectively.

Remember, the degree of a polynomial is the highest exponent present in the polynomial. By understanding this basic concept, you can easily handle more complex polynomial expressions and equations in your studies of algebra and beyond.


Grade 10 → 2.1.2


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Definition and Types of Polynomials
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Zeros of a Polynomial

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Degree of a Polynomial

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