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


Algebra is a fundamental branch of mathematics, and in Class 10, one of the important topics you will encounter is polynomials. This comprehensive guide aims to build a strong understanding of polynomials, their properties, and operations. By the end of it, you will be comfortable working with polynomials both academically and practically.

What is a polynomial?

A polynomial is a mathematical expression made up of variables, coefficients, and the operations of addition, subtraction, and multiplication. A polynomial can be thought of as the sum of several terms, where each term consists of a variable raised to a power and a coefficient multiplied by that variable.

Polynomial structure

The general structure of a polynomial is expressed as follows:

a n x n + a n-1 x n-1 + ... + a 2 x 2 + a 1 x + a 0

Here:

  • a n , a n-1 , ..., a 0 are called coefficients.
  • x denotes the variable.
  • Every exponent of x is a non-negative integer.

Examples of polynomials

  • 3x 2 + 2x + 1 is a polynomial of degree 2.
  • x 3 - 4x + 7 is a polynomial of degree 3.
  • 5 is a polynomial of degree 0.

Degree of a polynomial

The degree of a polynomial is the highest power of the variable in the polynomial. It tells us about the most significant effect on the value of the polynomial when the variable is increased or decreased.

Identifying the degree

To find the degree, simply identify the largest exponent in the polynomial. Here are some examples:

  • The power of 7x 5 - 3x 3 + x is 5, because the highest power is 5.
  • The degree of 2x 4 + x 3 - 5 is 4.
  • The power of 9x 2 + x + 6 is 2.

Types of polynomials

Polynomials are named based on their degree. Here are some common types:

  1. Constant polynomial: A polynomial with degree 0, such as 5 or -2.
  2. Linear polynomial: A polynomial of degree 1, e.g., 2x + 1.
  3. Quadratic polynomial: A polynomial with degree 2, such as x 2 - 4x + 4.
  4. Cubic polynomial: A polynomial of degree 3, such as x 3 + 2x + 1.
  5. Quartic polynomial: A polynomial of degree 4, for example, x 4 - 2x 2 + x.

Operations on polynomials

Polynomials can be added, subtracted, multiplied, and divided, and each operation has specific rules.

Add

To add polynomials, add like terms. Like terms are those in which the same variable is raised to the same power.

Example:

(2x 2 + 3x + 5) + (x 2 + 4x + 2)
  • Combine the terms: 2x 2 + x 2 = 3x 2
  • Combine the terms: 3x + 4x = 7x
  • Combine the terms: 5 + 2 = 7

Result: 3x 2 + 7x + 7

2x² , x² 3x , 4 x 5 , 2 = 3x² + 7x + 7

Subtraction

To subtract polynomials, distribute the negative signs across the terms of the polynomial to be subtracted and then combine like terms.

Example:

(3x 3 + 2x - 5) - (x 3 - 4x + 3)
  • Distribute the negative sign: - (x 3 - 4x + 3) = -x 3 + 4x - 3
  • Combine the terms: 3x 3 - x 3 = 2x 3
  • Combine the terms: 2x + 4x = 6x
  • Combine the terms: -5 - 3 = -8

Result: 2x 3 + 6x - 8

Multiplication

When multiplying polynomials, use the distributive property, often known as the FOIL method for binomials.

Example:

(x + 2)(x + 3)

Apply the distributive property:

  • x * (x + 3) = x 2 + 3x
  • 2 * (x + 3) = 2x + 6

Combine all the parts: x 2 + 3x + 2x + 6

Combine like terms: x 2 + 5x + 6

x(x + 3) = x² + 3x 2(x + 3) = 2x + 6 = x² + 5x + 6

Polynomial division

Dividing polynomials can be more complicated than addition, subtraction, or multiplication. A common method of division is polynomial long division.

Example of polynomial division

Let's try dividing x 2 + 2x + 3 by x + 1:

  1. Divide the first term of the dividend by the first term of the divisor: x 2 /x = x.
  2. Multiply the entire denominator by this result: x(x + 1) = x 2 + x.
  3. Subtract the result from the original polynomial: x 2 + 2x + 3 - (x 2 + x) = x + 3.
  4. Repeat the process for the remaining x + 3 using x + 1.

This process will continue until you cannot do any further division, and the remainder you get will be less than the order of the divisor.

Graphing polynomials

Graphically, polynomials are represented by smooth, continuous curves on the coordinate plane. The degree and coefficients of the polynomial affect the shape and orientation of the graph.

Example: Graph of a quadratic polynomial

The graph of y = x 2 - 4 is a parabola opening upward.

y = x² – 4

Roots of polynomials

The roots (or zeros) of a polynomial are those values of the variable that make the polynomial equal to zero. Graphically, they are the points where the polynomial crosses or touches the x-axis.

Factorization of polynomials

Factoring is the process of breaking down a polynomial into simpler terms (products of other polynomials) that when multiplied give the original polynomial.

Example: Factoring a quadratic polynomial

Consider x 2 - 5x + 6 The factors are the two numbers that add up to -5 and multiply by 6, which are -3 and -2. Thus:

x 2 - 5x + 6 = (x - 3)(x - 2)

Uses of polynomials

Polynomials have widespread use in science, engineering, and other mathematical fields. They are used in calculations involving curves and conditions, such as trajectories and predictive models.

Conclusion

Polynomials are a fundamental concept in algebra, which provides immense utility in solving mathematical problems. Understanding their structure, operations (such as addition, subtraction, multiplication and division), factorization and their graphical representation is crucial to master algebra and apply it in various real-world situations. With practice, you will be able to handle polynomials with ease.


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