Class 10 Maths

Triangles Triangles : Formulas, Theorems & Examples (Class 10 Maths) Triangles and Similarity Triangles are fundamental geometric shapes that form the building blocks for many other polygons. In this chapter, we will delve into the fascinating world of similar triangles. Similarity is a concept that describes shapes that are the same in proportion but may differ in size. Understanding similar triangles is crucial in geometry as it has wide applications in various fields like architecture, engineering, map-making, and even art!In this chapter, you will learn about:• Similar Figures and Similar Polygons• Basic Proportionality Theorem (Thales’ Theorem)• Criteria for Similarity of Triangles (AAA, AA, SSS, SAS)• Areas of Similar Triangles• Pythagoras Theorem and its Converse   1. SIMILAR FIGURES • DEFINITION: Similar figures are figures that have the same shape but may have different sizes. This means one is an enlargement or reduction of the other.• Examples:o Two circles of different radii.o Two squares of different side lengths.o Photographs of different sizes of the same object.o Maps and scale models of buildings.• Congruent vs. Similar:o Important Note: All congruent figures are always similar (they have the same shape and the same size!).o However, the converse is not true: similar figures are not necessarily congruent (they have the same shape but may have different sizes).   2. SIMILAR POLYGONS • DEFINITION: Two polygons are said to be similar to each other if:o (i) Corresponding Angles are Equal: This means that if you match up the vertices of the two polygons in the ‘same order’, the angles at those corresponding vertices must be equal.o (ii) Lengths of Corresponding Sides are Proportional: This means the ratio of the lengths of any two corresponding sides is the same for all pairs of corresponding sides. If polygon A and polygon B are similar, then:(Side 1 of A) / (Corresponding Side 1 of B) = (Side 2 of A) / (Corresponding Side 2 of B) = … = a constant ratio (called the scale factor). • Examples:o Line Segments: Any two line segments are similar because their lengths are always proportional.o Circles: Any two circles are similar because the ratio of their circumferences to their diameters is always π, and the ratio of their radii acts as the scale factor.o Squares: Any two squares are similar because all angles are 90 degrees and the ratio of their side lengths is constant. o Example (Non-Similar Rectangles): Consider a rectangle with sides 2cm and 4cm, and another with sides 3cm and 5cm. While all angles are 90 degrees, the ratio of sides is not constant (2/3 ≠ 4/5). Therefore, these rectangles are NOT similar. Key Takeaways: Similar Polygons* Same shape, sides are proportional, corresponding angles are equal.* Conditions for similarity: (i) Equal corresponding angles, (ii) Proportional corresponding sides.* Not all polygons with equal angles or proportional sides are necessarily similar (need both conditions to be met). 3. SIMILAR TRIANGLES • Deduction from Similar Polygons: Two triangles are similar if:o (i) Corresponding Angles are Equal:If ΔABC and ΔPQR are similar, then:∠A = ∠P, ∠B = ∠Q, ∠C = ∠Ro (ii) Corresponding Sides are Proportional:If ΔABC and ΔPQR are similar, then:AB/PQ = AC/PR = BC/QR Key Takeaways: Similar Triangles (Definition)* Two triangles are similar if: * (i) All corresponding angles are equal (∠A=∠P, ∠B=∠Q, ∠C=∠R) * (ii) All corresponding sides are proportional (AB/PQ = AC/PR = BC/QR)* Both conditions MUST be satisfied for triangles to be similar (by definition). 4. THEOREM 1: BASIC PROPORTIONALITY THEOREM (THALES’ THEOREM) • Statement: If a line is drawn parallel to one side of a triangle to intersect the other two sides at distinct points, then it divides the other two sides in the same ratio.• Diagram:• Given: In ΔABC, DE || BC, where D is a point on AB and E is a point on AC.• To prove: AD/DB = AE/EC• Construction: Draw EM ⊥ AD and DN ⊥ AE. Join B to E and C to D. (To use the area of triangles to establish ratios, we draw perpendiculars (altitudes) EM and DN to sides AD and AE respectively. Joining BE and CD helps us form triangles ABDE and ACDE for comparison.) • Proof:o Step 1: Area Ratios in ΔADE and ΔBDE:In ΔADE and ΔBDE,ar(ΔADE) / ar(ΔBDE) = (1/2 × AD × EM) / (1/2 × DB × EM) [Area of a triangle = 1/2 × base × height]= AD / DB …(i)[Explanation: Here, we consider AD and DB as bases of triangles ADE and BDE respectively. Notice that both triangles share the same altitude EM from vertex E to the line containing AB.]o Step 2: Area Ratios in ΔADE and ΔCDE:In ΔADE and ΔCDE,ar(ΔADE) / ar(ΔCDE) = (1/2 × AE × DN) / (1/2 × EC × DN) [Area of a triangle = 1/2 × base × height]= AE / EC …(ii)[Explanation: Similarly, here AE and EC are bases of triangles ADE and CDE, and they share the same altitude DN from vertex D to the line containing AC.]o Step 3: Equal Areas:ar(ΔBDE) = ar(ΔCDE) …(iii)[Reason: ΔBDE and ΔCDE are on the same base DE and between the same parallel lines DE and BC. Triangles on the same base and between the same parallels are equal in area.]o Step 4: Equating Ratios:From (i), (ii) and (iii),Since ar(ΔBDE) = ar(ΔCDE), we have:AD/DB = AE/ECHence Proved. THEOREM 1: BASIC PROPORTIONALITY THEOREM (BPT) / THALES’ THEOREM• Example:In triangle ABC, DE || BC. If AD = 2cm, DB = 3cm, and AE = 3cm, find EC. AD=2, DB=3, AE=3, EC=?Solution:Since DE || BC, by Thales’ Theorem (BPT), we have:AD/DB = AE/ECSubstituting the given values:2/3 = 3/ECCross-multiplying:2 * EC = 3 * 32 * EC = 9EC = 9/2 = 4.5 cmTherefore, EC = 4.5 cm. Key Takeaways: Thales’ Theorem (BPT)* If a line is parallel to one side of a triangle and intersects the other two sides, it divides those sides proportionally.* Formula: AD/DB = AE/EC (when DE || BC in ΔABC, and D is on AB, E is on AC)* Useful for finding unknown side

Arithmetic Progression Arithmetic Progression : Formulas, Theorems & Examples (Class 10 Maths) Sequences, Series, and Progressions: Building Blocks of AP Before we understand Arithmetic Progression, let’s clarify some fundamental terms: Sequences: Definition: A sequence is an ordered list of numbers. Each number in the sequence is called a term. Sequences follow a specific rule or pattern to determine the next term. Types: Finite Sequence: A sequence with a limited number of terms. It has a last term. Example: 2, 4, 6, 8, 10 (This is a finite sequence with 5 terms) Infinite Sequence: A sequence with an unlimited number of terms. It continues indefinitely and does not have a last term. Example: 1, 2, 3, 4, 5, … (The “…” indicates that the sequence continues infinitely) Examples of Sequences: Sequence of even numbers: 2, 4, 6, 8, 10, … Sequence of odd numbers: 1, 3, 5, 7, 9, … Sequence of squares of natural numbers: 1, 4, 9, 16, 25, … Sequence defined by a rule: Let the nth term be given by an = 2n + 1. Then the sequence is: 3, 5, 7, 9, 11, … (by substituting n=1, 2, 3, 4, 5, …) Series: Definition: A series is the sum of the terms of a corresponding sequence. Formation: If a sequence is a1, a2, a3, … , an, …, then the series is a1 + a2 + a3 + … + an + … Types (corresponding to sequences): Finite Series: Sum of terms of a finite sequence. Example: For the sequence 2, 4, 6, 8, 10, the series is 2 + 4 + 6 + 8 + 10 = 30 Infinite Series: Sum of terms of an infinite sequence. Example: For the sequence 1, 2, 3, 4, 5, …, the series is 1 + 2 + 3 + 4 + 5 + … Examples of Series: Series of first five even numbers: 2 + 4 + 6 + 8 + 10 Series of natural numbers: 1 + 2 + 3 + 4 + 5 + … Progressions: Definition: A progression is a special type of sequence where the terms follow a specific mathematical pattern that can be expressed using a formula for the general term (or nth term). Essentially, there’s a predictable rule to generate the terms. Examples of Progressions: Arithmetic Progression (AP): The difference between consecutive terms is constant (what we’ll focus on). Geometric Progression (GP): The ratio between consecutive terms is constant. Harmonic Progression (HP): The reciprocals of the terms are in Arithmetic Progression. Arithmetic Progression (AP) Definition of Arithmetic Progression (AP): An Arithmetic Progression (AP) is a progression (and therefore a sequence) in which the difference between any two consecutive terms is constant. This constant difference is the defining characteristic of an AP. Common Difference (d): This constant difference is called the common difference, denoted by ‘d’. Understanding Common Difference: If a1, a2, a3, a4, … is an AP, then: a2 – a1 = d a3 – a2 = d a4 – a3 = d and so on… In general, a(n+1) – an = d for any term an and its succeeding term a(n+1). Example of Arithmetic Progression: Consider the sequence: 2, 5, 8, 11, 14, … Let’s check the difference between consecutive terms: 5 – 2 = 3 8 – 5 = 3 11 – 8 = 3 14 – 11 = 3 The difference is consistently 3. Therefore, 2, 5, 8, 11, 14, … is an Arithmetic Progression with a common difference (d) = 3. More Examples of Arithmetic Progressions: Example 1: 10, 8, 6, 4, 2, 0, -2, … Common difference: 8 – 10 = -2, 6 – 8 = -2, and so on. Here, d = -2. Example 2: -3, -3, -3, -3, -3, … Common difference: -3 – (-3) = 0, -3 – (-3) = 0, and so on. Here, d = 0. Example 3: 0.5, 1.0, 1.5, 2.0, 2.5, … Common difference: 1.0 – 0.5 = 0.5, 1.5 – 1.0 = 0.5, and so on. Here, d = 0.5.   Impact of Common Difference (d) on AP: Positive Common Difference (d > 0): The AP is increasing. Each term is greater than the previous term. Example: 2, 5, 8, 11, … (d = 3) Zero Common Difference (d = 0): The AP is constant. All terms are the same. Example: -3, -3, -3, -3, … (d = 0) Negative Common Difference (d < 0): The AP is decreasing. Each term is smaller than the previous term. Example: 10, 8, 6, 4, … (d = -2) Finite and Infinite Arithmetic Progressions: Bounded and Unbounded Finite AP: Definition: A finite Arithmetic Progression is an AP that has a finite (countable) number of terms. It has a last term. Example: The AP: 2, 5, 8, …, 32, 35, 38 Here, the first term is 2, the common difference is 3, and the last term is 38. We can count the number of terms (though it might be tedious). Infinite AP: Definition: An infinite Arithmetic Progression is an AP that has an infinite (uncountable) number of terms. It continues indefinitely and does not have a last term. Example: The AP: 2, 5, 8, 11, … Here, the first term is 2, the common difference is 3, and the “…” indicates that it goes on forever. Key Difference: Finite AP: Has a last term, number of terms is countable. Infinite AP: No last term, number of terms is uncountable (infinite). General Term (Nth Term) of an AP: The Formula to Find Any Term Sometimes, we need to find a specific term in an AP, like the 10th term, 50th term, or even the 1000th term. Listing out all terms to reach that position would be very time-consuming. The general term formula provides a direct way to calculate any term in an AP without listing all preceding terms. Formula for the Nth Term (Tn or an): Let: a be the first term of the AP. d be the common difference. n be the

Quadratic Equations Quadratic Equations : Formulas, Theorems & Examples (Class 10 Maths) 1.1 What is a Polynomial? Before diving into quadratic polynomials, let’s briefly remember what a polynomial is: A Polynomial is an algebraic expression made up of variables and coefficients, combined using addition, subtraction, and multiplication, where the exponents of the variables are always non-negative integers (whole numbers). Examples of Polynomials: 3x² + 2x – 5, y³ – 7y + 1, 8 (constant polynomial) Non-Examples (Not Polynomials): x⁻¹ + 2, √x + 3, x/y + 1 (because of negative exponents, fractional exponents, or variables in the denominator) 1.2 Definition of a Quadratic Polynomial: A Quadratic Polynomial is a specific type of polynomial. It is a polynomial of degree 2. This means the highest power of the variable in the polynomial is 2. The standard form of a quadratic polynomial in variable ‘x’ is: P(x) = ax² + bx + c Where: ‘a’, ‘b’, and ‘c’ are real numbers. These are the coefficients. ‘a’ is the coefficient of x² ‘b’ is the coefficient of x ‘c’ is the constant term Crucially, ‘a’ must not be equal to 0 (a ≠ 0). If ‘a’ were 0, the x² term would disappear, and the polynomial would become linear (bx + c) or constant (c), not quadratic. In simpler words: A quadratic polynomial is like an expression with three main parts: something times x-squared, plus something times x, plus a constant number. The most important part is the x-squared term, and it must be there (meaning ‘a’ can’t be zero). 1.3 Examples of Quadratic Polynomials (with Coefficients Identified): Let’s look at some examples and identify the coefficients ‘a’, ‘b’, and ‘c’ in each case: Example 1: P(x) = 2x² + 5x – 3 Here, a = 2, b = 5, c = -3 Example 2: Q(x) = -x² + 7x + 10 Here, a = -1 (remember -x² is the same as -1x²), b = 7, c = 10 Example 3: R(x) = x² – 4x Here, a = 1 (x² is 1x²), b = -4, c = 0 (there’s no constant term, so c is 0) Example 4: S(x) = 3x² + 9 Here, a = 3, b = 0 (there’s no ‘x’ term, so b is 0), c = 9 Example 5: T(x) = -5x² Here, a = -5, b = 0, c = 0 (only the x² term is present) 1.4 Non-Examples (Not Quadratic Polynomials): U(x) = 4x + 1 (This is a linear polynomial, degree 1, because the highest power of x is 1. It’s of the form bx + c, where a=0) V(x) = x³ – 2x² + x – 6 (This is a cubic polynomial, degree 3, because the highest power of x is 3. It has an x³ term, so it’s not quadratic) W(x) = 7 (This is a constant polynomial, degree 0, because there’s no variable ‘x’. It’s just a constant term ‘c’, where a=0 and b=0) 2. ZEROS OF A QUADRATIC POLYNOMIAL 2.1 What are Zeros? A zero of a polynomial P(x) is a value of ‘x’ for which the polynomial becomes equal to zero. In other words, if we substitute a certain value for ‘x’ into P(x) and the result is 0, then that value is a zero of the polynomial. For a quadratic polynomial P(x) = ax² + bx + c, a zero is a value of ‘x’ that satisfies the equation: ax² + bx + c = 0 2.2 Finding Zeros Graphically: We can find the zeros of a quadratic polynomial graphically by looking at its graph. As we learned earlier, the graph of y = P(x) = ax² + bx + c is a parabola. The zeros of the polynomial are the x-coordinates of the points where the parabola intersects the x-axis. These are the points where y = 0, which is exactly when P(x) = 0. Two Intersections: If the parabola intersects the x-axis at two distinct points, the quadratic polynomial has two distinct real zeros. One Intersection (Vertex on x-axis): If the parabola touches the x-axis at exactly one point (its vertex lies on the x-axis), the quadratic polynomial has one real zero (a repeated zero or equal zeros). No Intersections: If the parabola does not intersect the x-axis at all, the quadratic polynomial has no real zeros. In this case, the zeros are complex numbers (which you will study in higher classes). 2.3 Relationship between Zeros and Coefficients of a Quadratic Polynomial This is a very important relationship in quadratic polynomials! Let’s say α (alpha) and β (beta) are the two zeros of the quadratic polynomial P(x) = ax² + bx + c. Then, there are direct formulas connecting these zeros to the coefficients a, b, and c: Sum of Zeros (α + β): α + β = -b/a (Negative of the coefficient of x divided by the coefficient of x²) Product of Zeros (αβ): αβ = c/a (Constant term divided by the coefficient of x²) These relationships are extremely useful for: Verifying if you’ve found the zeros correctly. Finding the zeros if you know the sum and product. Constructing a quadratic polynomial if you know its zeros. Example 6: Verifying the Relationship Consider the quadratic polynomial P(x) = x² – 5x + 6. We can factorize it as P(x) = (x – 2)(x – 3). Therefore, the zeros are x = 2 and x = 3. Let α = 2 and β = 3. Let’s verify the relationships: Sum of Zeros (α + β) = 2 + 3 = 5 From the formula: -b/a = -(-5)/1 = 5 (Verified!) Product of Zeros (αβ) = 2 * 3 = 6 From the formula: c/a = 6/1 = 6 (Verified!) 3. FINDING ZEROS OF A QUADRATIC POLYNOMIAL There are different methods to find the zeros of a quadratic polynomial. We’ll discuss two common methods for Class 10: 3.1 Factorization Method (Splitting the Middle Term) This method works if the quadratic polynomial can be easily factorized into two

Linear Equations Notes Linear Equations Notes : Formulas, Theorems & Examples (Class 10 Maths) Ch 3 : PAIR OF LINEAR EQUATIONS IN TWO VARIABLES LESSON OBJECTIVES: Framing Equations: Learn to create pairs of linear equations in two variables from real-life situations and word problems. Graphical Solutions: Solve pairs of linear equations graphically and understand the geometric interpretations: Intersecting Lines (Unique Solution) Parallel Lines (No Solution) Coincident Lines (Infinitely Many Solutions) Consistent and Inconsistent Systems Algebraic Solutions: Master algebraic methods to solve pairs of linear equations: Substitution Method Elimination Method Cross-Multiplication Method CONTENTS OF LESSON NOTES: 1. LINEAR EQUATIONS: DEFINITION: A linear equation in two variables is an equation that can be written in the form ax + by + c = 0, where: x and y are the variables. a, b, and c are real numbers, and a and b are not both zero simultaneously. (Meaning at least one of ‘a’ or ‘b’ must be non-zero). a is the coefficient of x, b is the coefficient of y, and c is the constant term. WHY “LINEAR”? The term “linear” comes from the fact that when you graph a linear equation in two variables, it always represents a straight line. EXAMPLES: 2x + 3y – 6 = 0 (Here, a=2, b=3, c=-6) x – y = 5 (Can be rewritten as 1x – 1y – 5 = 0, so a=1, b=-1, c=-5) y = 2x + 1 (Can be rewritten as -2x + 1y – 1 = 0, so a=-2, b=1, c=-1) 3x = 7 (Can be rewritten as 3x + 0y – 7 = 0, so a=3, b=0, c=-7. Note ‘b’ can be zero, but ‘a’ is not zero in this case) 2y = -4 (Can be rewritten as 0x + 2y + 4 = 0, so a=0, b=2, c=4. Note ‘a’ can be zero, but ‘b’ is not zero in this case) 2. PAIR OF LINEAR EQUATIONS: DEFINITION: A pair of linear equations in two variables is a set of two linear equations that involve the same two variables (usually x and y). WHY “PAIR”? We need two equations to solve for two unknowns (variables). One linear equation alone has infinitely many solutions. GENERAL FORM: A pair of linear equations in two variables can be represented as: a₁x + b₁y + c₁ = 0 (Equation 1) a₂x + b₂y + c₂ = 0 (Equation 2) Where a₁, b₁, c₁, a₂, b₂, c₂ are real numbers, and a₁ and b₁ are not both zero, and a₂ and b₂ are not both zero. EXAMPLES: Equation 1: x + y = 10 Equation 2: x – y = 4 Equation 1: 2x – 3y = 7 Equation 2: 4x + y = -2 3. GRAPHICAL METHOD OF SOLUTION OF A PAIR OF LINEAR EQUATIONS: GEOMETRIC INTERPRETATION: Each linear equation in a pair represents a straight line on a graph. When we solve a pair of linear equations graphically, we are essentially finding the point(s) of intersection of these two lines. POSSIBLE CASES: When you graph two lines in a plane, there are three possibilities: (a) INTERSECTING LINES: Graphical Representation: The two lines intersect each other at exactly one point. Number of Solutions: There is exactly one solution (unique solution). The coordinates of the point of intersection (x, y) give the solution to the pair of equations. Consistent Pair: Such a pair of equations is called a consistent pair of equations because it has at least one solution. (b) COINCIDENT LINES: Graphical Representation: The two lines coincide or overlap each other completely. This means they are essentially the same line. Number of Solutions: There are infinitely many solutions. Every point on the line is a solution to both equations. Dependent and Consistent Pair: Such a pair of equations is called a dependent and consistent pair of equations. “Consistent” because there are solutions, and “dependent” because one equation can be derived from the other (they are essentially the same). (c) PARALLEL LINES: Graphical Representation: The two lines are parallel to each other. They never intersect. Number of Solutions: There is no solution. There is no point that lies on both lines simultaneously. Inconsistent Pair: Such a pair of equations is called an inconsistent pair of equations because it has no solution. 4. ALGEBRAIC METHODS OF SOLVING A PAIR OF LINEAR EQUATIONS: (a) SUBSTITUTION METHOD: PROCESS: In this method, we solve for one variable in terms of the other from one equation and then substitute this expression into the second equation to solve for the remaining variable. STEPS: Step 1: Express one variable in terms of the other: Choose either equation and solve for one variable (say, y) in terms of the other variable (x). Pick the equation that looks simpler to manipulate. Step 2: Substitute: Substitute this expression for y (in terms of x) into the other This will result in an equation in just one variable (x). Step 3: Solve for x: Solve the new linear equation in x to find the value of x. Step 4: Substitute back to find y: Substitute the value of x you just found back into either of the original equations (or the expression for y in terms of x from Step 1) to find the value of y. Step 5: Check your solution (optional but recommended): Substitute the values of x and y you found into both original equations to verify that they satisfy both equations. EXAMPLE: Solve the pair of equations: x + y = 14 (Equation 1) x – y = 4 (Equation 2) Step 1: From Equation 1, express y in terms of x: y = 14 – x Step 2: Substitute this into Equation 2: x – (14 – x) = 4 Step 3: Solve for x: x – 14 + x = 4 => 2x – 14 = 4 => 2x = 18 => x = 9 Step 4: Substitute x = 9 back into y = 14 – x: y = 14 – 9 = 5 Solution: x =

Polynomials Polynomials Notes: Formulas, Theorems & Examples (Class 10 Maths) Ch 2 : POLYNOMIALS These notes are designed to help you understand Polynomials easily, even if you’re learning on your own. They cover everything in the CBSE/NCERT Class 10 syllabus and will give you a quick but detailed overview of the chapter for effective revision. CONTENT OF LESSON: Variables Constants Algebraic Expressions Polynomials Degree of a Polynomial Types of Polynomials Geometrical Representation of Polynomials Zeroes of a Polynomial Relationship between Zeroes and Coefficients of Quadratic and Cubic Polynomials Factorization of Quadratic Polynomials Division Algorithm for Polynomials   CHAPTER CONCEPTS: 1. VARIABLES: DEFINITION: Variables are symbols (usually letters like x, y, z) that represent values that can change or vary. In polynomial expressions, they are the unknowns we work with. EXAMPLES: In the expression 2x + 3, ‘x’ is the variable. In xy – 5, ‘x’ and ‘y’ are variables. 2. CONSTANTS: DEFINITION: Constants are fixed numerical values that do not change. They are the numbers in polynomial expressions. EXAMPLES: In the expression 2x + 3, ‘3’ and ‘2’ are constants. In xy – 5, ‘-5’ is a constant. Other examples: 5, -7, 1/2, √2, π are all constants. 3. ALGEBRAIC EXPRESSIONS: DEFINITION: An algebraic expression is an expression made up of variables, constants, and mathematical operations (+, -, ×, ÷, exponents, roots, etc.). They are the foundation for polynomials. KEY POINT: Algebraic expressions become equations when they are set equal to something (e.g., 3x + 2 = 5 is an equation, while 3x + 2 is an algebraic expression). EXAMPLES: 3x²y + 4xy + 5x + 6 √x + 7 (x + 1) / (x – 2) 5x⁻¹ + 2 TERMS: An algebraic expression is a sum of terms, which are considered to be building blocks for expressions. Terms are separated by ‘+’ or ‘-‘ signs. Constant terms: (e.g., 5, -3) Variable terms: (e.g., 2x, -4xy, 3x²) Like terms: (terms with the same variables raised to the same powers, e.g., 2x and 5x, 3x²y and -x²y) 4. POLYNOMIALS: DEFINITION: A polynomial is a special type of algebraic expression in which the exponent on every variable is a whole number (0, 1, 2, 3,…). WHOLE NUMBERS: Remember, whole numbers are non-negative integers: 0, 1, 2, 3, … KEY DIFFERENCE from Algebraic Expressions: In general algebraic expressions, exponents can be any rational number. In polynomials, exponents are restricted to whole numbers. EXAMPLES OF POLYNOMIALS: 5x³ + 3x + 1 (Exponents are 3, 1, and 0 for the constant term) x² – 7x + 10 (Exponents are 2, 1, and 0) 7 (Constant polynomial, can be written as 7x⁰, exponent is 0) -2y⁴ + y² – 5y + 8 EXAMPLES OF NON-POLYNOMIALS (Algebraic Expressions but NOT Polynomials): 2x + 3√x because √x = x^(1/2), and 1/2 is NOT a whole number. x⁻¹ + 2x because x⁻¹ = 1/x, and -1 is NOT a whole number. 3 + 2/x because 2/x = 2x⁻¹, and -1 is NOT a whole number. x² + x^(1.5) – 1 because 1.5 is NOT a whole number. 5. DEGREE OF A POLYNOMIAL: DEFINITION: For a polynomial in one variable, the degree of the polynomial is the highest exponent on the variable in the polynomial. EXAMPLES: x² + 2x + 3: Degree is 2 (highest power of x is 2). 5x³ – 4x² + x – 7: Degree is 3 (highest power of x is 3). 2x – 9: Degree is 1 (highest power of x is 1). 6: Degree is 0 (constant polynomial, can be written as 6x⁰). 0: Degree is not defined (zero polynomial). 6. TYPES OF POLYNOMIALS: Polynomials can be classified based on two criteria: (a) Based on the NUMBER OF TERMS: MONOMIAL: A polynomial with just one term. Examples: 2x, 6x², 9xy, -7x⁵, 10, ½y³, -√3z⁷ In essence: Single term polynomials. BINOMIAL: A polynomial with two terms. Examples: 4x² + x, 5x + 4, x¹⁰ – 1, 3y + 7y², √2z – 5, a + b In essence: Two terms joined by ‘+’ or ‘-‘. TRINOMIAL: A polynomial with three terms. Examples: x² + 3x + 4, x² + 2xy + y², p³ – 4p + 9, x² + x + 1, a² + b² + c² In essence: Three terms joined by ‘+’ or ‘-‘. (b) Based on the DEGREE of the polynomial: LINEAR POLYNOMIAL: A polynomial whose degree is one. General form: ax + b (where a ≠ 0) Examples: 2x + 1, -x + 5, 7y, ½z – 3, x – √7 In essence: Highest power of variable is 1. QUADRATIC POLYNOMIAL: A polynomial of degree two. General form: ax² + bx + c (where a ≠ 0) Examples: 3x² + 8x + 5, -2x² + 3x – 1, y² + 9, √5z², x² – 4x + 4 In essence: Highest power of variable is 2. CUBIC POLYNOMIAL: A polynomial of degree three. General form: ax³ + bx² + cx + d (where a ≠ 0) Examples: 2x³ + 5x² + 9x + 15, x³ – 1, 4y³ + 2y² – y + 6, -z³ + 8z, x³ + 3x² + 3x + 1 In essence: Highest power of variable is 3. CONSTANT POLYNOMIAL: A polynomial of degree zero (a non-zero constant). General form: c (where c ≠ 0) Examples: 5, -3, √2, π, -1/7 In essence: Just a number (not zero). ZERO POLYNOMIAL: The polynomial 0. Its degree is not defined. 7. GRAPHICAL REPRESENTATIONS OF POLYNOMIALS: REPRESENTING EQUATIONS ON A GRAPH: Any equation can be represented as a graph on the Cartesian plane (x-y plane). Each point on the graph represents the (x, y) coordinates that satisfy the equation. LINEAR POLYNOMIAL (y = ax + b): The graph of a linear polynomial is always a straight line. It cuts the x-axis at exactly one point (unless it’s a horizontal line not on the x-axis, or a vertical line which is not a function of x in this form). WHY one x-intercept? A linear polynomial has degree 1, so at most one

Real Numbers Real Numbers Notes : Formulas, Theorems & Examples (Class 10 Maths) Ch 1 : REAL NUMBERS Welcome to the world of Real Numbers! In this chapter, we’ll explore the fascinating system of numbers we use every day. From counting to complex calculations, Real Numbers are the foundation of mathematics. We’ll understand different types of real numbers and some important ideas about them. SUMMARY OF TOPICS WE’LL COVER: What are Real Numbers? Rational Numbers and their Properties Irrational Numbers and their Properties The Fundamental Theorem of Arithmetic (and its applications) Decimal Expansions of Rational Numbers (Terminating and Non-Terminating Repeating) HCF and LCM of Numbers (including Rational Numbers) 1. RATIONAL NUMBERS: DEFINITION: A rational number is any number that can be expressed in the form p/q, where: ‘p’ and ‘q’ are INTEGERS (whole numbers and their negatives, including zero). ‘q’ is NOT EQUAL TO ZERO (q ≠ 0). Think about it: Why can’t ‘q’ be zero? Division by zero is undefined in mathematics! EXAMPLES: Fractions: 1/2, -2/3, 7/9, 15/4 Integers: 0, 1, -5, 10 (because any integer ‘n’ can be written as n/1) Terminating Decimals: 0.5 (which is 1/2), 1.75 (which is 7/4), -0.2 (which is -1/5) Repeating Decimals: 0.333… (which is 1/3), 1.666… (which is 5/3), 0.142857142857… (which is 1/7) Key Point: Rational numbers, when expressed as decimals, are either TERMINATING (end after a certain number of digits) or NON-TERMINATING REPEATING (digits repeat in a pattern). 2. IRRATIONAL NUMBERS: DEFINITION: Irrational numbers are numbers that CANNOT be written in the form p/q, where p and q are integers and q ≠ 0. EXAMPLES: Square roots of non-perfect squares: √2, √3, √5, √7, √15 (and many more!) Pi (π): Approximately 3.14159…, but its decimal representation goes on forever without repeating. Non-terminating, Non-repeating decimals: 0.10110111011110… (This pattern ensures it never repeats). Why are numbers like √2 irrational? It’s a bit more complex to prove, but essentially, you can’t find two integers that, when you divide them, will exactly equal √2. DECIMAL EXPANSION OF IRRATIONAL NUMBERS: Irrational numbers have decimal expansions that are NON-TERMINATING AND NON-REPEATING. They go on forever without any repeating pattern. 3. REAL NUMBERS: DEFINITION: The collection of ALL RATIONAL NUMBERS AND ALL IRRATIONAL NUMBERS together makes up the set of Real Numbers. Think of it like this: Real Numbers are the “big family” that includes both Rational and Irrational numbers. REAL NUMBER LINE: All real numbers can be represented on a number line. 4. THE FUNDAMENTAL THEOREM OF ARITHMETIC: STATEMENT: Every composite number (a whole number greater than 1 that is not prime) can be expressed (factorized) as a product of prime numbers. This factorization is UNIQUE, except for the order in which the prime factors occur. Why is this “Fundamental”? It’s a cornerstone of number theory! It tells us that prime numbers are the building blocks of all composite numbers. PRIME NUMBERS: Numbers greater than 1 that are only divisible by 1 and themselves (e.g., 2, 3, 5, 7, 11, 13…). COMPOSITE NUMBERS: Numbers greater than 1 that are not prime (e.g., 4, 6, 8, 9, 10, 12…). EXAMPLES: 12 = 2 × 2 × 3 = 22 × 3 (Prime factors are 2 and 3) 30 = 2 × 3 × 5 (Prime factors are 2, 3, and 5) 100 = 2 × 2 × 5 × 5 = 22 × 52 (Prime factors are 2 and 5) APPLICATIONS: The Fundamental Theorem of Arithmetic is used to find: HCF (Highest Common Factor) LCM (Lowest Common Multiple) 5. REVISITING IRRATIONAL NUMBERS (THEOREMS): THEOREM 1.3: Let ‘p’ be a prime number. If ‘p’ divides a2, then ‘p’ also divides ‘a’ (where ‘a’ is a positive integer). Example: If 3 divides 62 (which is 36), then 3 also divides 6. THEOREM 1.4: √2 is irrational. (Similarly, √3, √5, √7, etc., are also irrational). This is a very important result! It confirms that numbers like √2 are indeed irrational and cannot be expressed as simple fractions. 6. REVISITING RATIONAL NUMBERS AND THEIR DECIMAL EXPANSIONS (THEOREMS): THEOREM 1.5: Let ‘x’ be a rational number whose decimal expansion terminates. Then ‘x’ can be expressed in the form p/q, where p and q are co-prime (have no common factors other than 1), and the prime factorization of ‘q’ is of the form 2n 5m (where n and m are non-negative integers). In simpler terms: If a rational number has a terminating decimal, its denominator (in simplest form) will only have prime factors of 2 and/or 5. Example: 0.25 = 25/100 = 1/4. Denominator 4 = 22 (form 2n 5m where n=2, m=0). THEOREM 1.6: Let x = p/q be a rational number such that the prime factorization of ‘q’ is of the form 2n 5m (where n, m are non-negative integers). Then ‘x’ has a decimal expansion which terminates. Converse of Theorem 1.5: If the denominator (in simplest form) of a rational number only has prime factors of 2 and/or 5, then its decimal expansion will terminate. Example: 7/8. Denominator 8 = 23. Decimal expansion is 0.875 (terminates). THEOREM 1.7: Let x = p/q be a rational number where p and q are co-prime. If the prime factorization of ‘q’ is NOT of the form 2n 5m, then ‘x’ has a decimal expansion which is non-terminating repeating (recurring). If the denominator has prime factors other than 2 and 5, the decimal will repeat. Example: 1/3. Denominator 3 (not of the form 2n 5m). Decimal expansion is 0.333… (repeating). 7. HCF AND LCM OF NUMBERS: FOR ANY TWO POSITIVE INTEGERS ‘A’ AND ‘B’: HCF (a, b): Product of the smallest power of each common prime factor in the numbers. LCM (a, b): Product of the greatest power of each common and uncommon prime factor in the numbers. Relationship: HCF (a, b) × LCM (a, b) = Product of the numbers (a × b) Example: Let a = 12 (22 × 3) and b = 18 (2 × 32) HCF (12, 18) = 21 × 31 = 6 (Common prime factors with smallest powers) LCM (12, 18) = 22 × 32 = 36 (All prime factors with greatest powers) HCF × LCM = 6 × 36 = 216. Product of numbers = 12 × 18 = 216. Relationship verified! LCM AND HCF OF RATIONAL NUMBERS (FRACTIONS): LCM of Rational numbers = (LCM of numerators) / (HCF of denominators) HCF of Rational