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Grade 10 - Math Notes, Study notes of Mathematics

Mathematics

Study notes for grade 10, contains lessons from first to fourth quarter.

Typology: Study notes

2023/2024

Uploaded on 06/10/2024

kairah-nicole-santos 🇵🇭

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MATH 10

1st Quarter

Table of Contents:

1. Sequences

• Introduction to Sequences

• Terms and Notation

• Recursive and Explicit Definitions

• Finding the nth Term

• Arithmetic Sequences

• Geometric Sequences

• Other Types of Sequences

2. Polynomials and Polynomial Equations

• Introduction to Polynomials

• Polynomial Degree and Terms

• Adding and Subtracting

Polynomials

• Multiplying Polynomials

• Division of Polynomials

• The Remainder Theorem

• The Factor Theorem

• Factoring Polynomials

• Solving Polynomial Equations

➢ Introduction to Sequences:

A sequence is an ordered list of numbers or

terms that follow a particular pattern. In

mathematics, sequences are essential in

understanding patterns, making predictions,

and solving various problems.

➢ Definition of Sequences:

A sequence is a function that maps natural

numbers (including zero) to a set of numbers

or terms. It can be represented using the

general notation: {a₁, a₂, a₃, ...}, where "a₁"

represents the first term, "a₂" the second

term, and so on.

1. Types of Sequences: Sequences can be

classified into different types based on

the patterns they exhibit:

a. Arithmetic Sequences:

• Definition:

Arithmetic sequences have a common

difference between consecutive terms.

• Example:

{2, 5, 8, 11, ...} with a common difference of 3.

Lesson 1: Sequences

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MATH 10

1 st^ Quarter Table of Contents:

1. Sequences - Introduction to Sequences - Terms and Notation - Recursive and Explicit Definitions - Finding the nth Term - Arithmetic Sequences - Geometric Sequences - _Other Types of Sequences

Polynomials and Polynomial Equations_
- Introduction to Polynomials
- Polynomial Degree and Terms
- Adding and Subtracting Polynomials
- Multiplying Polynomials
- Division of Polynomials
- The Remainder Theorem
- The Factor Theorem
- Factoring Polynomials
- Solving Polynomial Equations ➢ Introduction to Sequences : A sequence is an ordered list of numbers or terms that follow a particular pattern. In mathematics, sequences are essential in understanding patterns, making predictions, and solving various problems. ➢ Definition of Sequences: A sequence is a function that maps natural numbers (including zero) to a set of numbers or terms. It can be represented using the general notation: {a₁, a₂, a₃, ...}, where "a₁" represents the first term, "a₂" the second term, and so on.
Types of Sequences: Sequences can be classified into different types based on the patterns they exhibit: a. Arithmetic Sequences:

Definition: Arithmetic sequences have a common difference between consecutive terms.
Example: {2, 5, 8, 11, ...} with a common difference of 3. Lesson 1: Sequences

b. Geometric Sequences:

Definition: Geometric sequences have a common ratio between consecutive terms.
Example: {3, 6, 12, 24, ...} with a common ratio of 2. c. Other Types of Sequences:
Definition: Other sequences may exhibit different patterns or combinations of arithmetic and geometric sequences.
Example: {1, 4, 9, 16, 25, ...} represents the sequence of perfect squares.

Characteristics of Sequences: Sequences possess certain characteristics that help us analyze and understand them: a. Order: The terms in a sequence have a specific order. b. Pattern: Sequences follow a particular pattern or rule. c. Term Position: Each term has a position or index within the sequence. d. Common Difference or Ratio: Certain types of sequences have a common difference or ratio between terms.
History of Sequences: The study of sequences dates back to ancient civilizations, where patterns in numbers were observed and recorded. Indian mathematicians like Pingala and Bhaskara made significant contributions to the study of sequences, which later influenced European mathematicians like Fibonacci.
Examples of Sequences: a. Arithmetic Sequence Example: Consider the arithmetic sequence {2, 5, 8, 11, ...}. The common difference is 3. Find the 10th term. Solution: To find the nth term of an arithmetic sequence, we use the formula an = a₁ + (n – 1 ). Plugging in the values, we have a 10 = 2 + (10 - 1)3 = 29. b. Geometric Sequence Example: Consider the geometric sequence {3, 6, 12, 24, ...}. The common ratio is 2. Find the 6th term. Solution: To find the nth term of a geometric sequence, we use the formula an = a₁ * r^(n - 1). Plugging in the values, we have a6 = 3 * 2^(6 - 1) = 3 * 32 = 96.
Recursive and Explicit Definitions:

b. Linear Polynomials: Degree 1 polynomials with one term involving the variable raised to the power of 1. c. Quadratic Polynomials: Degree 2 polynomials with one term involving the variable raised to the power of 2. d. Cubic Polynomials: Degree 3 polynomials with one term involving the variable raised to the power of 3. e. Higher Degree Polynomials: Polynomials with degrees greater than 3.

Characteristics of Polynomials: Polynomials possess several characteristics that help us analyze and understand them: a. Degree: The degree of a polynomial is the highest power of the variable in the polynomial. b. Coefficients: The coefficients are the numbers multiplying the variable terms. c. Constant Term: The constant term is the term without any variable. d. Leading Coefficient: The coefficient of the term with the highest power of the variable. e. Zero Polynomial: A polynomial with all zero coefficients.
History of Polynomials: Polynomials have a rich history dating back to ancient civilizations like Babylonians and Greeks. Notable mathematicians like Al- Khwarizmi and Descartes made significant contributions to the study of polynomials, laying the foundation for modern algebra.
Examples of Polynomials and Polynomial Equations: a. Polynomial Example: Consider the polynomial P(x) = 3x³ - 2x² + 5x - 1. This polynomial is a degree 3 polynomial with coefficients 3, - 2, 5, and - 1. b. Polynomial Equation Example: Solve the equation P(x) = 0, where P(x) = x² + 3x - 4. Solution: To solve the equation, we set P(x) equal to zero and find the values of x that satisfy the equation. By factoring or using the quadratic formula, we can find the solutions to be x = 1 and x = - 4.

2 nd^ Quarter Table of Contents:

1. Polynomial Functions - Introduction to Polynomial Functions - Degree and Leading Coefficient - Graphing Polynomial Functions - Roots and Zeroes of Polynomial Functions - Polynomial Division - _Synthetic Division

Circles_
- Introduction to Circles
- Parts of a Circle: Center, Radius, Diameter, and Circumference
- Area of a Circle
- Circle Terminology
- _Circle Notation and Equations
Chords, Arcs, and Central Angles_
- Chords in a Circle
- Arcs in a Circle
- Central Angles and Their Measures
- Inscribed Angles and Their Measures
- Relationships Between Chords, _Arcs, and Central Angles
Arcs and Inscribed Angles_
- Inscribed Angles and Intercepted Arcs
- Angle Measures of Inscribed Angles - Relationships Between Inscribed Angles and Intercepted Arcs - _Circumference and Arc Length
Tangents and Secants of a Circle_

Tangent to a Circle
Secant of a Circle
Properties of Tangents and Secants
Tangents and Secants Intersecting Outside the Circle
_Tangents and Secants Intersecting Inside the Circle

Tangent and Secant Segments_

Tangent-Secant Theorem
Secant-Secant Theorem
Segment Lengths Involving Tangents, Secants, and Chords
_Tangent-Secant and Secant- Secant Power Theorems

Plane Coordinate Geometry_

Introduction to Plane Coordinate Geometry
Distance Formula
Midpoint Formula
_Coordinate Proof in Geometry

The Equation of a Circle_

Standard Form of the Equation of a Circle
Center-Radius Form of the Equation of a Circle
Converting between Different Forms of the Equation of a Circle
Applying the Equation of a Circle

b. Linear Function Example: The linear function f(x) = 3x + 2 represents a degree 1 polynomial function with a leading coefficient of 3 and a constant term of 2. c. Quadratic Function Example: The quadratic function f(x) = x² - 4x + 3 is a degree 2 polynomial function with a leading coefficient of 1, a coefficient of - 4, and a constant term of 3.

Roots and Zeroes of Polynomial Functions: - Definition: The roots or zeroes of a polynomial function are the values of x for which the function evaluates to zero. In other words, they are the x-values that make the polynomial equal to zero.

Example: For the quadratic function f(x) = x² - 4x + 3, we can find the roots by setting f(x) = 0 and solving the quadratic equation x² - 4x + 3 = 0. By factoring or using the quadratic formula, we find that the roots of this polynomial function are x = 1 and x = 3.
Characteristics of Roots : Multiplicity: The multiplicity of a root refers to the number of times it appears as a solution to the polynomial equation. It corresponds to the number of times the corresponding linear factor appears in the factored form of the polynomial.

Polynomial Division:

Polynomial Long Division: Polynomial long division is a method used to divide one polynomial by another polynomial. It involves the steps of dividing, multiplying, subtracting, and bringing down terms to obtain the quotient and remainder.
Example: Divide the polynomial P(x) = 3x³ - 2x² + 5x - 1 by the polynomial Q(x) = x - 2 using polynomial long division. Solution: The long division process allows us to divide P(x) by Q(x). We divide the leading term of P(x) by the leading term of Q(x), which gives us the first term of the quotient. Then, we multiply Q(x) by this term, subtract it from P(x), and continue the process until we obtain the quotient and remainder.

Synthetic Division:

Synthetic Division: Synthetic division is a shorthand method used to divide a polynomial by a linear factor of the form (x - c), where "c" is a constant. It simplifies the process of polynomial division by eliminating the need to write down all the terms.
Example:

Divide the polynomial P(x) = 2x³ + 5x² - 4x + 1 by the factor (x - 3) using synthetic division. Solution: Synthetic division allows us to divide P(x) by (x - 3) by only writing down the coefficients of P(x). We perform a simplified process that involves bringing down the first coefficient, multiplying, adding, and repeating until we obtain the quotient polynomial. ➢ Introduction to Circles: Circles are fundamental geometric shapes that have been studied for centuries. ➢ Definition of Circles: A circle is a closed curve consisting of all points in a plane that are equidistant from a fixed point called the center. The distance from the center to any point on the circle is called the radius, and twice the radius is called the diameter. The circumference of a circle is the distance around its boundary.

Parts of a Circle:
- Center: The fixed point in the middle of the circle.
- Radius: The distance from the center to any point on the circle.
- Diameter: The distance across the circle passing through the center.
- Circumference: The distance around the boundary of the circle.
- Area: The measure of the region enclosed by the circle.
Area of a Circle:
- Formula: The area (A) of a circle is given by the formula A = πr², where r is the radius. Lesson 2 : Circles

➢ Introduction to Chords, Arcs, and Central Angles: Chords, arcs, and central angles are key elements of circles.

Chords in a Circle:
- Definition: A chord is a line segment connecting any two points on the boundary of a circle.

Characteristics: The diameter is a special chord that passes through the center of the circle. Chords can be of different lengths and orientations within the circle.

Arcs in a Circle:

Definition : An arc is a portion of the circle's boundary between two points, often defined by the chords or radii that intersect the arc.
Types of Arcs:
Minor Arc: An arc that is less than a semicircle.
Major Arc: An arc that is greater than a semicircle.
Semicircle: An arc that is exactly half the circumference of the circle.

Central Angles and Their Measures:

Definition: A central angle is an angle formed by two radii of a circle, with the vertex at the center of the circle.
Characteristics: The measure of a central angle is equal to the measure of the intercepted arc. The central angle determines the degree of rotation between the radii.

Inscribed Angles and Their Measures:

Definition: An inscribed angle is an angle formed by two chords or a chord and a tangent, with the vertex on the circle.
Characteristics: An inscribed angle is half the measure of the intercepted arc. The measure of an inscribed angle can change depending on the position of the chord or tangent.

Relationships Between Chords, Arcs, and Central Angles:

Central Angle and Intercepted Arc: The measure of a central angle is equal to the measure of the intercepted arc.
Inscribed Angle and Intercepted Arc: The measure of an inscribed angle is half the measure of the intercepted arc.
Relationships with Chords: Chords that intersect in a circle create several interesting relationships between angles and arcs.

History of Chords, Arcs, and Central Angles: The study of chords, arcs, and central angles has a long history dating back to ancient civilizations. Mathematicians like Archimedes and Euclid made significant contributions to the understanding of these concepts. Lesson 3 : Chords, Arcs, and Central Angles

Examples of Chords, Arcs, and Central Angles: - Example 1: Consider a circle with radius 6 cm. Find the length of a chord that is 5 cm away from the center of the circle. Solution: Using the Pythagorean theorem, we can determine that the chord's length is 8 cm.

Example 2: In a circle with a central angle of 60 degrees, find the measure of the intercepted arc. Solution: Since the measure of a central angle is equal to the measure of the intercepted arc, the intercepted arc is also 60 degrees. ➢ Introduction to Arcs and Inscribed Angles: Arcs and inscribed angles are important concepts in circle geometry.

Inscribed Angles and Intercepted Arcs:

Definition: An inscribed angle is an angle formed by two chords or a chord and a tangent, with the vertex on the circle.
Characteristics: The measure of an inscribed angle is half the measure of its intercepted arc. Inscribed angles that intercept the same arc have equal measures.

Relationships between Inscribed Angles and Intersected Arcs:

Theorem: If two inscribed angles of a circle intercept the same arc, then the angles are congruent.
Proof: By considering the intercepted arcs and using the theorem that states angles in the same segment are equal, we can prove the congruence of inscribed angles.

Angle Measures of Inscribed Angles:

Example 1: In a circle with a central angle of 60 degrees, find the measure of the intercepted arc and the measure of the corresponding inscribed angle. Solution: Since the measure of a central angle is equal to the measure of the intercepted arc, Lesson 4 : Arcs and Inscribed Angles

➢ Introduction to Tangents and Secants of a Circle: Tangents and secants are important concepts in circle geometry.

Tangent to a Circle:
- Definition: A tangent is a line that intersects a circle at exactly one point, known as the point of tangency. It is perpendicular to the radius at the point of tangency.

Characteristics: The tangent line and the radius at the point of tangency form a right angle. Tangent lines are unique for every point on the circle.

Secant of a Circle:

Definition: A secant is a line that intersects a circle at two distinct points. It can either be a line passing through the circle or a line segment that extends beyond the circle.
Characteristics: Secants can intersect the circle at two points, creating various properties and relationships with chords, arcs, and angles.

Properties of Tangents and Secants:

Tangent-Secant Theorem: If a tangent and a secant are drawn from the same external point to a circle, the square of the length of the tangent is equal to the product of the lengths of the secant and its external segment.
Secant-Secant Theorem: If two secants are drawn from the same external point to a circle, the product of the lengths of one secant and its external segment is equal to the product of the lengths of the other secant and its external segment.

Tangents and Secants Intersecting Outside the Circle:

Characteristics: When a tangent and a secant intersect outside the circle, certain angle Lesson 5 : Tangents and Secants of a Circle

relationships and segment lengths are formed.

Examples:

Tangents and Secants Intersecting Inside the Circle:

Characteristics: When a tangent and a secant intersect inside the circle, unique relationships are formed between angles, segments, and intercepted arcs.
Examples:

History of Tangents and Secants: The study of tangents and secants traces back to ancient mathematicians like Euclid and Archimedes, who made significant contributions to the understanding of these concepts. ➢ Introduction to Tangent and Secant Segments: Tangent and secant segments are important elements in circle geometry.
Tangent-Secant Theorem:

Definition: The Tangent-Secant Theorem states that if a tangent and a secant are drawn from the same external point to a circle, the square of the length of the tangent is equal to the product of the lengths of the secant and its external segment.
Formula: If the length of the tangent segment is represented as "a," the length of the secant segment as "b," and the length of the external segment as "c," the theorem can be written as a² = b × c. Lesson 6 : Tangent and Secant Segments

➢ Introduction to Plane Coordinate Geometry: Plane coordinate geometry, also known as Cartesian coordinate geometry, is a branch of mathematics that studies the relationships between points, lines, and shapes on a coordinate plane.

Definition of the Coordinate Plane:
- The coordinate plane consists of two perpendicular number lines, the x-axis (horizontal) and the y- axis (vertical), intersecting at their common origin (0,0). It divides the plane into four quadrants: I, II, III, and IV.
Cartesian Coordinates:
- Definition : Cartesian coordinates are used to identify points in the coordinate plane. Each point is represented by an ordered pair (x, y), where "x" is the horizontal distance from the y-axis (the x- coordinate), and "y" is the vertical distance from the x-axis (the y-coordinate). 3. Graphing Points: - Plotting Points: To graph a point, locate its x-coordinate on the x-axis and its y-coordinate on the y-axis. The coordinates determine the point's position on the coordinate plane. - Example: Graph the point (3, - 2) on the coordinate plane. Solution: Start at the origin (0,0), move 3 units to the right along the x-axis, and then move 2 units downward along the y-axis. Plot the point at the intersection of these coordinates.
Distance Formula:

Definition: The distance formula is used to find the distance between two points in the coordinate plane.
Formula: The distance between two points (x₁, y₁) and (x₂, y₂) is given by the formula √((x₂ - x₁)² + (y₂
y₁)²). Lesson 7 : Plane Coordinate Geometry

Example: Find the distance between the points (2, 3) and (-1, 5). Solution: Using the distance formula, we have √((- 1 - 2)² + (5 - 3)²) = √((-3)² + 2²) = √(9 + 4) = √13 units (approximate).

Midpoint Formula:

Definition: The midpoint formula is used to find the coordinates of the midpoint between two points in the coordinate plane.
Formula: The midpoint between two points (x₁, y₁) and (x₂, y₂) is given by the formula ((x₁ + x₂)/2, (y₁

y₂)/2).

Example: Find the midpoint between the points (4, 6) and (-2, - 1). Solution: Using the midpoint formula, we have ((4 + (-2))/2, (6 + (-1))/2) = (1, 2.5).

Equation of a Line:

Definition: The equation of a line represents the relationship between the x and y coordinates of the points lying on that line.
Slope-Intercept Form: The equation of a line in slope- intercept form is y = mx + b, where "m" is the slope and "b" is the y-intercept.
Example: Write the equation of a line with a slope of 2 and a y-intercept of - 3. Solution: The equation can be written as y = 2x - 3.

History of Plane Coordinate Geometry: Plane coordinate geometry was developed independently by René Descartes and Pierre de Fermat in the 17th century. Their work laid the foundation for analytic geometry, which connects algebra and geometry.

radius form and identify its center and radius. Solution: Expanding the equation gives x²

4x + 4 + y² - 6y + 9 = 9. Rearranging terms, we have (x²
y²) + 4x - 6y + 4 = 0. Comparing this with the general form, we find that the equation is equivalent to (x + 2)² + (y - 3)² = 3², with a center at (-2, 3) and a radius of 3.

Applying the Equation of a Circle:
- Graphing Circles: The equation of a circle allows us to graph circles by plotting the center and measuring the radius.
- Identifying Circle Properties: The equation provides information about the center and radius of the circle, which can be used to determine its size, position, and other characteristics.
- Example: Given the equation (x - 1)² + (y + 2)² = 25, identify the center, radius, and graph the circle. Solution: Comparing the equation with the standard form, we find that the center is (1, - 2) and the radius is √25 = 5. By plotting the center and measuring a distance of 5 in all directions, we can graph the circle. 3 rd^ Quarter Table of Contents: 1. Permutation and Combinations

Permutations
_Combinations

Permutations_

Definition of Permutations
Permutation Formula
Permutation Examples and Applications
_Permutation with Repetition

Combinations_

Definition of Combinations
Combination Formula
Combination Examples and Applications
_Combination with Repetition

Probability of Compound Events_

Introduction to Probability
Compound Events
Probability of Independent Events
_Probability of Dependent Events

Independent and Dependent Events_

Independent Events: Definition and Examples
Dependent Events: Definition and Examples
Relationship between Independence and Dependence
Conditional Probability and its Applications

6. Conditional Probability - Definition of Conditional Probability - Conditional Probability Formula - Examples of Conditional Probability - Applications of Conditional Probability ➢ Introduction to Permutations and Combinations: Permutations and combinations are mathematical concepts used to count and analyze different arrangements and selections of objects.

Permutations:

Definition: Permutations are arrangements of objects in a specific order, without repetition, where the order matters.
Characteristics: The number of permutations depends on the number of objects and the positions in which they can be arranged.
Example: Consider the letters "A," "B," and "C." The permutations of these letters include "ABC," "ACB," "BAC," "BCA," "CAB," and "CBA."

Combinations:

Definition: Combinations are selections of objects without regard to the order, without repetition. Lesson 1 : Permutation and Combinations

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