Precalculus Tutorials for Senior High School and College Students

Below are the tutorials on Precalculus Tutorials for Senior High School and College Students. All links point videos on the Sipnayan Youtube channel.  The link to the complete Playlist can be found here.

Lesson 1.00: Introduction to Precalculus for Senior High School
Lesson 1.01: A Brief Review of the Cartesian Plane
Lesson 1.02: The Conic Sections – Circle, Ellipse, Parabola, Hyperbola
Lesson 1.03: Equation of a Circle with Center at the Origin – Practice Exercises
Lesson 1.04: Equation of a Circle with Center (h,k)
Lesson 1.05: Equation of a Circle with Center at the Origin and Radius
Lesson 1.06: Writing the Equation of a Circle with Center (h,k) and the Radius
Lesson 1.07: Finding the Center and Radius of a Circle Given its Equation
Lesson 1.08: Standard and General Forms of the Equation of a Circle
Lesson 1.09: Converting Equations of a Circle From Standard Form to General
Form

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Operations on Polynomial Expressions

Below are the tutorials on Operations on Polynomial Expressions. All links point videos on the Sipnayan Youtube channel.  The link to the complete Playlist can be found here.

PE01 Introduction to Polynomial Expressions
PE02 Like Terms and Unlike Terms
PE03 Addition of Polynomial Expressions Part 1
PE04 Addition of Polynomial Expressions Part 2
PE05 Addition of Polynomial Expressions Part 3
PE06 Subtraction of Polynomial Expressions Part 1
PE07 Subtraction of Polynomial Expressions Part 2
PE08 Multiplication of Polynomial Expressions Part 1
PE09 Multiplication of Polynomial Expressions Part 2
PE10 Multiplication of Polynomial Expressions Part 3
PE11 Multiplication of Polynomial Expressions Part 4
PE12 Division of Polynomial Expressions Part 1
PE13 Division of Polynomial Expressions Part 2
PE14 Division of Polynomial Expressions Part 3
PE15 Division of Polynomial Expressions Part 4
PE16 Division of Polynomial Expressions Part 5
PE17 Division of Polynomial Expressions Part 6

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Factoring Tutorial Series

Below are the tutorials on Factoring. All links point videos on the Sipnayan Youtube channel.  The link to the complete Playlist can be found here.

Lesson 1: Basic Concepts of Factoring
Lesson 2: Factoring Polynomials with Common Monomial Factor
Lesson 3: Factoring Polynomials by Grouping Part 1
Lesson 4: Factoring Polynomials by Grouping Part 2
Lesson 5: Factoring Perfect Square Trinomials
Lesson 6: Factoring General Trinomials Part 1
Lesson 7: Factoring General Trinomials Part 2
Lesson 8: Factoring General Trinomials Part 3
Lesson 9: Factoring Difference of Two Squares
Lesson 10: Factoring Sum and Difference of Two Cubes
Lesson 11: Factoring Polynomials with More than 2 Factors

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Clock Word Problem Solving Series

Below are the tutorials on Clock Word Problem Series. All links point videos on the Sipnayan Youtube channel.  The link to the complete Playlist can be found here.

Lesson 1: Introduction to the Clock Problems Tutorial Series
Lesson 2: Clock Problems: Formula for the Amount of Rotation of the Hour Hand
Lesson 3: DERIVATION of the Amount of Rotation of the Hour Hand Given the Time
Lesson 4: Derivation of the Formula for the Amount of Rotation of the Minute Hand
Lesson 5: Derivation of the Formula for Calculating the Angle Between the Hands of the Clock
Lesson 6: Applying the Clock Angle Problem Formulas

to be continued…

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Angle Sum Tutorial Series

Below are the tutorials on Angle Sum Theorem. All links point videos on the Sipnayan Youtube channel.  The link to the complete Playlist can be found here.

Lesson 1: A Brief Review of the Concept of Angles
Lesson 2: Proof that The Sum of the Interior Angles of Triangles is 180 degrees
Lesson 3: Proof that The Sum of the Interior Angles of a Quadrilateral is 360 degrees
Lesson 4: Proof that The Sum of the Interior Angles of a Pentagon is 540 degrees
Lesson 5: Sum of the Interior Angles of Any Polygon (Derivation)

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REX05 Reducing Rational Expressions to Lowest Terms Part 1

In the previous video, we have learned about domain of rational functions. In this video, we are going to learn about how to reduce rational expressions particularly fractions to lowest terms.  Watch the Tagalog math tutorial video above and then answer the exercises below. The complete video playlist of the Sipnayan tutorials on Rational Expressions can be found here.

Practice Exercise

Reduce the following to lowest terms.

1.) \dfrac{8}{12}

2.) \dfrac{15}{25}.

3.) \dfrac{9}{27}.

4.) \dfrac{6}{24}.

5.) $latex  \dfrac{10}{22}$.

Answers

1.) \dfrac{2}{3}

2.) \dfrac{3}{5}.

3.) \dfrac{1}{3}.

4.) \dfrac{1}{4}.

5.) $latex  \dfrac{5}{11}$.
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REX04 Domain of Rational Functions

In the previous video, we have learned to rule out values of a variable in a rational expression. In this video, we are going to learn to find the domain of a rational function. This is very similar to the concept of ruling out values of rational expressions.  Watch the Tagalog math tutorial video above and then answer the exercises below. The complete video playlist of the Sipnayan tutorials on Rational Expressions can be found here.

Practice Exercise

Find the domain of the following rational functions.

1.) f(x) = \dfrac{x - 9}{x - 3}

2.) f(x) = \dfrac{2x + 1}{x^2 + 3x + 2}.

3.) g(x) = \dfrac{x^2 - 1}{x}.

4.) h(x) = \dfrac{x^2 + 2x}{x^2 - 9}.

5.) j(x) = \dfrac{x + 6}{3 - x}.

Answers

1. \{x|x \neq 3\}

2. \{x|x \neq -2, x \neq -1\}

3. \{x|x \neq 0 \}

4. \{ x|x \neq 3, x \neq -3 \}

5. \{ x \neq 3 \}

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REX03 Ruling Out the Values of the Variable

In the previous video, we have learned how to evaluate rational expressions. In this video, we are going to learn how to rule out the values of the variable in rational expressions. This concept is important because it is connected to the domain of rational functions  Watch the Tagalog math tutorial video above and then answer the exercises below. The complete video playlist of the Sipnayan tutorials on Rational Expressions can be found here.

Practice Exercise

Rule out the values of the variable of the following rational expressions.

1.) \dfrac{x + 4}{x}

2.) \dfrac{2x - 1}{x + 2}.

3.) \dfrac{x^2 - 1}{2x + 5}.

4.) \dfrac{x^2 + 2x}{x^2 - 9}.

5.) \dfrac{x - 9}{x^2 - 5x + 4}.

Answers

1. x \neq 0

2. x \neq -2

3. x \neq -\frac{5}{2}

4. x \neq 3, x \neq -3

5. x \neq 1, x \neq 4

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REX02 Evaluating Rational Expressions

In the previous video, we have learned about rational expressions. In this video, we learn how to evaluate rational expressions .  Watch the Tagalog math tutorial video above and then answer the exercises below. The complete video playlist of the Sipnayan tutorials on Rational Expressions can be found here.

Practice Exercise

Evaluate the following rational expressions.

1.)  If x = -2, find the value of

\dfrac{3x - 1}{x - 4}

2.) If y = 3, find the value of

\dfrac{y^2 + 5y - 6}{5y + 1}.

3.) The function f(x) = \frac{2x + 5}{-x + 7} is a rational function. Find f(0).

4.) If p = -4, q = 3, and r = 1, what is the value of

\dfrac{p^2 - qr + q^2}{p - r}

5.) If x = \frac{1}{2}, what is the value of

\dfrac{4x - 1}{2x^2 + 1}

Answers

1.  7/6
2. – 3/4
3. 5/7
4. – 22/5
5. 7/4

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REX01 Introduction to Rational Expressions

In this video, we learn the definition of rational numbers, rational expressions, and rational functions .  Watch the Tagalog math tutorial video above and then answer the exercises below. The complete video playlist of the Sipnayan tutorials on Rational Expressions can be found here.

Practice Exercise

True or False

1.)  The fraction \frac{\pi}{3} is a rational number.

2.) The fraction \frac{2x - 1}{5x + 1} is a rational expression.

3.) The fraction \frac{\sqrt{x} + 3}{x^2 + 3} is a rational expression.

4.) The function f(x) = \frac{1}{x} is a rational function.

5.) The fraction \frac{3}{8} is a rational number.

Answers

1.False. The numerator and denominator of a rational number must be integers.
2. True
3. False. \sqrt{x} + 3 is not a polynomial expression.
4. True
5. True

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AE06 Evaluating Algebraic Expressions

In the previous video, we learned how to represent division with algebraic expressions. In this video, we will learn how to evaluate algebraic expressions.  Watch the Tagalog math tutorial video above and then answer the exercises below. The complete video playlist of the Sipnayan tutorials on Algebraic expressions can be found here.

Practice Exercise

Evaluate the following algebraic expressions.

1.)  3x - 4 where x = 3

2.) 5y^2 - 3 where y =2

3.) - 6m + 4 where m = \frac{1}{3}

4.) \frac{4a}{3} + 2 where a = -6

5.) 10t - 14 where t = 0.5

6.) 3r + 2s where r = -2 and s = 5

Answers

1. 5
2. 17
3. 2
4. -6
5. -9
6. 4

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AE05 Representing Division with Algebraic Expressions

In the previous video, we learned how to represent multiplication with algebraic expressions. In this video, we will learn how to represent division with algebraic expression.  Watch the Tagalog math tutorial video above and then answer the exercises below. The complete video playlist of the Sipnayan tutorials on Algebraic expressions can be found here.

Practice Exercise

Represent the following division with algebraic expression

1.)  12 ÷ a

2.) 3x ÷ 4

3.) 2y + 5 ÷ x

4.) (2y + 5) ÷ x

5.) 3 ÷ (2a – 3b)

Answers

1. \dfrac{12}{a}
2. \dfrac{3x}{4}
3. 2y + \dfrac{5}{x}
4. \dfrac{2y + 5}{x}
5. \dfrac{3}{2a - 3b}

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AE04 Representing Exponentiation with Algebraic Expressions

In the previous video, we learned how to represent multiplication with algebraic expressions. In this video, we will learn how to represent exponentiation with algebraic expression.  Watch the Tagalog math tutorial video above and then answer the exercises below. The complete video playlist of the Sipnayan tutorials on Algebraic expressions can be found here.

Practice Exercise

Represent the following algebraic expressions using exponentiation.

1.)  5 · a ·  a ·  a ·  a

2.) m · m · m · m ·  n · n

3.) -8 · x · x · x ·  y ·  y ·  z

4.) 2 ·  m ·  m ·  m

5.) 2 · p · p ·  p ·  3 ·  q ·  q

Answers

1. 5a^4
2. m^3n^2
3. -8x^3y^2z
4. 2m^3
5. 6p^3q^2

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AE03 Representing Multiplication of Algebraic Expression

In the previous video, we learned how to represent quantities using algebraic expressions. In this video, we will introduce another notation on multiplying algebraic expression omitting the × symbol.  Watch the Tagalog math tutorial video above and then answer the exercises below. The complete video playlist of the Sipnayan tutorials on Algebraic expressions can be found here.

Practice Exercise

Represent the following using algebraic expressions without the × symbol.

1.)  a × 3

2.) p × 2 × r

3.) -8 × (z + 6)

4.) y × 3/4

5.) 0.5 × n

Answers

1. 3a
2. 2pr
3. 3 × n
4. 8(z + 6)
5. 0.5n

Note that n here are positive integers.

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AE02 Representation Using Algebraic Expressions

In the previous video, we learned about algebraic expressions. In this video, we will learn how to represent quantities using algebraic expressions. We will use real life situations and translate them to algebraic expressions. Watch the Tagalog math tutorial video above and then answer the exercises below. The complete video playlist of the Sipnayan tutorials on Algebraic expressions can be found here.

Practice Exercise

Represent the following using algebraic expressions.

1.) a number x decreased by 3

2.) the cost of 4 books at p pesos each

3.) the total weight of 3 sacks of rice at n kilograms per sack

4.) an increase in height h in inches of 3 inches

5.) a loss of 5 fruits at k pesos each

Answers

1. x – 3
2. 4 × p
3. 3 × n
4. h + 3
5. -5 × k

Note that n here are positive integers.

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AE01 Introduction to Algebraic Expressions

This is the first part of our series on algebraic expressions. In this video, we will learn how to generalize patterns to understand the meaning of algebraic expressions. Watch the Taglish math tutorial video above and then answer the exercises below. The complete video playlist of the Sipnayan tutorials on algebraic expressions can be found here.

Practice Exercise

Write an algebraic expression for each of the following sequence of integers. Use n as a variable.

1.) 2, 4, 6, 8, 10, …

2.) 5, 10, 15, 20, 25, 30, …

3.) 4, 9, 14, 19, 24, 29

4.) 1, 4, 9, 16, 25, 36, 49, …

5.) 5, 8, 11, 14, 17, 20, 23

Answers

1. 2n
2. 5n
3. 5n - 1
4. n^2
5. 3n + 2

Note that n here are positive integers.

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