5 3 Partial Fractions
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This document discusses partial fraction decomposition, which is the process of breaking down rational expressions into sums of simpler fractional components. It explains that a rational expression can be broken into "partial fractions" with distinct linear factors in the denominator, as well as cases where factors are repeated or where the denominator contains quadratic terms that are prime or repeated. Examples are provided for each of these different cases of partial fraction decomposition.
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Quadratic equations
Quadratic Equations
In One Variable
1. Quadratic Equation
an equation of the form
ax2 + bx + c = 0
where a, b, and c are real numbers
2.Types of Quadratic Equations
Complete Quadratic
3x2 + 5x + 6 = 0
Incomplete/Pure Quadratic Equation
3x2 - 6 = 0
3.Solving an Incomplete Quadratic
4.Example 1. Solve: x2 – 4 = 0
Solution:
x2 – 4 = 0
x2 = 4
√x² = √4
x = ± 2
5.Example 2. Solve: 5x² - 11 = 49
Solution:
5x² - 11 = 49
5x² = 49 + 11
5x² = 60
x² = 12
x = ±√12
x = ±2√3
6.Solving Quadratic Equation
7.By Factoring
Place all terms in the left member of the equation, so that the right member is zero.
Factor the left member.
Set each factor that contains the unknown equal to zero.
Solve each of the simple equations thus formed.
Check the answers by substituting them in the original equation.
8.Example: x² = 6x - 8
Solution:
x² = 6x – 8
x² - 6x + 8 = 0
(x – 4)(x – 2) = 0
x – 4 = 0 | x – 2 = 0
x = 4 x = 2
9.By Completing the Square
Write the equation with the variable terms in the left member and the constant term in the right member.
If the coefficient of x² is not 1, divide every term by this coefficient so as to make the coefficient of x² equal to 1.
Take one-half the coefficient of x, square this quantity, and add the result to both members.
Find the square root of both members, placing a ± sign before the square root of the right member.
Solve the resulting equation for x.
10.Example: x² - 8x + 7 = 0
11.By Quadratic Formula
Example: 3x² - 2x - 7 = 0
12.Solve the following:
1. x² - 15x – 56 = 0
2. 7x² = 2x + 6
3. 9x² - 3x + 8 = 0
4. 8x² + 9x -144 = 0
5. 2x² - 3 + 12x
13.Activity:
Solve the following quadratic formula.
By Factoring By Quadratic Formula
1. x² - 5x + 6 = 0 1. x² - 7x + 6 = 0
2. 3 x² = x + 2 2. 10 x² - 13x – 3 = 0
3. 2 x² - 11x + 12 = 0 3. x (5x – 4) = 2
By Completing the Square
1. x² + 6x + 5 = 0
2. x² - 8x + 3 = 0
3. 2 x² + 3x – 5 = 0
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It then lists the four properties of equations: adding/subtracting the same quantity to both sides, multiplying/dividing both sides by the same quantity.
Next, it provides examples of how to solve different types of equations (those involving addition, subtraction, multiplication, or division of the variable) using both the traditional method of operations and the shortcut method of transposing terms.
Finally, it gives examples of solving equations with variables on both sides, word problems involving equations, and equations from applied contexts like age.
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This document discusses simplifying rational expressions by dividing out common factors, factoring numerators and denominators, and dividing common factors. It provides examples of simplifying various rational expressions step-by-step and explains how to identify excluded values that would make denominators equal to zero.
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Quadratic equations
Quadratic Equations
In One Variable
1. Quadratic Equation
an equation of the form
ax2 + bx + c = 0
where a, b, and c are real numbers
2.Types of Quadratic Equations
Complete Quadratic
3x2 + 5x + 6 = 0
Incomplete/Pure Quadratic Equation
3x2 - 6 = 0
3.Solving an Incomplete Quadratic
4.Example 1. Solve: x2 – 4 = 0
Solution:
x2 – 4 = 0
x2 = 4
√x² = √4
x = ± 2
5.Example 2. Solve: 5x² - 11 = 49
Solution:
5x² - 11 = 49
5x² = 49 + 11
5x² = 60
x² = 12
x = ±√12
x = ±2√3
6.Solving Quadratic Equation
7.By Factoring
Place all terms in the left member of the equation, so that the right member is zero.
Factor the left member.
Set each factor that contains the unknown equal to zero.
Solve each of the simple equations thus formed.
Check the answers by substituting them in the original equation.
8.Example: x² = 6x - 8
Solution:
x² = 6x – 8
x² - 6x + 8 = 0
(x – 4)(x – 2) = 0
x – 4 = 0 | x – 2 = 0
x = 4 x = 2
9.By Completing the Square
Write the equation with the variable terms in the left member and the constant term in the right member.
If the coefficient of x² is not 1, divide every term by this coefficient so as to make the coefficient of x² equal to 1.
Take one-half the coefficient of x, square this quantity, and add the result to both members.
Find the square root of both members, placing a ± sign before the square root of the right member.
Solve the resulting equation for x.
10.Example: x² - 8x + 7 = 0
11.By Quadratic Formula
Example: 3x² - 2x - 7 = 0
12.Solve the following:
1. x² - 15x – 56 = 0
2. 7x² = 2x + 6
3. 9x² - 3x + 8 = 0
4. 8x² + 9x -144 = 0
5. 2x² - 3 + 12x
13.Activity:
Solve the following quadratic formula.
By Factoring By Quadratic Formula
1. x² - 5x + 6 = 0 1. x² - 7x + 6 = 0
2. 3 x² = x + 2 2. 10 x² - 13x – 3 = 0
3. 2 x² - 11x + 12 = 0 3. x (5x – 4) = 2
By Completing the Square
1. x² + 6x + 5 = 0
2. x² - 8x + 3 = 0
3. 2 x² + 3x – 5 = 0
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It then lists the four properties of equations: adding/subtracting the same quantity to both sides, multiplying/dividing both sides by the same quantity.
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2) Examples are provided of finding the derivative of functions like f(x) = 6x^3, which is f'(x) = 18x^2.
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2) There are several types of first order linear differential equations, including separable, homogeneous, exact, and linear equations.
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3. Key concepts such as the absolute value, distance between complex numbers, and the interpretation of multiplication in polar form.
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- Properties of i including i2 = -1.
- Converting complex numbers between Cartesian, polar, trigonometric and exponential forms.
- Operations on complex numbers such as addition, subtraction, multiplication and division.
- Comparing real and imaginary parts of complex numbers when solving equations.
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Workshop ptk
Dokumen tersebut terdiri dari 3 bagian:
1) Skripsi tentang penerapan flipped classroom dengan media blog untuk meningkatkan hasil belajar siswa SMA tentang penanganan limbah
2) Skripsi tentang peningkatan motivasi belajar siswa SMP melalui pemberian reward dan punishment untuk materi aljabar dan fungsi
3) Surat keterangan IPGI untuk guru yang mengikuti workshop Penelitian Tindakan Kelas selama 40 jam
Polynomial function
This document discusses polynomial functions and the remainder theorem. It provides examples of using long division and the remainder theorem to find the remainder when dividing one polynomial by another. Specifically, it shows:
1) How to perform long division of polynomials by working from highest to lowest degree terms and subtracting appropriately. There may be a remainder left over.
2) The remainder theorem states that if a polynomial P(x) is divided by (x-c), the remainder is equal to P(c).
3) Examples of using long division and directly applying the remainder theorem to find the remainder of dividing one polynomial by another.
Patterns and sequences
The document discusses arithmetic and geometric sequences and their key characteristics. An arithmetic sequence has a common difference, while a geometric sequence has a common ratio. Several example sequences are provided and the reader is asked to identify whether they are arithmetic, geometric, or neither and determine their common differences or ratios.
Ge6161 lab manual
This document is a lab manual for the course GE6161 Computer Practices Laboratory at Arunai Engineering College in Tiruvannamalai, India. It was prepared by M. Ashokkumar, R. Karthick, and T. Selvaganapathy. The document lists 10 experiments covering topics like using MS Office/Open Office, creating charts and graphs, problem solving with flowcharts and programming, arrays, strings, functions, recursion, structures, and unions. Each experiment includes an aim, algorithm, program code where applicable, output, and a self-exercise section for students.
Arithmetic sequence proj.
The document describes a patient's recovery plan after knee surgery. The doctor recommends jogging for 12 minutes per day for the first week, then increasing the time by 6 minutes each week thereafter. To determine how many weeks it will take to reach jogging for 60 minutes per day, the patient uses the concept of an arithmetic sequence, where each term is increased by a fixed amount. By increasing the weekly jogging time by 6 minutes each week, the patient calculates it will take 8 weeks to reach 60 minutes of daily jogging.
Partial fractions
This document discusses two methods for obtaining partial fraction decompositions of rational functions. Method 1 writes the decomposition in a form that avoids complex numbers. Method 2 is preferable when there are distinct or repeated first-degree factors, as it eliminates the need to solve simultaneous equations. The document illustrates the two methods using examples and explains that experience will guide the appropriate choice of method for different situations.
Unit 7.4
Partial fraction decomposition involves writing rational functions as the sum of simpler fractional components. It is useful for integration in calculus. A rational function is broken into partial fractions by factoring the denominator and grouping terms with each linear or irreducible quadratic factor. Examples demonstrate decomposing fractions with linear factors and repeated or quadratic factors. Partial fraction decomposition allows rational functions to be integrated and graphs of rational functions to be sketched.
Matrices
This document is a mathematics form on matrices that covers 7 objectives: understanding matrix concepts; equal and not equal matrices; addition and subtraction of matrices; multiplication of matrices; identity matrices; inverse matrices; and solving simultaneous equations using matrices. Examples are provided for each objective to illustrate key concepts and procedures related to matrices.
Remainder and Factor Theorem
The document discusses the Remainder Theorem and Factor Theorem. The Remainder Theorem states that if a polynomial p(x) is divided by a factor x - a, the remainder will be zero if x - a is a factor of p(x). The Factor Theorem is the reverse - if dividing a polynomial by x = a gives a zero remainder, then x - a is a factor of the polynomial. Both theorems relate the remainder of polynomial division to the factors of the polynomial.
Linear Algebra
This document provides an overview of topics that will be covered in a linear algebra course, including matrices, vectors, and their applications. Some example applications discussed are cryptography, space exploration, game programming, and electrical networks. It also provides examples of how matrices and vectors are used in games, specifically in representing game properties like position, velocity, acceleration, distance, and line of sight for objects in 2D and 3D space.
Matrix transformation
Using matrices to transform geometric figures, including translations, dilations, reflections, and rotations. Translations use a matrix with the distances of movement in each row. Dilations multiply coordinates by a scalar factor. Reflections across an axis involve changing the sign of coordinates on one side of the axis. Rotation matrices involve trigonometric functions to rotate the figure a specified number of degrees clockwise or counterclockwise. Examples show setting up and performing each type of transformation on sample polygons.
Sequence and series
The document defines a sequence as a set of numbers written in order. It provides the example of a sequence where each term is obtained by adding 1 to the preceding term. This rule is expressed by the equation an = an-1 + 1, where an is the nth term and an-1 is the preceding term. It asks the reader to write the first 5 terms of such a sequence and derive the rule based on the pattern.
Factor Theorem and Remainder Theorem
Factor Theorem and Remainder Theorem. Mathematics10 Project under Mrs. Marissa De Ocampo. Prepared by Danielle Diva, Ronalie Mejos, Rafael Vallejos and Mark Lenon Dacir of 10- Einstein. CNSTHS.
Business maths
This document provides information about matrices and determinants. It begins with defining what a matrix is and discusses the order, types (square, row, column, zero, diagonal, scalar, unit), and multiplication of matrices by scalars. The key points covered are:
1) A matrix is a rectangular array of numbers or functions represented by symbols in rows and columns.
2) The order of a matrix refers to the number of rows and columns.
3) Types of matrices include square, row, column, zero, diagonal, scalar, and unit matrices.
4) A scalar matrix results from multiplying a diagonal matrix by a scalar value.
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5 3 Partial Fractions
- 1. Section 5.3 Partial Fractions
- 2. The Idea Behind Partial Fraction Decomposition
- 3. Each of the two fractions on the right is called a partial fraction. The sum of these fractions is called the partial fraction decomposition of the rational expression on the left-hand side.
- 5. The Partial Fraction Decomposition of a Rational Expression with Distinct Linear Factors in the Denominator
- 8. Example
- 9. The Partial Fraction Decomposition of a Rational Expression with Linear Factors in the Denominator, Some of Which are Repeated
- 12. Example
- 13. The Partial Fraction Decomposition of a Rational Expression with Prime, Nonrepeated Quadratic Factors in the Denominator
- 16. Example
- 17. The Partial Fraction Decomposition of a Rational Expression with a Prime, Repeated Quadratic Factor in the Denominator
- 20. Example
- 21. (a) (b) (c) (d)
- 22. (a) (b) (c) (d)