Unit 1 lesson 4 notes 2
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The document discusses the difference of squares factoring method. It explains that for a binomial to be factorable using this method, it must have two terms separated by a subtraction sign, where the first term is a perfect square and the second term is a perfect square. If these four requirements are met, the binomial can be factored into the form (a+b)(a-b). It provides examples of factoring x^2 - 100 and x^2 - 1 using this difference of squares method.
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The first derivative of a function f(x) indicates where the function is increasing, decreasing, or constant. If f'(x) is positive, f(x) is increasing, and if f'(x) is negative, f(x) is decreasing. If f'(x) is 0, f(x) is constant. To determine where a function is increasing or decreasing, identify the critical numbers and determine the sign of f'(x) in intervals between them. The first derivative test also allows identification of relative extrema - if f'(x) changes from negative to positive at a critical number c, then f(c) is a relative minimum, and if it changes from positive to negative, f(c
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The document discusses three special factoring formulas:
1) Perfect square trinomials can be factored using the difference of two squares formula.
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3) The sum or difference of two cubes can be factored into a binomial times a trinomial using specific formulas.
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Youtube Link: https://youtu.be/C2qctvyjG7U
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Matlab lecture 7 – regula falsi or false position method@taj
Matlab lecture 7 – regula falsi or false position method@taj
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This lecture is from my class lectures in IIT-JU.
.
Find me:
Website: https://www.tajimiitju.blogspot.com
Linked in: https://www.linkedin.com/in/tajimiitju
Researchgate: https://www.researchgate.net/profile/Tajim_Md_Niamat_Ullah_Akhund
Youtube: https://www.youtube.com/tajimiitju?sub_confirmation=1
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3. The key characteristics for an expression to be factorable as the difference of two squares are: it has two terms, the first term is a perfect square, it uses subtraction, and the last term is a perfect square.
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Matlab lecture 7 – regula falsi or false position method@taj
Matlab lecture 7 – regula falsi or false position method@taj
.
This lecture is from my class lectures in IIT-JU.
.
Find me:
Website: https://www.tajimiitju.blogspot.com
Linked in: https://www.linkedin.com/in/tajimiitju
Researchgate: https://www.researchgate.net/profile/Tajim_Md_Niamat_Ullah_Akhund
Youtube: https://www.youtube.com/tajimiitju?sub_confirmation=1
Slideshare: https://www.slideshare.net/TajimMdNiamatUllahAk
Facebook: https://www.facebook.com/tajim.mohammad
Gitlab: https://gitlab.com/users/tajimiitju
Google+: plus.google.com/+tajimiitju
Email: tajim.mohammad.3@gmail.com
Twitter: https://twitter.com/Tajim53
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This document provides guidance on sketching graphs of functions by considering key features such as symmetries, intercepts, extrema, asymptotes, concavity, and inflection points. It then works through an example of sketching the graph of the function f(x) = (2x^2 - 8)/(x^2 - 16). Key steps include finding vertical and horizontal asymptotes, critical points and inflection points, intervals of increase/decrease, and finally sketching the graph.
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The document discusses concurrence of lines, which refers to three or more lines passing through a common point. It provides examples of showing lines are concurrent by:
1) Taking two line equations and solving them simultaneously to find the point of intersection.
2) Substituting the point of intersection into the remaining line(s) to prove it lies on those line(s) as well.
3) Making a statement that the lines are concurrent if they pass through the same point.
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This document discusses various graphical and numerical methods for finding the roots or zeros of equations, including interval methods like bisection, the false position method, open methods like Newton-Raphson and secant methods, and fixed point methods. Interval methods use initial lower and upper bounds to repeatedly narrow the range containing the root. Open methods take initial values and generate successive approximations converging on the root through techniques like drawing tangents or divided differences. Fixed point methods analyze convergence based on the form of iteration used to find a solution.
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This document discusses different methods for finding the equation of a line, including:
1) Given the slope and y-intercept
2) Using a graph
3) Given a point and the slope
4) Given two points
It provides examples of how to write the line equation using each method.
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This document discusses synthetic division, a shortcut method for long division of polynomials when dividing by divisors of the form x - k. Synthetic division works by setting up an array and adding terms in columns, multiplying the results by -k. The remainder obtained has an important interpretation related to the Remainder Theorem - evaluating the polynomial function at x = k. The Factor Theorem also relates dividing a polynomial by (x - k) to determining if (x - k) is a factor. Examples demonstrate using synthetic division and these theorems to factor polynomials and find zeros.
Math Q1 - Week 2.pdf
The document provides a lesson on factoring the sum and difference of two cubes. It begins with definitions of perfect cubes and examples of factoring expressions like x3 - 27 and x3 + 27 using the rule that (x - a)(x2 + ax + a2) = x3 - a3 and (x + a)(x2 - ax + a2) = x3 + a3. Students are then asked to determine if additional expressions are the sum or difference of two cubes and factor them accordingly. The next section covers factoring perfect square trinomials, defining them as expressions of the form a2 + 2ab + b2 and providing the factoring pattern (a + b)2. Examples
Paso 2 profundizar y contextualizar el conocimiento de la unidad 1 ff
This document provides an overview of algebraic expressions and factorization techniques covered in Unit 1. It defines basic algebraic expressions, polynomials, monomials, binomials, trinomials and more. It then describes 10 cases of factorization, including common factors, difference of squares, sum/difference of cubes, and more. Examples are provided for each case to demonstrate the factorization processes.
Factoring polynomials
1. There are several methods for factoring polynomials outlined in the document: factoring using the distributive property, factoring the difference of two squares/cubes, factoring a perfect square trinomial, and factoring general trinomials using trial and error or grouping.
2. Factoring trinomials involves determining the signs in the factors based on the signs of the terms, then finding two factors of the constant term that satisfy the middle term.
3. More complex polynomials can be factored by grouping like terms or using special factoring patterns like the difference of squares/cubes.
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This document provides 10 problems involving quadratic functions and equations: (1) graphing quadratic equations; (2) solving quadratic equations by factoring; (3) identifying true statements about quadratic equations; (4) solving quadratic equations using the quadratic formula; (5) writing an equation to model profit from bracelet sales; (6) writing an equation for the sum of squares of two consecutive even numbers; (7) analyzing a graph of a quadratic function; (8) finding measurements of diagonals of a rhombus using its area; (9) determining the number of students in a classroom using time per student; (10) analyzing the height of a hot air balloon over time.
Mth 4108-1 b
This document provides 10 problems involving quadratic functions and equations: (1) graphing quadratic equations; (2) solving quadratic equations by factoring; (3) identifying true statements about quadratic equations; (4) solving quadratic equations using the quadratic formula; (5) writing an equation to model profit from bracelet sales; (6) writing an equation for the sum of squares of two consecutive even numbers; (7) analyzing a graph of a quadratic function; (8) solving for measurements of a rhombus given information about its diagonals and area; (9) solving for the number of students in a classroom given information about class time and changes in time per student; (10) solving application problems involving the height
Factoring cases - Andrés.pptx
1. The document discusses 7 main cases of factorizing polynomials: 1) common factor, 2) common factor by grouping terms, 3) difference of perfect squares, 4) perfect square trinomial, 5) trinomial of the form x^2 + bx + c, 6) trinomial of the form ax^2 + bx + c, and 7) sum and difference of perfect cubes.
2. For each case, the document provides examples and outlines the step-by-step process for factorizing polynomials that fall under that particular case. This includes extracting common factors, grouping terms, taking square roots or cube roots, and using multiplication properties.
3. The document is intended to serve as a reference
Solving systems by substitution
Here are the steps to solve this problem:
Let x = one number
Let y = the other number
x + y = 84
x = 3y
x + 3y = 84
4y = 84
y = 21
x = 3(21) = 63
The numbers are 21 and 63.
Expresiones Algebraicas, Factorización y Radicación.
This document provides information on various mathematical concepts including:
1. A monomial is the product of a real number and one or more variables. A polynomial contains more than one variable, constants, and exponents.
2. When adding or subtracting polynomials, like terms are combined. When multiplying polynomials, each term in one polynomial is multiplied by each term in the other and the results are added.
3. Fractions can be added by finding a common denominator and adding the numerators. Fractions are multiplied by multiplying the numerators and multiplying the denominators.
4. Equations can be solved using techniques like reduction, substitution, and factoring polynomials.
5. Factoring is the process of writing an
Q1-W1-Factoring Polynomials.pptx
This document provides instruction on factoring polynomials. It covers several factoring methods including common monomial factoring, difference of squares, sum and difference of cubes, perfect square trinomials, and general trinomials. Examples are provided for each method. The objectives are to determine appropriate factoring methods, factor polynomials completely using various techniques, and solve problems involving polynomial factors.
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- 1. Unit 1 Lesson 4 Notes 2 Factoring Differences of Squares How to determine if you can factor using the difference of squares method: 1. There has to be two terms. 2. There is a subtraction sign 3. The first term is a perfect square 4. The second term is a perfect square If these requirements are all met then you can factor the binomial. (a² - b²) = (a+b)(a-b)
- 2. EXAMPLE 1 (x² - 100) How to determine if you can factor using the difference of squares method: 1. There has to be two terms. 2. There is a subtraction sign 3. The first term is a perfect square 4. The second term is a perfect square If these requirements are all met then you can factor the binomial. (x² - 100) = (x+10)(x-10)
- 3. EXAMPLE 2 (x² - 1) How to determine if you can factor using the difference of squares method: 1. There has to be two terms. 2. There is a subtraction sign 3. The first term is a perfect square 4. The second term is a perfect square If these requirements are all met then you can factor the binomial. (x² - 1) = (x+1)(x-1)