Expresiones algebraicas
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En esta presentación de slideshare se vera las diferentes formas de realizar las expresiones algebraicas
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บทที่4
The document discusses basic C programming concepts like data types, operators, precedence, assignments, and increment/decrement operators. It explains:
1) Common data types in C like int and float.
2) Arithmetic operators like +, -, *, /, and modulus %. It also covers increment/decrement operators ++ and --.
3) Operator precedence which determines the order of operations in an expression.
2) Simple and compound assignments using =, +=, -=, *=, /=, and %= operators.
3) The difference between prefix and postfix increment/decrement operators and their order of operations.
Matherrors
1) When adding like terms with the same variable exponent, you add the coefficients and keep the same exponent, not change the exponent. x + x = 2x, not x^2.
2) Terms are only like terms if they have the same variables raised to the same exponents. 3x + 2 cannot be simplified to 5x since x and 3 are not like terms.
3) When distributing, you must distribute to each term in the parentheses. 2(3x - 5) = 6x - 10, not 6x - 5.
4) Be careful of signs and what number the variable is being added or subtracted to. x + 3 = -4 becomes x = -
2.3 linear equations
The document discusses linear equations and how to graph them. It defines linear equations as having variables with exponents of 1 that are added or subtracted. It explains how to identify the slope and y-intercept of a linear equation in slope-intercept form (y=mx+b) in order to graph it as a line on the coordinate plane. Key steps include solving for y and identifying the slope (m) and y-intercept (b). Examples are provided to demonstrate finding intercepts and graphing various linear equations.
Funciones1
1. The document describes methods for completing squares to graph functions and find their domains and ranges.
2. Examples are provided of using the method to graph several functions and find their corresponding domains and ranges.
3. For each function, the method involves completing the square of the variables, finding the x- and y-intercepts, and stating the domain and range.
9 6 Factoring Special Polynomials
This document discusses factoring special polynomials, including perfect square polynomials and the difference of two squares. Perfect square polynomials have a first term and last term that are perfect squares, and the middle term is twice the product of the square roots of the first and last terms. The difference of two squares can be factored by taking the square root of the first term and finding the positive and negative factors of the last term.
Nov. 17 Rational Inequalities
The document discusses solving rational inequalities. It explains the steps to solve rational inequalities which are to simplify the expression so zero is on one side, factor any quadratics, place critical numbers on a number line, test points in intervals to determine if the expression is positive or negative, and state the solution intervals. It then works through examples of solving various rational inequalities graphically and algebraically, placing the critical numbers on a number line and determining the intervals where the expressions are positive or negative.
Limites trigonométricos
1. The document is about trigonometric limits and contains 7 pages with 34 solved limits. It uses fundamental limit laws and techniques like factorizing and applying trigonometric identities to find the limits.
2. Some of the limits involve factors like sinx/x, tanx/x, secx/x as x approaches 0 and techniques are shown to deal with indeterminate forms like 0/0.
3. The solutions find limits of combinations of trigonometric functions like sinx, cosx, tanx as the input x approaches values like 0, π/4, a using standard trigonometric limits and properties of limits.
M.c.qs.- 3(functions & their graphs)
This document contains 28 multiple choice questions about functions and their graphs. The questions cover topics such as slope, linear and quadratic functions, intercepts, domains and ranges. Sample questions ask about the slope of a given line, the opening direction of a parabola, and the x and y-intercepts of an equation. The answers to the questions are provided at the end.
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บทที่4
The document discusses basic C programming concepts like data types, operators, precedence, assignments, and increment/decrement operators. It explains:
1) Common data types in C like int and float.
2) Arithmetic operators like +, -, *, /, and modulus %. It also covers increment/decrement operators ++ and --.
3) Operator precedence which determines the order of operations in an expression.
2) Simple and compound assignments using =, +=, -=, *=, /=, and %= operators.
3) The difference between prefix and postfix increment/decrement operators and their order of operations.
Matherrors
1) When adding like terms with the same variable exponent, you add the coefficients and keep the same exponent, not change the exponent. x + x = 2x, not x^2.
2) Terms are only like terms if they have the same variables raised to the same exponents. 3x + 2 cannot be simplified to 5x since x and 3 are not like terms.
3) When distributing, you must distribute to each term in the parentheses. 2(3x - 5) = 6x - 10, not 6x - 5.
4) Be careful of signs and what number the variable is being added or subtracted to. x + 3 = -4 becomes x = -
2.3 linear equations
The document discusses linear equations and how to graph them. It defines linear equations as having variables with exponents of 1 that are added or subtracted. It explains how to identify the slope and y-intercept of a linear equation in slope-intercept form (y=mx+b) in order to graph it as a line on the coordinate plane. Key steps include solving for y and identifying the slope (m) and y-intercept (b). Examples are provided to demonstrate finding intercepts and graphing various linear equations.
Funciones1
1. The document describes methods for completing squares to graph functions and find their domains and ranges.
2. Examples are provided of using the method to graph several functions and find their corresponding domains and ranges.
3. For each function, the method involves completing the square of the variables, finding the x- and y-intercepts, and stating the domain and range.
9 6 Factoring Special Polynomials
This document discusses factoring special polynomials, including perfect square polynomials and the difference of two squares. Perfect square polynomials have a first term and last term that are perfect squares, and the middle term is twice the product of the square roots of the first and last terms. The difference of two squares can be factored by taking the square root of the first term and finding the positive and negative factors of the last term.
Nov. 17 Rational Inequalities
The document discusses solving rational inequalities. It explains the steps to solve rational inequalities which are to simplify the expression so zero is on one side, factor any quadratics, place critical numbers on a number line, test points in intervals to determine if the expression is positive or negative, and state the solution intervals. It then works through examples of solving various rational inequalities graphically and algebraically, placing the critical numbers on a number line and determining the intervals where the expressions are positive or negative.
Limites trigonométricos
1. The document is about trigonometric limits and contains 7 pages with 34 solved limits. It uses fundamental limit laws and techniques like factorizing and applying trigonometric identities to find the limits.
2. Some of the limits involve factors like sinx/x, tanx/x, secx/x as x approaches 0 and techniques are shown to deal with indeterminate forms like 0/0.
3. The solutions find limits of combinations of trigonometric functions like sinx, cosx, tanx as the input x approaches values like 0, π/4, a using standard trigonometric limits and properties of limits.
M.c.qs.- 3(functions & their graphs)
This document contains 28 multiple choice questions about functions and their graphs. The questions cover topics such as slope, linear and quadratic functions, intercepts, domains and ranges. Sample questions ask about the slope of a given line, the opening direction of a parabola, and the x and y-intercepts of an equation. The answers to the questions are provided at the end.
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The document provides formulas for squaring binomial expressions and factoring polynomials. It lists the formulas for squaring terms of the form (a ± b)2 and expressions involving addition or subtraction of terms. It also gives examples of factoring polynomials into their prime factors and performing arithmetic operations on polynomials.
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The document discusses perfect square trinomials and how to factor them. It provides examples of factoring various square trinomials using the properties that the first and last terms must be perfect squares, and the middle term must be twice the product of the square roots of the first and last terms. It then has students practice factoring several square trinomial examples on their own.
Alg2 lesson 8-7
The system of equations is solved as follows:
1) x + 2y = 5 and y = 3x - 1 are substituted into each other and simplified
2) This results in 7x = 7, so x = 1
3) Substituting x = 1 into y = 3x - 1 gives y = 2
4) Therefore, the solution is (1, 2).
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This document contains a 7-page document with 34 solved trigonometric limit problems. The problems use fundamental limit laws and techniques such as factoring, simplifying, and applying standard trigonometric limits to find the value of various trigonometric function limits as the variable approaches specific values.
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The document discusses linear equations and their two main forms - slope-intercept form (y=mx+b) and point-slope form (y-y1=m(x-x1)). It explains how to use each form to find the equation of a line given values for the slope and/or a point on the line. Examples are provided for finding linear equations using both forms.
Solving Quadratic Equations by Graphing
This document provides instructions for solving quadratic equations by graphing. It begins with an overview of standard form and then works through two example problems. For problem 1, it graphs the equation x^2 + 4x - 5 = 0 and finds the vertex, y-intercept, line of symmetry, and x-intercepts to check the solution. For problem 2, it similarly graphs x^2 + 4x = -6 and finds the relevant points. It concludes that if a parabola does not intersect the x-axis, it represents no solution to the equation.
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2. Each system contains 2-4 equations with 2-5 unknown variables.
3. The solutions to each system of linear equations are not shown.
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This document provides instructions for graphing linear equations. It begins with a "Do Now" activity to find and correct mistakes in a sample function table and graph. It then shows examples of graphing different linear equations by plotting points from their function tables on coordinate grids. The document concludes with a prompt to explain to an absent student how to determine from a graph if a mistake was made in a problem.
Formula
The document shows mathematical expressions and their results. It includes:
1) Fifteen different mathematical expressions with their equal results shown in columns below.
2) The expressions include addition, subtraction, multiplication, division and exponent operations with integers and variables.
3) The results of calculating each expression are provided to demonstrate their equivalency.
January 11
1) The document discusses graphing linear inequalities on a number line and coordinate plane. It provides examples of solving inequalities for y and graphing the corresponding boundary lines, shading the appropriate regions.
2) Methods for graphing inequalities include solving for y, graphing the boundary line, and shading the correct region based on whether the inequality is <, ≤, >, or ≥.
3) An example problem models an inequality describing the maximum number of nickels and dimes that can be had with less than $5.00, graphing the solution on the n-d plane.
March 20th, 2014
1. The document discusses factoring polynomials. It covers factoring trinomials of the form x^2 + bx + c by finding two binomial factors with a sum of b and product of c.
2. The steps for factoring completely are: look for the greatest common factor, look for special cases like difference of squares or perfect square trinomial, find two different binomial factors if not in a special form, and factor by grouping if there are 4 terms.
3. Examples show factoring trinomials and using the FOIL method in reverse to factor. Factoring requires understanding the relationship between factors and terms in a polynomial.
Graphing a line
The document describes graphing a line using the equation y = mx + b, where m is the slope and b is the y-intercept. It provides an example of the line y = 2x + 3, with a slope of m = 2 and a y-intercept of b = 3. A table and graph are included to show the line passing through the points (0,3) and (1,5).
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This document contains corrections to problems on a chapter 1 test. It provides the work and solutions to problems involving fractions, solving equations, simplifying algebraic expressions, identifying counterexamples, and writing verbal expressions algebraically and vice versa.
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This document provides instructions on how to sketch parabolas by identifying their key features:
1) Roots occur where the curve cuts the x-axis (where y = 0)
2) Turning points can be found by completing the square of the equation
3) The y-intercept occurs where x = 0
It demonstrates finding these features for equations such as y = x^2 - 2x - 15, identifying the minimum turning point as (1, -16), roots as (5, 0) and (-3, 0), and y-intercept as (0, -15). Finally, it asks the key question of sketching y = x^2 - 6x + 5, showing its
Polynomial functions!
This document discusses key concepts about polynomial functions including that they have multiple terms, zeros are found by setting the polynomial equal to 0 and switching signs, x-intercepts are the same as zeros and are where the graph crosses the x-axis, standard form arranges terms by degree with highest degree first and factored form groups common factors together, multiplicity refers to repeated factors, and end behavior depends on whether the highest degree term is even or odd and if its coefficient is positive or negative.
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The document contains solutions to three differential equations:
1) y'' - 3y' = 8e^3x + 4sinx with solution Y = (C1 + C2e^3x) + (8/3)Xe^3x + (6/5)cosx - (2/5)sinx
2) y'' + y = xcosx - cosx with solution Y = C1cosx + C2sinx + (1/4)xcosx - (1/2)xsinx + (1/4)x^2sinx
3) y'' + 4y = 4cosx + 3sinx - 8 with solution Y
Completing the square
1) The document discusses completing the square, which involves rewriting quadratic expressions in the form (x + a)2 + b to find maximum and minimum values.
2) Examples are provided of completing the square for expressions like x2 + 8x + 3 and 2x2 + 4x + 11.
3) The technique of setting the expression equal to 0 and solving for x is described as a way to find the minimum value and the corresponding x-value that produces it.
Identidades
This document lists various trigonometric identities including:
1) Definitions of trig functions like sine, cosine, tangent, cotangent, secant, and cosecant
2) Identities involving negative angles
3) Sum and difference identities for trig functions of summed or subtracted angles
4) Double angle and half angle identities
5) Product-to-sum and sum-to-product identities
6) Pythagorean identities relating trig functions to their squares
Mathematica6
The document provides instructions for using Mathematica 6.0 to create direction field plots. It explains how to access Mathematica 6.0 from ITaP machines, how notebooks work in Mathematica, and provides the commands to generate a direction field plot for the differential equation y=x^2-1 between x=-3 and x=3. An example output of these commands in a Mathematica notebook is also shown.
Factoring polynomials
Factoring polynomials involves finding common factors that can be divided out of terms, similar to factoring numbers but with variables; this is done by looking for a single variable or number that is a common factor of all terms that can be pulled out in front of parentheses. The document provides examples of different types of factoring polynomials including using the greatest common factor, difference of squares, grouping, and perfect squares and cubes.
Systems%20of%20 three%20equations%20substitution
The document provides 12 systems of 3 equations each that can be solved by substitution. For each system, the steps to solve by substitution are shown, along with the unique solution. The solutions are provided in the form (x, y, z) where x, y, z are the values of the variables that satisfy all 3 equations simultaneously. System 8 is noted as having no unique solution.
indice-ppt.ppt
1. Indices, surds, logarithms and exponential functions are covered. Key points include laws of indices, rationalizing surds, laws of logarithms including change of base, and the general forms of exponential functions as ax and ex.
2. Example questions demonstrate working with indices, simplifying surds, solving logarithmic and exponential equations using appropriate logarithm laws and change of variable techniques.
3. Solutions to example equations involve algebraic manipulation and setting logarithmic and exponential terms equal to solve for variables.
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Factoring Perfect Square Trinomial
The document discusses perfect square trinomials and how to factor them. It provides examples of factoring various square trinomials using the properties that the first and last terms must be perfect squares, and the middle term must be twice the product of the square roots of the first and last terms. It then has students practice factoring several square trinomial examples on their own.
Alg2 lesson 8-7
The system of equations is solved as follows:
1) x + 2y = 5 and y = 3x - 1 are substituted into each other and simplified
2) This results in 7x = 7, so x = 1
3) Substituting x = 1 into y = 3x - 1 gives y = 2
4) Therefore, the solution is (1, 2).
93311880 limites-trigonometricos
This document contains a 7-page document with 34 solved trigonometric limit problems. The problems use fundamental limit laws and techniques such as factoring, simplifying, and applying standard trigonometric limits to find the value of various trigonometric function limits as the variable approaches specific values.
Edu 290 powerpoint1
The document discusses linear equations and their two main forms - slope-intercept form (y=mx+b) and point-slope form (y-y1=m(x-x1)). It explains how to use each form to find the equation of a line given values for the slope and/or a point on the line. Examples are provided for finding linear equations using both forms.
Solving Quadratic Equations by Graphing
This document provides instructions for solving quadratic equations by graphing. It begins with an overview of standard form and then works through two example problems. For problem 1, it graphs the equation x^2 + 4x - 5 = 0 and finds the vertex, y-intercept, line of symmetry, and x-intercepts to check the solution. For problem 2, it similarly graphs x^2 + 4x = -6 and finds the relevant points. It concludes that if a parabola does not intersect the x-axis, it represents no solution to the equation.
Aljabar Linear Soal01
1. The document provides 11 systems of linear equations to solve.
2. Each system contains 2-4 equations with 2-5 unknown variables.
3. The solutions to each system of linear equations are not shown.
Day 3 graphing linear equations
This document provides instructions for graphing linear equations. It begins with a "Do Now" activity to find and correct mistakes in a sample function table and graph. It then shows examples of graphing different linear equations by plotting points from their function tables on coordinate grids. The document concludes with a prompt to explain to an absent student how to determine from a graph if a mistake was made in a problem.
Formula
The document shows mathematical expressions and their results. It includes:
1) Fifteen different mathematical expressions with their equal results shown in columns below.
2) The expressions include addition, subtraction, multiplication, division and exponent operations with integers and variables.
3) The results of calculating each expression are provided to demonstrate their equivalency.
January 11
1) The document discusses graphing linear inequalities on a number line and coordinate plane. It provides examples of solving inequalities for y and graphing the corresponding boundary lines, shading the appropriate regions.
2) Methods for graphing inequalities include solving for y, graphing the boundary line, and shading the correct region based on whether the inequality is <, ≤, >, or ≥.
3) An example problem models an inequality describing the maximum number of nickels and dimes that can be had with less than $5.00, graphing the solution on the n-d plane.
March 20th, 2014
1. The document discusses factoring polynomials. It covers factoring trinomials of the form x^2 + bx + c by finding two binomial factors with a sum of b and product of c.
2. The steps for factoring completely are: look for the greatest common factor, look for special cases like difference of squares or perfect square trinomial, find two different binomial factors if not in a special form, and factor by grouping if there are 4 terms.
3. Examples show factoring trinomials and using the FOIL method in reverse to factor. Factoring requires understanding the relationship between factors and terms in a polynomial.
Graphing a line
The document describes graphing a line using the equation y = mx + b, where m is the slope and b is the y-intercept. It provides an example of the line y = 2x + 3, with a slope of m = 2 and a y-intercept of b = 3. A table and graph are included to show the line passing through the points (0,3) and (1,5).
Chapter 1 test corrections (algebra)
This document contains corrections to problems on a chapter 1 test. It provides the work and solutions to problems involving fractions, solving equations, simplifying algebraic expressions, identifying counterexamples, and writing verbal expressions algebraically and vice versa.
Sketching parabolas
This document provides instructions on how to sketch parabolas by identifying their key features:
1) Roots occur where the curve cuts the x-axis (where y = 0)
2) Turning points can be found by completing the square of the equation
3) The y-intercept occurs where x = 0
It demonstrates finding these features for equations such as y = x^2 - 2x - 15, identifying the minimum turning point as (1, -16), roots as (5, 0) and (-3, 0), and y-intercept as (0, -15). Finally, it asks the key question of sketching y = x^2 - 6x + 5, showing its
Polynomial functions!
This document discusses key concepts about polynomial functions including that they have multiple terms, zeros are found by setting the polynomial equal to 0 and switching signs, x-intercepts are the same as zeros and are where the graph crosses the x-axis, standard form arranges terms by degree with highest degree first and factored form groups common factors together, multiplicity refers to repeated factors, and end behavior depends on whether the highest degree term is even or odd and if its coefficient is positive or negative.
Ecuaciones Diferenciales[1]
The document contains solutions to three differential equations:
1) y'' - 3y' = 8e^3x + 4sinx with solution Y = (C1 + C2e^3x) + (8/3)Xe^3x + (6/5)cosx - (2/5)sinx
2) y'' + y = xcosx - cosx with solution Y = C1cosx + C2sinx + (1/4)xcosx - (1/2)xsinx + (1/4)x^2sinx
3) y'' + 4y = 4cosx + 3sinx - 8 with solution Y
Completing the square
1) The document discusses completing the square, which involves rewriting quadratic expressions in the form (x + a)2 + b to find maximum and minimum values.
2) Examples are provided of completing the square for expressions like x2 + 8x + 3 and 2x2 + 4x + 11.
3) The technique of setting the expression equal to 0 and solving for x is described as a way to find the minimum value and the corresponding x-value that produces it.
Identidades
This document lists various trigonometric identities including:
1) Definitions of trig functions like sine, cosine, tangent, cotangent, secant, and cosecant
2) Identities involving negative angles
3) Sum and difference identities for trig functions of summed or subtracted angles
4) Double angle and half angle identities
5) Product-to-sum and sum-to-product identities
6) Pythagorean identities relating trig functions to their squares
Mathematica6
The document provides instructions for using Mathematica 6.0 to create direction field plots. It explains how to access Mathematica 6.0 from ITaP machines, how notebooks work in Mathematica, and provides the commands to generate a direction field plot for the differential equation y=x^2-1 between x=-3 and x=3. An example output of these commands in a Mathematica notebook is also shown.
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The document provides 12 systems of 3 equations each that can be solved by substitution. For each system, the steps to solve by substitution are shown, along with the unique solution. The solutions are provided in the form (x, y, z) where x, y, z are the values of the variables that satisfy all 3 equations simultaneously. System 8 is noted as having no unique solution.
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1.Select the correct description of right-hand and left-hand beh.docx
1.
Select the correct description of right-hand and left-hand behavior of the graph of the polynomial function.
ƒ(x) = 4x
2
- 5x + 4
[removed]
Falls to the left, rises to the right.
[removed]
Falls to the left, falls to the right.
[removed]
Rises to the left, rises to the right.
[removed]
Rises to the left, falls to the right.
[removed]
Falls to the left.
QUESTION 2
1.
Describe the right-hand and the left-hand behavior of the graph of
t(x) = 4x
5
- 7x
3
- 13
[removed]
Because the degree is odd and the leading coefficient is positive, the graph falls to the left and rises to the right.
[removed]
Because the degree is odd and the leading coefficient is positive, the graph rises to the left and rises to the right.
[removed]
Because the degree is odd and the leading coefficient is positive, the graph falls to the left and falls to the right.
[removed]
Because the degree is odd and the leading coefficient is positive, the graph rises to the left and falls to the right.
[removed]
Because the degree is even and the leading coefficient is positive, the graph rises to the left and rises to the right.
QUESTION 3
1.
Select the correct description of right-hand and left-hand behavior of the graph of the polynomial function.
ƒ(x) = 3 - 5x + 3x
2
- 5x
3
[removed]
Falls to the left, rises to the right.
[removed]
Falls to the left, falls to the right.
[removed]
Rises to the left, rises to the right.
[removed]
Rises to the left, falls to the right.
[removed]
Falls to the left.
QUESTION 4
1.
Select from the following which is the polynomial function that has the given zeroes.
2,-6
[removed]
f(x) = x
2
- 4x + 12
[removed]
f(x) = x
2
+ 4x + 12
[removed]
f(x) = -x
2
-4x - 12
[removed]
f(x) = -x
2
+ 4x - 12
[removed]
f(x) = x
2
+ 4x - 12
QUESTION 5
1.
Select from the following which is the polynomial function that has the given zeroes.
0,-2,-4
[removed]
f(x) = -x
3
+ 6x
2
+ 8x
[removed]
f(x) = x
3
- 6x
2
+ 8x
[removed]
f(x) = x
3
+ 6x
2
+ 8x
[removed]
f(x) = x
3
- 6x
2
- 8x
[removed]
f(x) = x
3
+ 6x
2
- 8x
QUESTION 6
1.
Sketch the graph of the function by finding the zeroes of the polynomial.
f(x) = 2x
3
- 10x
2
+ 12x
[removed]
0,2,3
[removed]
0,2,-3
[removed]
0,-2,3
[removed]
0,2,3
[removed]
0,-2,-3
QUESTION 7
1.
Select the graph of the function and determine the zeroes of the polynomial.
f(x) = x
2
(x-6)
[removed]
0,6,-6
[removed]
0,6
[removed]
0,-6
[removed]
0,6
[removed]
0,-6
QUESTION 8
1.
Use the Remainder Theorem and Synthetic Division to find the function value.
g(x) = 3x
6
+ 3x
4
- 3x
2
+ 6, g(0)
[removed]
6
[removed]
3
[removed]
-3
[removed]
8
[removed]
7
QUESTION 9
1.
Use the Remainder Theorem and Synthetic Division to find the function value.
f(x) = 3x
3
- 7x + 3, f(5)
[removed]
-343
[removed]
343
[removed]
345
[removed]
340
[removed]
344
QUESTION 10
1.
Use the Remainder Theorem and Synthetic Division to find the function value.
h(x) = x
3
- 4x
2
- 9x + 7, h(4)
[removed] ...
Limits of a function: Introductory to Calculus
This document discusses limits and continuity in functions. It defines limits intuitively and formally. Some key theorems on limits are presented, including the limit laws for sums, products, quotients, and other operations. Examples are provided to demonstrate evaluating limits using direct substitution and the limit laws. One-sided limits and limits of piecewise functions are also briefly introduced.
เอกนาม
This document contains examples of algebraic expressions and equations. It shows how to combine like terms, factor expressions, and solve simple equations. For example, it shows that (x + 2)(x + 3) can be factored as x^2 + 5x + 6. It also provides step-by-step workings for subtracting expressions like 2y - (-5y) = 2y + 5y = 7y.
Chapter 2.5
The document discusses the distributive property and combining like terms in algebra. It defines key terms such as terms, coefficients, and like terms. It then explains the distributive property using examples of distributing a number over terms in parentheses. Finally, it provides practice problems for students to work through using the distributive property to combine like terms.
Ecs lineales
This document contains 26 equations to solve for the variable x. The equations involve adding, subtracting, multiplying and dividing terms with x. Students are instructed to copy the equations into their notebook and solve for x in each case. The solutions provided range from simple integers to more complex fractional values of x.
Chapter 2(limits)
This document provides information on calculating limits using limit laws and discusses one-sided limits and limits at infinity. It includes theorems on limit laws and examples of applying the laws to calculate limits. There are also 36 practice problems with answers provided to find specific limits algebraically or using limit laws for rational functions, functions with noninteger or negative powers, and limits approaching positive or negative infinity.
Texto de matemática y lógica
The document contains examples of algebraic expressions and equations. Some expressions are set equal to numbers to form equations. Steps are shown to solve equations for unknown variables by isolating them on one side of the equal sign.
Texto de matemática y lógica
The document contains examples of algebraic expressions and equations. Some expressions are set equal to numbers to form equations. Several examples involve solving simple equations for unknown variables. Patterns and properties of numbers, expressions, and equations are demonstrated throughout the examples.
System dynamics 3rd edition palm solutions manual
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Expresiones algebraicas
- 4. Eliminamos los paréntesis, el signo operacional suma ++ no afecta a los signos de los monomios encerrados, la expresión quedaría simplemente así: 8x+4x–3y– 5y+2z+z=(8+4)x+(−3−5)y+(2+1)z=12x−8y+3z8x +4x–3y– 5y+2z+z=(8+4)x+(−3−5)y+(2+1)z=12x−8y+3z Al retirar los paréntesis, el signo ++ no afecta a los signos operacionales de los términos de los polinomios encerrados quedando: 6x+z+2x+3y−y−5z6x+z+2x+3y−y−5z Reuniendo y reduciendo términos semejantes, tenemos: 6x+2x+3y−y+z−5z=(6+2)x+(3−1)y+(z−5z)=8x+2y −4z6x+2x+3y−y+z−5z=(6+2)x+(3−1)y+(z−5z)=8x +2y−4z
- 6. Eliminando los paréntesis, resulta: 4a+2a+3b+5b–2c–c4a+2a+3b+5b–2c–c Reduciendo términos semejantes: 6a+8b–3c6a+8b–3c (Eliminando paréntesis se cambian los signos de 2m−5n2m−5n a −2m+5n−2m+5n y −p−p a pp: 8m+6n−2m+5n+p8m+6n−2m+5n+p Reduciendo términos semejantes: 6m+11n+p
- 8. Ejemplo:
- 16. Ejemplo: