9-4 Notes
•
0 likes•589 views
This document discusses division of polynomials and the remainder theorem in three sections. It begins with two warm-up problems dividing polynomials and checking the answers by substituting values for x. It then shows the division of 4/12x worked out with coefficients and evaluates the quotient polynomial at x=5, obtaining the expected remainder of 0.
Report
Share
Report
Share
Download to read offline
Recommended
Math 6
This document discusses rules for adding and subtracting fractions and mixed numbers with different denominators. It provides examples for:
1) Adding fractions by finding the least common denominator (LCD), converting the fractions to equivalent fractions with the LCD, and adding the numerators.
2) Subtracting fractions using the same process of finding the LCD, converting to equivalent fractions, and subtracting the numerators.
3) Adding and subtracting mixed numbers by first combining the whole numbers and then applying the fraction rules.
3.2 mult & div 1
The document provides instructions and examples for solving equations using multiplication and division. It introduces equations of the form ax = b, where a and b are constants and x is the variable. It explains that the same operation must be applied to both sides of the equation to isolate the variable. Examples are provided such as -3x = 15, which is solved by dividing both sides by -3 to get x = -5. Students are given practice problems to solve equations and justify solution steps.
Adding and subtracting mixed numbers
1. To add or subtract mixed numbers with different denominators, first add or subtract the whole numbers. Then find the LCD and use it to convert the fractions to equivalent fractions with a common denominator. Finally, add or subtract the fractions and reduce if possible.
2. Examples provided demonstrate finding the LCD, converting to equivalent fractions with the common denominator, then adding or subtracting the fractions and reducing if needed. Rules for adding and subtracting mixed numbers with different denominators are explained.
Subtracting similar fractions
This document discusses subtracting similar fractions. It states that to subtract similar fractions, you subtract the numerators and copy the denominator, reducing the fraction to its lowest terms if possible. It provides examples of subtracting fractions like 5/8 - 1/8 = 1/2 and 9/15 - 3/15 = 2/5, along with exercises to subtract fractions and reduce the results to their lowest forms.
Dividing fractions
1. Dividing fractions involves turning the dividing fraction upside down and changing the division sign to a multiplication sign. Then you multiply the numerators and denominators.
2. For mixed numbers, you first change the mixed number to an improper fraction before dividing. Then cancel down and change back to a mixed number if needed.
3. When dividing a whole number by a fraction, the whole number has an implied denominator of 1. Then you follow the same process of multiplying the numerators and denominators. Division of fractions can result in answers that are greater than the original numbers.
Adding And Subtracting Fraction Notes For Web
To add or subtract fractions with like denominators, add or subtract the numerators and keep the same denominators, simplifying the resulting fraction if needed. There are two types of fractions: proper fractions where the numerator is smaller than the denominator, and improper fractions where the numerator is greater than or equal to the denominator. For improper fractions, the answer must be simplified by dividing.
Multiplying and dividing fractions
1) This document provides instructions on multiplying and dividing fractions. It explains how to multiply and divide fractions by multiplying or dividing their numerators and denominators.
2) Visual representations are used to demonstrate multiplying fractions, such as fractions multiplied by whole numbers or other fractions. Mixed numbers are also covered.
3) Cancelling terms before and after calculations is discussed as a way to simplify fractions. Dividing fractions is explained as turning the second fraction upside down and multiplying instead of dividing.
Integrated 2 Section 6-5
The document defines key terms related to functions, including domain, range, and linear functions. It provides an example of writing a function to represent the relationship between the number of cups of coffee made and the total number of spoonfuls needed. The example function is f(c) = 2c + 5, where c is the independent variable representing cups and s is the dependent variable for spoonfuls. A table of values and graph of this linear function are included.
Recommended
Math 6
This document discusses rules for adding and subtracting fractions and mixed numbers with different denominators. It provides examples for:
1) Adding fractions by finding the least common denominator (LCD), converting the fractions to equivalent fractions with the LCD, and adding the numerators.
2) Subtracting fractions using the same process of finding the LCD, converting to equivalent fractions, and subtracting the numerators.
3) Adding and subtracting mixed numbers by first combining the whole numbers and then applying the fraction rules.
3.2 mult & div 1
The document provides instructions and examples for solving equations using multiplication and division. It introduces equations of the form ax = b, where a and b are constants and x is the variable. It explains that the same operation must be applied to both sides of the equation to isolate the variable. Examples are provided such as -3x = 15, which is solved by dividing both sides by -3 to get x = -5. Students are given practice problems to solve equations and justify solution steps.
Adding and subtracting mixed numbers
1. To add or subtract mixed numbers with different denominators, first add or subtract the whole numbers. Then find the LCD and use it to convert the fractions to equivalent fractions with a common denominator. Finally, add or subtract the fractions and reduce if possible.
2. Examples provided demonstrate finding the LCD, converting to equivalent fractions with the common denominator, then adding or subtracting the fractions and reducing if needed. Rules for adding and subtracting mixed numbers with different denominators are explained.
Subtracting similar fractions
This document discusses subtracting similar fractions. It states that to subtract similar fractions, you subtract the numerators and copy the denominator, reducing the fraction to its lowest terms if possible. It provides examples of subtracting fractions like 5/8 - 1/8 = 1/2 and 9/15 - 3/15 = 2/5, along with exercises to subtract fractions and reduce the results to their lowest forms.
Dividing fractions
1. Dividing fractions involves turning the dividing fraction upside down and changing the division sign to a multiplication sign. Then you multiply the numerators and denominators.
2. For mixed numbers, you first change the mixed number to an improper fraction before dividing. Then cancel down and change back to a mixed number if needed.
3. When dividing a whole number by a fraction, the whole number has an implied denominator of 1. Then you follow the same process of multiplying the numerators and denominators. Division of fractions can result in answers that are greater than the original numbers.
Adding And Subtracting Fraction Notes For Web
To add or subtract fractions with like denominators, add or subtract the numerators and keep the same denominators, simplifying the resulting fraction if needed. There are two types of fractions: proper fractions where the numerator is smaller than the denominator, and improper fractions where the numerator is greater than or equal to the denominator. For improper fractions, the answer must be simplified by dividing.
Multiplying and dividing fractions
1) This document provides instructions on multiplying and dividing fractions. It explains how to multiply and divide fractions by multiplying or dividing their numerators and denominators.
2) Visual representations are used to demonstrate multiplying fractions, such as fractions multiplied by whole numbers or other fractions. Mixed numbers are also covered.
3) Cancelling terms before and after calculations is discussed as a way to simplify fractions. Dividing fractions is explained as turning the second fraction upside down and multiplying instead of dividing.
Integrated 2 Section 6-5
The document defines key terms related to functions, including domain, range, and linear functions. It provides an example of writing a function to represent the relationship between the number of cups of coffee made and the total number of spoonfuls needed. The example function is f(c) = 2c + 5, where c is the independent variable representing cups and s is the dependent variable for spoonfuls. A table of values and graph of this linear function are included.
Integrated 2 Section 6-4
The document provides examples for writing and graphing linear inequalities in two variables. It defines key vocabulary like open and closed half-planes, boundary lines, and test points. It then works through examples of determining if points satisfy given inequalities and graphing inequalities on a coordinate plane by plotting boundary lines and shading the appropriate half-plane.
Iu12
The document discusses misconceptions about teaching math in the 21st century. The first misconception is that technology needs bells and whistles like animations and distractions, when the focus should be on whether the technology enhances the educational content. The second misconception is that math instruction needs to be entertaining rather than focusing on developing understanding. Effective technology use and making the content meaningful are more important goals for teaching math.
AA Section 11 8
This document discusses solving polynomial equations. It begins by working through examples of solving quadratic equations. It then introduces concepts like the Fundamental Theorem of Algebra, which states that every polynomial equation has at least one complex number solution. It discusses double roots and the multiplicity of roots. Finally, it works through examples of determining the number of real solutions a polynomial equation would have based on its degree.
Integrated Math 2 Section 4-2
Simulations are used to relate probabilities by modeling complex situations as simple experiments. They involve running trials to represent the complex situation. Examples given include using coins, cards, dice or spinners. The document discusses using the TI-83/84 calculator's ProbSim application to simulate batting averages over multiple at bats and determine the probability of getting a certain number of hits. It provides an example of simulating 10 at bats for someone with a .250 batting average to find the probability of getting exactly 3 hits. Homework assigned is problems 1-10 and odd problems 11-25 on page 155.
AA Section 6-8
The document is notes from a class on imaginary numbers. It begins with examples of simplifying expressions with imaginary numbers. It then defines imaginary numbers as solutions to equations where the variable is squared and equals a negative number. Examples are provided to show how to take the square root of a negative number results in an imaginary number. Further examples demonstrate operations with imaginary numbers like addition, subtraction, multiplication and simplification.
AA Section 5-3
The document contains examples and explanations of solving systems of equations by substitution. In Example 1, a system with two equations and two variables is solved to find the solution (2,4). In Example 2, a real-world word problem is modeled with a system of three equations with three variables to represent the number of different types of tickets printed for a play. The system is solved to find the numbers of adult (A=500), student (S=1000), and children's (C=250) tickets printed.
Notes 7-4
The document discusses that there are at least 12 permutations of the letters A, E, P, R, and S that are valid words in Scrabble. These 5 letters form more valid Scrabble words than any other set of 5 letters. It then lists 12 valid permutations of those 5 letters.
Notes 4-3
This document provides examples and explanations of trigonometric functions including sine, cosine, and tangent. It discusses how to find the quadrant an image point lies in after a rotation about the origin. It also shows how to use trig functions to find the coordinates of a point on the unit circle after a rotation, and examples of evaluating trig functions with various angle measures. Finally, it gives an example of using trig functions to find the heights of the hour and minute hands of a clock at a certain time.
Integrated 2 Section 2-6
This document discusses simplifying variable expressions. It begins with an essential question about adding, subtracting, multiplying, and dividing variable expressions. It then provides the vocabulary for order of operations, including grouping symbols, exponents, multiplication, division, addition, and subtraction. Examples are provided to demonstrate simplifying expressions using order of operations. The examples include solving for variables and writing expressions in terms of variables.
AA Section 11-7
The document discusses the rational-zero theorem and examples of applying it to find rational roots of polynomials. Specifically, it states that if p/q is a rational root of a polynomial with integer coefficients, then p must be a factor of the constant term and q must be a factor of the leading coefficient. Two examples are worked through finding possible rational roots by considering factors of the constant and leading coefficients.
AA Section 11-10
The document provides examples of modeling data with polynomials. Example 1 shows data of distance traveled by a ball down an inclined plane over time. A quadratic polynomial model is found to fit this data, with the formula d = 3t^2, where d is distance and t is time. Example 2 shows additional data over time and x values, to be fitted with a polynomial model.
Integrated Math 2 Section 3-5
Here are the key steps to solving proportions:
1. Write the proportion as a fraction equal sign.
2. Isolate the variable by multiplying/dividing both sides by the same value.
3. Solve for the variable.
4. Check your solution by plugging it back into the original proportion.
Practice these steps on your homework problems. Let me know if you have any other questions!
Integrated Math 2 Section 4-5
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
AA Section 2-9
The document contains examples and explanations of combined and joint variation. It begins with four warm-up problems that demonstrate combined variation, where the dependent variable varies directly with one independent variable and inversely with another. It explains that to find the constant of variation k, one variable is held constant while solving for k. The document then provides an example of finding k and solving a combined variation word problem. It also defines and provides an example of joint variation, where the dependent variable varies directly with two or more independent variables. It explains finding k is done in the same way as in combined variation problems.
Geometry Section 5-3 11-12
This document discusses inequalities in one triangle. It defines inequalities and lists properties of inequalities for real numbers. It presents theorems about exterior angle inequality and angle-side relationships in triangles. Examples demonstrate applying these concepts to identify angles and sides of triangles based on given information. The document provides practice problems for students to check their understanding.
Geometry Section 5-6 1112
The document discusses the Hinge Theorem and its converse for comparing sides and angles of triangles. It provides examples of applying the Hinge Theorem and its converse to determine if one side or angle is greater than the other. It also gives an example problem of proving that one side is less than the other using the Hinge Theorem and properties of alternate interior angles for parallel lines cut by a transversal. The document concludes with assigning practice problems related to applying the Hinge Theorem and its converse.
Geometry Section 3-2 1112
This document discusses angles and parallel lines. It presents three postulates and theorems about corresponding angles, alternate interior angles, and consecutive interior angles when two parallel lines are cut by a transversal. It then provides three multi-part examples that apply these postulates and theorems to find angle measurements or variables in diagrams. The examples demonstrate using vertical angles, supplementary angles, congruent corresponding and alternate interior angles, and algebra to solve for variables.
Geometry Section 1-5
The document defines and provides examples of angle relationships including adjacent angles, linear pairs, vertical angles, complementary angles, and supplementary angles. It defines each term and provides examples of identifying angle pairs that satisfy each relationship. It also includes examples of using properties of these angle relationships to solve problems, such as finding missing angle measures.
Geometry Section 1-4
The document defines various terms related to angle measure including ray, angle, vertex, acute angle, obtuse angle, and angle bisector. It then provides examples measuring angles in a figure and solving an equation involving angle measures.
Geometry Section 1-3
This document discusses finding points and midpoints on line segments. It defines midpoint as the point halfway between two endpoints and provides the formula to calculate it. Several examples are given to demonstrate how to find the midpoint of a segment, locate a point at a fractional distance from one endpoint, and find a point where the ratio of distances from the endpoints is a given ratio. The key concepts covered are calculating midpoints using averages of x- and y-coordinates, and setting up and solving equations to locate interior points using fractional or ratio distances along a segment.
Geometry Section 1-2
This document defines key vocabulary terms related to line segments and distance. It defines a line segment as a portion of a line distinguished by endpoints, and defines betweenness of points and the term "between." It also defines congruent segments, constructions, distance, and irrational numbers. Several examples are provided to demonstrate calculating distances between points on a number line, using a ruler to measure segments, and applying the Pythagorean theorem.
Geometry Section 1-2
This document defines key vocabulary terms related to line segments and distance, including line segment, betweenness of points, congruent segments, and distance formula. It provides examples of calculating distances between points on number lines and using the Pythagorean theorem to find distances between points graphed on a coordinate plane. Examples include measuring line segments with a ruler, finding distances by adding or subtracting measures, and applying the distance formula and Pythagorean theorem to solve for unknown distances.
More Related Content
Viewers also liked
Integrated 2 Section 6-4
The document provides examples for writing and graphing linear inequalities in two variables. It defines key vocabulary like open and closed half-planes, boundary lines, and test points. It then works through examples of determining if points satisfy given inequalities and graphing inequalities on a coordinate plane by plotting boundary lines and shading the appropriate half-plane.
Iu12
The document discusses misconceptions about teaching math in the 21st century. The first misconception is that technology needs bells and whistles like animations and distractions, when the focus should be on whether the technology enhances the educational content. The second misconception is that math instruction needs to be entertaining rather than focusing on developing understanding. Effective technology use and making the content meaningful are more important goals for teaching math.
AA Section 11 8
This document discusses solving polynomial equations. It begins by working through examples of solving quadratic equations. It then introduces concepts like the Fundamental Theorem of Algebra, which states that every polynomial equation has at least one complex number solution. It discusses double roots and the multiplicity of roots. Finally, it works through examples of determining the number of real solutions a polynomial equation would have based on its degree.
Integrated Math 2 Section 4-2
Simulations are used to relate probabilities by modeling complex situations as simple experiments. They involve running trials to represent the complex situation. Examples given include using coins, cards, dice or spinners. The document discusses using the TI-83/84 calculator's ProbSim application to simulate batting averages over multiple at bats and determine the probability of getting a certain number of hits. It provides an example of simulating 10 at bats for someone with a .250 batting average to find the probability of getting exactly 3 hits. Homework assigned is problems 1-10 and odd problems 11-25 on page 155.
AA Section 6-8
The document is notes from a class on imaginary numbers. It begins with examples of simplifying expressions with imaginary numbers. It then defines imaginary numbers as solutions to equations where the variable is squared and equals a negative number. Examples are provided to show how to take the square root of a negative number results in an imaginary number. Further examples demonstrate operations with imaginary numbers like addition, subtraction, multiplication and simplification.
AA Section 5-3
The document contains examples and explanations of solving systems of equations by substitution. In Example 1, a system with two equations and two variables is solved to find the solution (2,4). In Example 2, a real-world word problem is modeled with a system of three equations with three variables to represent the number of different types of tickets printed for a play. The system is solved to find the numbers of adult (A=500), student (S=1000), and children's (C=250) tickets printed.
Notes 7-4
The document discusses that there are at least 12 permutations of the letters A, E, P, R, and S that are valid words in Scrabble. These 5 letters form more valid Scrabble words than any other set of 5 letters. It then lists 12 valid permutations of those 5 letters.
Notes 4-3
This document provides examples and explanations of trigonometric functions including sine, cosine, and tangent. It discusses how to find the quadrant an image point lies in after a rotation about the origin. It also shows how to use trig functions to find the coordinates of a point on the unit circle after a rotation, and examples of evaluating trig functions with various angle measures. Finally, it gives an example of using trig functions to find the heights of the hour and minute hands of a clock at a certain time.
Integrated 2 Section 2-6
This document discusses simplifying variable expressions. It begins with an essential question about adding, subtracting, multiplying, and dividing variable expressions. It then provides the vocabulary for order of operations, including grouping symbols, exponents, multiplication, division, addition, and subtraction. Examples are provided to demonstrate simplifying expressions using order of operations. The examples include solving for variables and writing expressions in terms of variables.
AA Section 11-7
The document discusses the rational-zero theorem and examples of applying it to find rational roots of polynomials. Specifically, it states that if p/q is a rational root of a polynomial with integer coefficients, then p must be a factor of the constant term and q must be a factor of the leading coefficient. Two examples are worked through finding possible rational roots by considering factors of the constant and leading coefficients.
AA Section 11-10
The document provides examples of modeling data with polynomials. Example 1 shows data of distance traveled by a ball down an inclined plane over time. A quadratic polynomial model is found to fit this data, with the formula d = 3t^2, where d is distance and t is time. Example 2 shows additional data over time and x values, to be fitted with a polynomial model.
Integrated Math 2 Section 3-5
Here are the key steps to solving proportions:
1. Write the proportion as a fraction equal sign.
2. Isolate the variable by multiplying/dividing both sides by the same value.
3. Solve for the variable.
4. Check your solution by plugging it back into the original proportion.
Practice these steps on your homework problems. Let me know if you have any other questions!
Integrated Math 2 Section 4-5
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
AA Section 2-9
The document contains examples and explanations of combined and joint variation. It begins with four warm-up problems that demonstrate combined variation, where the dependent variable varies directly with one independent variable and inversely with another. It explains that to find the constant of variation k, one variable is held constant while solving for k. The document then provides an example of finding k and solving a combined variation word problem. It also defines and provides an example of joint variation, where the dependent variable varies directly with two or more independent variables. It explains finding k is done in the same way as in combined variation problems.
Geometry Section 5-3 11-12
This document discusses inequalities in one triangle. It defines inequalities and lists properties of inequalities for real numbers. It presents theorems about exterior angle inequality and angle-side relationships in triangles. Examples demonstrate applying these concepts to identify angles and sides of triangles based on given information. The document provides practice problems for students to check their understanding.
Geometry Section 5-6 1112
The document discusses the Hinge Theorem and its converse for comparing sides and angles of triangles. It provides examples of applying the Hinge Theorem and its converse to determine if one side or angle is greater than the other. It also gives an example problem of proving that one side is less than the other using the Hinge Theorem and properties of alternate interior angles for parallel lines cut by a transversal. The document concludes with assigning practice problems related to applying the Hinge Theorem and its converse.
Geometry Section 3-2 1112
This document discusses angles and parallel lines. It presents three postulates and theorems about corresponding angles, alternate interior angles, and consecutive interior angles when two parallel lines are cut by a transversal. It then provides three multi-part examples that apply these postulates and theorems to find angle measurements or variables in diagrams. The examples demonstrate using vertical angles, supplementary angles, congruent corresponding and alternate interior angles, and algebra to solve for variables.
Viewers also liked (17)
More from Jimbo Lamb
Geometry Section 1-5
The document defines and provides examples of angle relationships including adjacent angles, linear pairs, vertical angles, complementary angles, and supplementary angles. It defines each term and provides examples of identifying angle pairs that satisfy each relationship. It also includes examples of using properties of these angle relationships to solve problems, such as finding missing angle measures.
Geometry Section 1-4
The document defines various terms related to angle measure including ray, angle, vertex, acute angle, obtuse angle, and angle bisector. It then provides examples measuring angles in a figure and solving an equation involving angle measures.
Geometry Section 1-3
This document discusses finding points and midpoints on line segments. It defines midpoint as the point halfway between two endpoints and provides the formula to calculate it. Several examples are given to demonstrate how to find the midpoint of a segment, locate a point at a fractional distance from one endpoint, and find a point where the ratio of distances from the endpoints is a given ratio. The key concepts covered are calculating midpoints using averages of x- and y-coordinates, and setting up and solving equations to locate interior points using fractional or ratio distances along a segment.
Geometry Section 1-2
This document defines key vocabulary terms related to line segments and distance. It defines a line segment as a portion of a line distinguished by endpoints, and defines betweenness of points and the term "between." It also defines congruent segments, constructions, distance, and irrational numbers. Several examples are provided to demonstrate calculating distances between points on a number line, using a ruler to measure segments, and applying the Pythagorean theorem.
Geometry Section 1-2
This document defines key vocabulary terms related to line segments and distance, including line segment, betweenness of points, congruent segments, and distance formula. It provides examples of calculating distances between points on number lines and using the Pythagorean theorem to find distances between points graphed on a coordinate plane. Examples include measuring line segments with a ruler, finding distances by adding or subtracting measures, and applying the distance formula and Pythagorean theorem to solve for unknown distances.
Geometry Section 1-1
This document introduces basic geometry concepts such as points, lines, planes, and their intersections. It defines a point as having no size or shape, a line as an infinite set of collinear points, and a plane as a flat surface extending indefinitely. Examples demonstrate identifying geometric shapes from real-world objects and graphing points and lines on a coordinate plane. The summary defines key terms and provides examples of geometric concepts and relationships.
Algebra 2 Section 5-3
The document discusses inverse functions and relations. It defines an inverse relation as one where the coordinates of a relation are switched, and an inverse function as one where the domain and range of a function are switched. It provides examples of finding the inverse of specific relations and functions by switching their coordinates or domain and range. It also discusses how to determine if two functions are inverses using their graphs and the horizontal line test.
Algebra 2 Section 5-2
The document discusses composition of functions. It defines composition of functions as using the output of one function as the input of another. It provides an example of composing two functions f and g, showing the steps of evaluating f(g(x)) and g(f(x)) at different values of x. Another example is given with two functions defined by sets of ordered pairs, finding the compositions f∘g and g∘f by evaluating them at different inputs and stating their domains and ranges.
Algebra 2 Section 5-1
The document discusses operations that can be performed on functions, including addition, subtraction, multiplication, and division. Definitions of each operation are provided, along with examples of applying the operations to specific functions. Addition of functions involves adding the outputs of each function, subtraction involves subtracting the outputs, multiplication involves multiplying the outputs, and division involves dividing the outputs given the denominator function is not equal to 0. Several examples are worked through applying the different operations to functions like f(x)=2x and g(x)=-x+5. The examples also demonstrate evaluating composite functions and restricting domains as needed.
Algebra 2 Section 4-9
The document discusses determining the number and type of roots of polynomial equations. It states that every polynomial with degree greater than zero has at least one root in the set of complex numbers according to the Fundamental Theorem of Algebra. Descartes' Rule of Signs is introduced, which relates the number of changes in sign of a polynomial's terms to its possible positive and negative real roots. An example problem is worked through applying these concepts to determine the possible number of positive, negative, and imaginary roots.
Algebra 2 Section 4-8
This document discusses synthetic division and the remainder and factor theorems. It provides examples of using synthetic division to evaluate functions, determine the number of terms in a sequence, and factor polynomials. The key steps of synthetic division are shown, along with checking the remainder and determining common factors. Three examples are worked through to demonstrate these concepts.
Algebra 2 Section 4-6
This document discusses solving polynomial equations by factoring polynomials. It begins with essential questions and vocabulary about factoring polynomials and solving polynomial equations by factoring. It then provides the number of terms in a polynomial and the corresponding factoring technique that can be used. Examples of factoring various polynomials are also provided. The document aims to teach students how to factor polynomials and solve polynomial equations by factoring.
Geometry Section 6-6
The document defines key terms and theorems related to trapezoids and kites. It provides definitions for trapezoid, bases, legs of a trapezoid, base angles, isosceles trapezoid, midsegment of a trapezoid, and kite. It also lists theorems about properties of isosceles trapezoids and kites. Two examples problems are included, one finding measures of an isosceles trapezoid and another showing a quadrilateral is a trapezoid.
Geometry Section 6-5
The document discusses rhombi and squares. It defines a rhombus as a parallelogram with four congruent sides and gives its properties. A square is defined as a parallelogram with four right angles and four congruent sides. The document provides theorems for identifying rhombi and squares. It then gives examples of using the properties and theorems to determine if a shape is a rhombus, rectangle, or square.
Geometry Section 6-4
The document discusses properties of rectangles. A rectangle is defined as a parallelogram with four right angles. The key properties are that opposite sides are parallel and congruent, opposite angles are congruent, and consecutive angles are supplementary. The diagonals of a rectangle bisect each other and are congruent. Theorems are presented regarding the diagonals of rectangles. Examples apply the properties of rectangles to find missing side lengths, angles, and diagonals. One example uses the distance formula and slope to determine if a quadrilateral is a rectangle.
Geometry Section 6-3
The document discusses properties of parallelograms and provides examples of determining if a quadrilateral is a parallelogram. It defines four theorems for identifying parallelograms based on opposite sides, opposite angles, bisecting diagonals, and parallel/congruent sides. Examples solve systems of equations to find values of variables such that the quadrilaterals satisfy parallelogram properties. One example uses slopes of side segments to show a quadrilateral is a parallelogram due to parallel opposite sides.
Geometry Section 6-2
The document discusses properties of parallelograms. It defines a parallelogram as a quadrilateral with two pairs of parallel sides. It then lists several properties of parallelograms: opposite sides are congruent, opposite angles are congruent, consecutive angles are supplementary, and if one angle is a right angle all angles are right angles. It also discusses properties of diagonals in parallelograms, including that diagonals bisect each other and divide the parallelogram into two congruent triangles. Several examples demonstrate using these properties to solve problems about parallelograms.
Geometry Section 6-1
The document summarizes key concepts about polygons, including:
- The sum of the interior angles of a polygon with n sides is (n-2)180 degrees.
- The sum of the exterior angles of a polygon is 360 degrees.
- Examples are provided to demonstrate calculating sums of interior/exterior angles and finding missing angle measures using angle sums.
- Regular polygons are defined by their number of sides.
Algebra 2 Section 4-5
The document discusses analyzing graphs of polynomial functions. It provides examples of locating real zeros of polynomials using the location principle and estimating relative maxima and minima. Example 1 analyzes the polynomial f(x) = x^4 - x^3 - 4x^2 + 1 and locates its real zeros between consecutive integer values. Example 2 graphs the polynomial f(x) = x^3 - 3x^2 + 5 and estimates the x-coordinates of relative maxima and minima.
Algebra 2 Section 4-4
This document discusses polynomial functions. It defines key terms like polynomial in one variable, leading coefficient, and polynomial function. It provides examples of power functions of varying degrees like quadratic, cubic, quartic and quintic functions. The document also includes examples of evaluating polynomial functions, finding degrees and leading coefficients, graphing polynomial functions from tables of values, and describing properties of graphs.
More from Jimbo Lamb (20)
Recently uploaded
20240605 QFM017 Machine Intelligence Reading List May 2024
Everything I found interesting about machines behaving intelligently during May 2024
“I’m still / I’m still / Chaining from the Block”
“An Outlook of the Ongoing and Future Relationship between Blockchain Technologies and Process-aware Information Systems.” Invited talk at the joint workshop on Blockchain for Information Systems (BC4IS) and Blockchain for Trusted Data Sharing (B4TDS), co-located with with the 36th International Conference on Advanced Information Systems Engineering (CAiSE), 3 June 2024, Limassol, Cyprus.
Let's Integrate MuleSoft RPA, COMPOSER, APM with AWS IDP along with Slack
Discover the seamless integration of RPA (Robotic Process Automation), COMPOSER, and APM with AWS IDP enhanced with Slack notifications. Explore how these technologies converge to streamline workflows, optimize performance, and ensure secure access, all while leveraging the power of AWS IDP and real-time communication via Slack notifications.
Generative AI Deep Dive: Advancing from Proof of Concept to Production
Join Maher Hanafi, VP of Engineering at Betterworks, in this new session where he'll share a practical framework to transform Gen AI prototypes into impactful products! He'll delve into the complexities of data collection and management, model selection and optimization, and ensuring security, scalability, and responsible use.
Essentials of Automations: The Art of Triggers and Actions in FME
In this second installment of our Essentials of Automations webinar series, we’ll explore the landscape of triggers and actions, guiding you through the nuances of authoring and adapting workspaces for seamless automations. Gain an understanding of the full spectrum of triggers and actions available in FME, empowering you to enhance your workspaces for efficient automation.
We’ll kick things off by showcasing the most commonly used event-based triggers, introducing you to various automation workflows like manual triggers, schedules, directory watchers, and more. Plus, see how these elements play out in real scenarios.
Whether you’re tweaking your current setup or building from the ground up, this session will arm you with the tools and insights needed to transform your FME usage into a powerhouse of productivity. Join us to discover effective strategies that simplify complex processes, enhancing your productivity and transforming your data management practices with FME. Let’s turn complexity into clarity and make your workspaces work wonders!
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024
In this session I delve into the encryption technology used in Microsoft 365 and Microsoft Purview. Including the concepts of Customer Key and Double Key Encryption.
Large Language Model (LLM) and it’s Geospatial Applications
Large Language Model (LLM) and it’s Geospatial Applications.
Introducing Milvus Lite: Easy-to-Install, Easy-to-Use vector database for you...
Join us to introduce Milvus Lite, a vector database that can run on notebooks and laptops, share the same API with Milvus, and integrate with every popular GenAI framework. This webinar is perfect for developers seeking easy-to-use, well-integrated vector databases for their GenAI apps.
Enchancing adoption of Open Source Libraries. A case study on Albumentations.AI
Enchancing adoption of Open Source Libraries. A case study on Albumentations.AIVladimir Iglovikov, Ph.D.
Presented by Vladimir Iglovikov:
- https://www.linkedin.com/in/iglovikov/
- https://x.com/viglovikov
- https://www.instagram.com/ternaus/
This presentation delves into the journey of Albumentations.ai, a highly successful open-source library for data augmentation.
Created out of a necessity for superior performance in Kaggle competitions, Albumentations has grown to become a widely used tool among data scientists and machine learning practitioners.
This case study covers various aspects, including:
People: The contributors and community that have supported Albumentations.
Metrics: The success indicators such as downloads, daily active users, GitHub stars, and financial contributions.
Challenges: The hurdles in monetizing open-source projects and measuring user engagement.
Development Practices: Best practices for creating, maintaining, and scaling open-source libraries, including code hygiene, CI/CD, and fast iteration.
Community Building: Strategies for making adoption easy, iterating quickly, and fostering a vibrant, engaged community.
Marketing: Both online and offline marketing tactics, focusing on real, impactful interactions and collaborations.
Mental Health: Maintaining balance and not feeling pressured by user demands.
Key insights include the importance of automation, making the adoption process seamless, and leveraging offline interactions for marketing. The presentation also emphasizes the need for continuous small improvements and building a friendly, inclusive community that contributes to the project's growth.
Vladimir Iglovikov brings his extensive experience as a Kaggle Grandmaster, ex-Staff ML Engineer at Lyft, sharing valuable lessons and practical advice for anyone looking to enhance the adoption of their open-source projects.
Explore more about Albumentations and join the community at:
GitHub: https://github.com/albumentations-team/albumentations
Website: https://albumentations.ai/
LinkedIn: https://www.linkedin.com/company/100504475
Twitter: https://x.com/albumentationsCommunications Mining Series - Zero to Hero - Session 1
This session provides introduction to UiPath Communication Mining, importance and platform overview. You will acquire a good understand of the phases in Communication Mining as we go over the platform with you. Topics covered:
• Communication Mining Overview
• Why is it important?
• How can it help today’s business and the benefits
• Phases in Communication Mining
• Demo on Platform overview
• Q/A
UiPath Test Automation using UiPath Test Suite series, part 6
Welcome to UiPath Test Automation using UiPath Test Suite series part 6. In this session, we will cover Test Automation with generative AI and Open AI.
UiPath Test Automation with generative AI and Open AI webinar offers an in-depth exploration of leveraging cutting-edge technologies for test automation within the UiPath platform. Attendees will delve into the integration of generative AI, a test automation solution, with Open AI advanced natural language processing capabilities.
Throughout the session, participants will discover how this synergy empowers testers to automate repetitive tasks, enhance testing accuracy, and expedite the software testing life cycle. Topics covered include the seamless integration process, practical use cases, and the benefits of harnessing AI-driven automation for UiPath testing initiatives. By attending this webinar, testers, and automation professionals can gain valuable insights into harnessing the power of AI to optimize their test automation workflows within the UiPath ecosystem, ultimately driving efficiency and quality in software development processes.
What will you get from this session?
1. Insights into integrating generative AI.
2. Understanding how this integration enhances test automation within the UiPath platform
3. Practical demonstrations
4. Exploration of real-world use cases illustrating the benefits of AI-driven test automation for UiPath
Topics covered:
What is generative AI
Test Automation with generative AI and Open AI.
UiPath integration with generative AI
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Why You Should Replace Windows 11 with Nitrux Linux 3.5.0 for enhanced perfor...
The choice of an operating system plays a pivotal role in shaping our computing experience. For decades, Microsoft's Windows has dominated the market, offering a familiar and widely adopted platform for personal and professional use. However, as technological advancements continue to push the boundaries of innovation, alternative operating systems have emerged, challenging the status quo and offering users a fresh perspective on computing.
One such alternative that has garnered significant attention and acclaim is Nitrux Linux 3.5.0, a sleek, powerful, and user-friendly Linux distribution that promises to redefine the way we interact with our devices. With its focus on performance, security, and customization, Nitrux Linux presents a compelling case for those seeking to break free from the constraints of proprietary software and embrace the freedom and flexibility of open-source computing.
Alt. GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using ...
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
20240609 QFM020 Irresponsible AI Reading List May 2024
Everything I found interesting about the irresponsible use of machine intelligence in May 2024
Presentation of the OECD Artificial Intelligence Review of Germany
Consult the full report at https://www.oecd.org/digital/oecd-artificial-intelligence-review-of-germany-609808d6-en.htm
Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdf
Monitoring and observability aren’t traditionally found in software curriculums and many of us cobble this knowledge together from whatever vendor or ecosystem we were first introduced to and whatever is a part of your current company’s observability stack.
While the dev and ops silo continues to crumble….many organizations still relegate monitoring & observability as the purview of ops, infra and SRE teams. This is a mistake - achieving a highly observable system requires collaboration up and down the stack.
I, a former op, would like to extend an invitation to all application developers to join the observability party will share these foundational concepts to build on:
Recently uploaded (20)
20240605 QFM017 Machine Intelligence Reading List May 2024
20240605 QFM017 Machine Intelligence Reading List May 2024
Let's Integrate MuleSoft RPA, COMPOSER, APM with AWS IDP along with Slack
Let's Integrate MuleSoft RPA, COMPOSER, APM with AWS IDP along with Slack
Generative AI Deep Dive: Advancing from Proof of Concept to Production
Generative AI Deep Dive: Advancing from Proof of Concept to Production
Essentials of Automations: The Art of Triggers and Actions in FME
Essentials of Automations: The Art of Triggers and Actions in FME
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024
Large Language Model (LLM) and it’s Geospatial Applications
Large Language Model (LLM) and it’s Geospatial Applications
Introducing Milvus Lite: Easy-to-Install, Easy-to-Use vector database for you...
Introducing Milvus Lite: Easy-to-Install, Easy-to-Use vector database for you...
Enchancing adoption of Open Source Libraries. A case study on Albumentations.AI
Enchancing adoption of Open Source Libraries. A case study on Albumentations.AI
Communications Mining Series - Zero to Hero - Session 1
Communications Mining Series - Zero to Hero - Session 1
UiPath Test Automation using UiPath Test Suite series, part 6
UiPath Test Automation using UiPath Test Suite series, part 6
Why You Should Replace Windows 11 with Nitrux Linux 3.5.0 for enhanced perfor...
Why You Should Replace Windows 11 with Nitrux Linux 3.5.0 for enhanced perfor...
Alt. GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using ...
Alt. GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using ...
20240609 QFM020 Irresponsible AI Reading List May 2024
20240609 QFM020 Irresponsible AI Reading List May 2024
Presentation of the OECD Artificial Intelligence Review of Germany
Presentation of the OECD Artificial Intelligence Review of Germany
Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdf
Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdf
9-4 Notes
- 1. SECTION 9-4 Division and the Remainder Theorem http://homepage.mac.com/shelleywalsh/MathArt/polyd2.gif
- 2. WARM-UP 1. Divide. Check your answer by substituting 5 for x. 4 12x 3 3x 2. Divide. Check your answer by substituting 2 for x. 4 3 12x − 6x 3 3x
- 3. WARM-UP 1. Divide. Check your answer by substituting 5 for x. 4 12x 3 = 4x 3x 2. Divide. Check your answer by substituting 2 for x. 4 3 12x − 6x 3 3x