This document provides examples and explanations of using Pascal's triangle to expand binomial expressions. It shows how to:
1) Expand binomial expressions like (x + 1)5 and (x - y)8 using the appropriate row of Pascal's triangle as coefficients.
2) Determine missing coefficients in expansions like (x + y)7.
3) Compare directly multiplying factors to using the binomial theorem to expand expressions like (x - 5)4.
4) Expand various binomial expressions like (x + 2)6, (x2 - 3)5, and (-2 + 2x)4 by applying the binomial theorem.
This test contains 19 multiple choice questions about quantitative methods. It will be administered on Friday April 25, 2008 for 60 minutes and is worth a total of 35 marks. The test covers topics such as derivatives, optimization, and linear programming.
The document provides step-by-step work to find the exact zeros of the polynomial function f(x) = 2x^5 - x^4 - 10x^3 + 5x^2 + 12x - 6. It begins by graphing the function to suggest a zero of 0.5 and then uses synthetic division to reduce the polynomial degree. This yields a quadratic function that is then factored to find the zeros of x = ±3 and x = ±2. The final zeros listed are 0.5, ±3, and ±2.
The document provides step-by-step instructions for factoring polynomials, finding inverse functions, simplifying rational expressions, and graphing rational functions. It includes examples of each type of problem worked out in detail from beginning to end. The examples range from relatively simple to more complex in order to demonstrate a variety of situations that may occur.
This document discusses finding the rational zeros of polynomials using the Rational Zeros Theorem. It provides examples of finding all rational zeros of polynomials by considering possible values of p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. It also discusses using synthetic division and the Quadratic Formula to find the exact zeros of polynomials when not all zeros are rational.
This document contains a mathematics exam with 30 multiple choice questions covering various topics in calculus, linear algebra, differential equations, real analysis, and probability. The exam has 75 total marks and is divided into sections on vectors and matrices, differential equations, real analysis, and probability. It provides the questions, response options, and asks test takers to select the correct option(s) and write them in the answer booklet.
The document summarizes how various Avengers calculate and solve different math problems related to their superpowers and equipment. Captain America finds the area of his shield as a function of circumference. Iron Man solves a quadratic equation to power his arc reactor. The Hulk smashes a wall and calculates how many pieces it breaks into. Hawkeye finds the equation of a parabola based on its vertex and a point. Thor analyzes the domain and graph of a skipping hammer's movement. Black Widow determines the domain and range of a radical function.
1) Cubic equations can have one real root or three real roots. They always have at least one real root.
2) Cubic equations can be solved by using synthetic division to reduce them to a quadratic equation if one root is known, or by spotting factors.
3) Graphs of cubic equations cross the x-axis at least once, showing they have at least one real root, and can help locate approximate solutions.
The document provides examples of derivatives and their corresponding anti-derivatives (indefinite integrals) for various functions. It also demonstrates rules for taking the anti-derivative of functions using u-substitution. Some key rules covered include adding +c to account for constants and applying power rules for integrals involving terms like 4x, 3x^2, or other polynomial functions. Examples are worked through step-by-step to illustrate properly applying u-substitution and integrating more complex expressions.
This test contains 19 multiple choice questions about quantitative methods. It will be administered on Friday April 25, 2008 for 60 minutes and is worth a total of 35 marks. The test covers topics such as derivatives, optimization, and linear programming.
The document provides step-by-step work to find the exact zeros of the polynomial function f(x) = 2x^5 - x^4 - 10x^3 + 5x^2 + 12x - 6. It begins by graphing the function to suggest a zero of 0.5 and then uses synthetic division to reduce the polynomial degree. This yields a quadratic function that is then factored to find the zeros of x = ±3 and x = ±2. The final zeros listed are 0.5, ±3, and ±2.
The document provides step-by-step instructions for factoring polynomials, finding inverse functions, simplifying rational expressions, and graphing rational functions. It includes examples of each type of problem worked out in detail from beginning to end. The examples range from relatively simple to more complex in order to demonstrate a variety of situations that may occur.
This document discusses finding the rational zeros of polynomials using the Rational Zeros Theorem. It provides examples of finding all rational zeros of polynomials by considering possible values of p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. It also discusses using synthetic division and the Quadratic Formula to find the exact zeros of polynomials when not all zeros are rational.
This document contains a mathematics exam with 30 multiple choice questions covering various topics in calculus, linear algebra, differential equations, real analysis, and probability. The exam has 75 total marks and is divided into sections on vectors and matrices, differential equations, real analysis, and probability. It provides the questions, response options, and asks test takers to select the correct option(s) and write them in the answer booklet.
The document summarizes how various Avengers calculate and solve different math problems related to their superpowers and equipment. Captain America finds the area of his shield as a function of circumference. Iron Man solves a quadratic equation to power his arc reactor. The Hulk smashes a wall and calculates how many pieces it breaks into. Hawkeye finds the equation of a parabola based on its vertex and a point. Thor analyzes the domain and graph of a skipping hammer's movement. Black Widow determines the domain and range of a radical function.
1) Cubic equations can have one real root or three real roots. They always have at least one real root.
2) Cubic equations can be solved by using synthetic division to reduce them to a quadratic equation if one root is known, or by spotting factors.
3) Graphs of cubic equations cross the x-axis at least once, showing they have at least one real root, and can help locate approximate solutions.
The document provides examples of derivatives and their corresponding anti-derivatives (indefinite integrals) for various functions. It also demonstrates rules for taking the anti-derivative of functions using u-substitution. Some key rules covered include adding +c to account for constants and applying power rules for integrals involving terms like 4x, 3x^2, or other polynomial functions. Examples are worked through step-by-step to illustrate properly applying u-substitution and integrating more complex expressions.
1. The limit as x approaches 4 of x4-16 is 0. When factored, the expression becomes (x-4)(x+4)(x2+4) which equals 0 as x approaches 4.
2. The limit as x approaches infinity of x7-x2+1 is 1. When factored, the leading terms are x7 for both the top and bottom expressions, which equals 1 as x approaches infinity.
3. The limit as x approaches -1 of x2-1 is 0. When factored, the expression becomes (x+1)(x-1) which equals 0 as the factors are 0 when x is -1.
1) The document provides definitions and formulas for calculating derivatives, including the derivative of a function at a point, the derivative function, and common derivatives of basic functions like polynomials, exponentials, logarithms, trigonometric functions, and inverse trigonometric functions.
2) Examples are given of calculating derivatives using the definition of the derivative, for functions like f(x)=3x^2, f(x)=x+1, and f(x)=1/x.
3) Rules are listed for calculating the derivatives of sums, products, quotients of functions using properties of derivatives.
This document discusses solving rational equations and functions. It provides examples of solving rational equations by multiplying both sides by the LCD, checking for extraneous solutions, and solving word problems involving rates and distances using rational functions. It also gives an example of solving a word problem about a round trip flight with headwinds and tailwinds to determine the wind speed.
This document provides examples and exercises on determining composite functions from given functions. It includes:
- Examples of determining possible functions f and g for composite functions like y = (x + 4)2 and y = √x + 5.
- A table sketching and finding domains and ranges for various composite functions like y = f(f(x)) and y = f(g(x)) given functions f(x) and g(x).
- Exercises to determine composite functions f(g(x)) and possible functions f, g, and h for more complex functions like y = x2 - 6x + 5.
- Questions about restrictions on variables and domains for composite functions
The document provides information about polynomials at Higher level, including:
- Definitions of polynomials and examples of polynomial expressions
- Evaluating polynomials using substitution and nested/synthetic methods
- The factor theorem and using it to factorize polynomials
- Finding missing coefficients in polynomials
- Finding the polynomial expression given its zeros
The document discusses binomial coefficients and the binomial theorem. It provides examples of expanding binomial expressions using Pascal's triangle and the binomial coefficient formula. The key points are:
1) Binomial coefficients describe the coefficients that arise when expanding binomial expressions using the binomial theorem.
2) Pascal's triangle provides a visual representation of the binomial coefficients.
3) The binomial theorem can be used to expand binomial expressions in terms of binomial coefficients and write individual terms.
The document introduces how to use Punnett Squares to multiply and factor polynomials. It demonstrates multiplying (x + 3)(x + 2) to get x^2 + 5x + 6. For factoring trinomials, it shows factoring x^2 + 5x + 6 into (x + 3)(x + 2) using a Punnett Square. Finally, it factors an example with a leading coefficient other than 1, factoring 8x^2 - 2x - 3 into (4x - 3)(2x + 1).
The document discusses curve sketching of functions by analyzing their derivatives. It provides:
1) A checklist for graphing a function which involves finding where the function is positive/negative/zero, its monotonicity from the first derivative, and concavity from the second derivative.
2) An example of graphing the cubic function f(x) = 2x^3 - 3x^2 - 12x through analyzing its derivatives.
3) Explanations of the increasing/decreasing test and concavity test to determine monotonicity and concavity from a function's derivatives.
This document contains an exercise set from a chapter on functions. It includes 35 multi-part math problems testing concepts like domains and ranges of functions, rates of change, and word problems involving temperature, speed, and geometric shapes. The problems cover skills like determining maximum/minimum values, solving equations, and sketching graphs of functions.
- The document discusses factoring expressions using the greatest common factor (GCF) method and grouping method.
- With the GCF method, the greatest common factor is determined for each term and factored out.
- The grouping method is used when there are 4 or more terms. The first two terms are factored and the second two terms are factored, then the common factor is extracted.
- Examples of applying both methods to factor polynomials are provided and worked through step-by-step.
Solving polynomial equations in factored formListeningDaisy
The document provides instructions for solving polynomial equations in factored form. It begins by explaining that to solve an equation like (x – 5)(x + 4) = 0, one should not use the FOIL method but rather split the equation into two separate problems that each equal zero: x – 5 = 0 and x + 4 = 0. It then works through several examples of solving factored polynomial equations by finding the values of x that make each factor equal to zero. The document also covers factoring out the greatest common factor from expressions.
The document contains 38 multiple choice questions from an unsolved mathematics past paper from 2007. The questions cover topics such as functions, relations, complex numbers, logarithms, trigonometry, matrices, integrals, conic sections, and coordinate geometry.
This document provides instruction on factoring polynomials. It begins with examples of factoring linear and quadratic expressions. It then discusses using the Factor Theorem and the Remainder Theorem to determine if a binomial is a factor of a polynomial. Additional examples demonstrate factoring polynomials by grouping like terms and using special rules to factor the sum and difference of cubes. An example applies these factoring techniques to model the volume of a storage box.
Jacob's and Vlad's D.E.V. Project - 2012Jacob_Evenson
The document provides steps to simplify a rational function and find its domain. It factors the numerator and denominator, finds the x-intercepts where the numerator is 0, finds the vertical asymptotes where the denominator is 0, and determines the horizontal asymptote by comparing the powers of the numerator and denominator. It then uses this information to sketch the graph and identify the domain as the intervals where the function is defined.
This document discusses various techniques for factoring polynomials, including:
1. Factoring using the greatest common factor (GCF).
2. Factoring polynomials with 4 or more terms by grouping.
3. Factoring trinomials using factors that add up to the coefficient of the middle term.
4. Using the "box method" to factor trinomials where the coefficient of the x^2 term is not 1.
5. Factoring the difference of two squares using the formula a^2 - b^2 = (a + b)(a - b).
The document discusses Taylor series expansions (desarrollos limitados) and provides 15 examples of applying the Taylor formula to common functions like exponential, logarithmic, trigonometric and hyperbolic functions. Specifically, it gives the Taylor series expansion of each function centered around 0 and in terms of powers of x, along with the little-oh notation describing the remainder term.
6.4 factoring and solving polynomial equationshisema01
The document provides examples and instructions for factoring polynomials of various types, including:
- Trinomials like x^2 - 5x - 12
- Sum and difference of cubes like x^3 + 8 and 8x^3 - 1
- Polynomials with a common monomial factor like 6x^2 + 15x
- Quadratics in the form of au^2 + bu + c
It also discusses using the zero product property to solve polynomial equations by factoring and setting each factor equal to zero.
This document discusses two methods for solving quadratic inequalities: graphing and using a sign diagram. For graphing, the inequality is graphed like a boundary line and the range where the inequality is true is shaded. For the sign diagram method, the zeros of the quadratic function are placed on a number line and the intervals where the function has the same sign as the inequality are determined to be the solution set. Examples of both methods are shown and key aspects like critical numbers and sign changes are explained.
This document provides an overview of mathematical functions in MATLAB, including:
1) Common math functions such as absolute value, rounding, floor/ceiling, exponents, logs, and trigonometric functions.
2) How to write custom functions and use programming constructs like if/else statements and for loops.
3) Data analysis functions including statistics and histograms.
4) Complex number representation and basic complex functions in MATLAB.
The document discusses various factoring techniques, including:
- Factoring by greatest common factor (GCF)
- Factoring trinomials of the form ax^2 + bx + c by finding two numbers that multiply to c and add to b
- Factoring the difference of two squares and perfect square trinomials
- Using the "box method" to factor trinomials by placing factors of the first and last terms in the boxes
- Factoring by grouping polynomials with four terms into two groups of two terms each.
1. The limit as x approaches 4 of x4-16 is 0. When factored, the expression becomes (x-4)(x+4)(x2+4) which equals 0 as x approaches 4.
2. The limit as x approaches infinity of x7-x2+1 is 1. When factored, the leading terms are x7 for both the top and bottom expressions, which equals 1 as x approaches infinity.
3. The limit as x approaches -1 of x2-1 is 0. When factored, the expression becomes (x+1)(x-1) which equals 0 as the factors are 0 when x is -1.
1) The document provides definitions and formulas for calculating derivatives, including the derivative of a function at a point, the derivative function, and common derivatives of basic functions like polynomials, exponentials, logarithms, trigonometric functions, and inverse trigonometric functions.
2) Examples are given of calculating derivatives using the definition of the derivative, for functions like f(x)=3x^2, f(x)=x+1, and f(x)=1/x.
3) Rules are listed for calculating the derivatives of sums, products, quotients of functions using properties of derivatives.
This document discusses solving rational equations and functions. It provides examples of solving rational equations by multiplying both sides by the LCD, checking for extraneous solutions, and solving word problems involving rates and distances using rational functions. It also gives an example of solving a word problem about a round trip flight with headwinds and tailwinds to determine the wind speed.
This document provides examples and exercises on determining composite functions from given functions. It includes:
- Examples of determining possible functions f and g for composite functions like y = (x + 4)2 and y = √x + 5.
- A table sketching and finding domains and ranges for various composite functions like y = f(f(x)) and y = f(g(x)) given functions f(x) and g(x).
- Exercises to determine composite functions f(g(x)) and possible functions f, g, and h for more complex functions like y = x2 - 6x + 5.
- Questions about restrictions on variables and domains for composite functions
The document provides information about polynomials at Higher level, including:
- Definitions of polynomials and examples of polynomial expressions
- Evaluating polynomials using substitution and nested/synthetic methods
- The factor theorem and using it to factorize polynomials
- Finding missing coefficients in polynomials
- Finding the polynomial expression given its zeros
The document discusses binomial coefficients and the binomial theorem. It provides examples of expanding binomial expressions using Pascal's triangle and the binomial coefficient formula. The key points are:
1) Binomial coefficients describe the coefficients that arise when expanding binomial expressions using the binomial theorem.
2) Pascal's triangle provides a visual representation of the binomial coefficients.
3) The binomial theorem can be used to expand binomial expressions in terms of binomial coefficients and write individual terms.
The document introduces how to use Punnett Squares to multiply and factor polynomials. It demonstrates multiplying (x + 3)(x + 2) to get x^2 + 5x + 6. For factoring trinomials, it shows factoring x^2 + 5x + 6 into (x + 3)(x + 2) using a Punnett Square. Finally, it factors an example with a leading coefficient other than 1, factoring 8x^2 - 2x - 3 into (4x - 3)(2x + 1).
The document discusses curve sketching of functions by analyzing their derivatives. It provides:
1) A checklist for graphing a function which involves finding where the function is positive/negative/zero, its monotonicity from the first derivative, and concavity from the second derivative.
2) An example of graphing the cubic function f(x) = 2x^3 - 3x^2 - 12x through analyzing its derivatives.
3) Explanations of the increasing/decreasing test and concavity test to determine monotonicity and concavity from a function's derivatives.
This document contains an exercise set from a chapter on functions. It includes 35 multi-part math problems testing concepts like domains and ranges of functions, rates of change, and word problems involving temperature, speed, and geometric shapes. The problems cover skills like determining maximum/minimum values, solving equations, and sketching graphs of functions.
- The document discusses factoring expressions using the greatest common factor (GCF) method and grouping method.
- With the GCF method, the greatest common factor is determined for each term and factored out.
- The grouping method is used when there are 4 or more terms. The first two terms are factored and the second two terms are factored, then the common factor is extracted.
- Examples of applying both methods to factor polynomials are provided and worked through step-by-step.
Solving polynomial equations in factored formListeningDaisy
The document provides instructions for solving polynomial equations in factored form. It begins by explaining that to solve an equation like (x – 5)(x + 4) = 0, one should not use the FOIL method but rather split the equation into two separate problems that each equal zero: x – 5 = 0 and x + 4 = 0. It then works through several examples of solving factored polynomial equations by finding the values of x that make each factor equal to zero. The document also covers factoring out the greatest common factor from expressions.
The document contains 38 multiple choice questions from an unsolved mathematics past paper from 2007. The questions cover topics such as functions, relations, complex numbers, logarithms, trigonometry, matrices, integrals, conic sections, and coordinate geometry.
This document provides instruction on factoring polynomials. It begins with examples of factoring linear and quadratic expressions. It then discusses using the Factor Theorem and the Remainder Theorem to determine if a binomial is a factor of a polynomial. Additional examples demonstrate factoring polynomials by grouping like terms and using special rules to factor the sum and difference of cubes. An example applies these factoring techniques to model the volume of a storage box.
Jacob's and Vlad's D.E.V. Project - 2012Jacob_Evenson
The document provides steps to simplify a rational function and find its domain. It factors the numerator and denominator, finds the x-intercepts where the numerator is 0, finds the vertical asymptotes where the denominator is 0, and determines the horizontal asymptote by comparing the powers of the numerator and denominator. It then uses this information to sketch the graph and identify the domain as the intervals where the function is defined.
This document discusses various techniques for factoring polynomials, including:
1. Factoring using the greatest common factor (GCF).
2. Factoring polynomials with 4 or more terms by grouping.
3. Factoring trinomials using factors that add up to the coefficient of the middle term.
4. Using the "box method" to factor trinomials where the coefficient of the x^2 term is not 1.
5. Factoring the difference of two squares using the formula a^2 - b^2 = (a + b)(a - b).
The document discusses Taylor series expansions (desarrollos limitados) and provides 15 examples of applying the Taylor formula to common functions like exponential, logarithmic, trigonometric and hyperbolic functions. Specifically, it gives the Taylor series expansion of each function centered around 0 and in terms of powers of x, along with the little-oh notation describing the remainder term.
6.4 factoring and solving polynomial equationshisema01
The document provides examples and instructions for factoring polynomials of various types, including:
- Trinomials like x^2 - 5x - 12
- Sum and difference of cubes like x^3 + 8 and 8x^3 - 1
- Polynomials with a common monomial factor like 6x^2 + 15x
- Quadratics in the form of au^2 + bu + c
It also discusses using the zero product property to solve polynomial equations by factoring and setting each factor equal to zero.
This document discusses two methods for solving quadratic inequalities: graphing and using a sign diagram. For graphing, the inequality is graphed like a boundary line and the range where the inequality is true is shaded. For the sign diagram method, the zeros of the quadratic function are placed on a number line and the intervals where the function has the same sign as the inequality are determined to be the solution set. Examples of both methods are shown and key aspects like critical numbers and sign changes are explained.
This document provides an overview of mathematical functions in MATLAB, including:
1) Common math functions such as absolute value, rounding, floor/ceiling, exponents, logs, and trigonometric functions.
2) How to write custom functions and use programming constructs like if/else statements and for loops.
3) Data analysis functions including statistics and histograms.
4) Complex number representation and basic complex functions in MATLAB.
The document discusses various factoring techniques, including:
- Factoring by greatest common factor (GCF)
- Factoring trinomials of the form ax^2 + bx + c by finding two numbers that multiply to c and add to b
- Factoring the difference of two squares and perfect square trinomials
- Using the "box method" to factor trinomials by placing factors of the first and last terms in the boxes
- Factoring by grouping polynomials with four terms into two groups of two terms each.
This document provides examples of finding Taylor and Maclaurin series expansions for various functions. It gives the step-by-step workings for finding the first few terms of series expansions centered at different points for functions like ln(x), 1/x, sin(x), x^4 + x^2, (x-1)e^x, and others. It also discusses using these expansions to approximate integrals and find sums of infinite series.
This document contains practice problems and solutions for combining functions. It includes:
1. Multiple choice questions about compositions of functions.
2. Explicit equations for compositions and composite functions using given functions f(x), g(x), h(x), and k(x).
3. Graphing composite functions and determining their domains.
4. Evaluating composite functions for given values of x.
5. Writing composite functions as sums or compositions of simpler functions.
PMR Form 3 Mathematics Algebraic FractionsSook Yen Wong
The document provides instructions for expanding and factorizing algebraic expressions involving single and double brackets. It explains how to expand brackets by distributing terms inside brackets to each term outside. For factorizing, it describes finding common factors and grouping terms. It also covers techniques for factorizing quadratic expressions, difference of squares, and grouping. Further sections cover simplifying algebraic fractions through factorizing numerators and denominators and combining like terms.
This document discusses algebraic fractions and polynomials. It covers dividing polynomials by monomials and other polynomials. The key steps of polynomial long division and Ruffini's rule for polynomial division are explained. Finding the quotient, remainder, and whether a polynomial is divisible are discussed. Finding the roots of polynomials and using the remainder theorem are also covered. Various techniques for factorizing polynomials are presented, including taking out common factors, using identities, the fundamental theorem of algebra, and Ruffini's rule.
This document provides a study guide for Chapter 6 of Algebra 2 covering evaluating expressions using laws of exponents, determining if functions are polynomials, describing polynomial end behavior, factoring polynomials, solving polynomial equations, dividing polynomials using long and synthetic division, finding zeros of polynomials, writing polynomials given their zeros, and graphing polynomials. It includes 79 problems to work through involving these topics.
This document provides a study guide for Chapter 6 of Algebra 2 covering evaluating expressions using laws of exponents, determining if functions are polynomials, describing polynomial end behavior, factoring polynomials, solving polynomial equations, dividing polynomials using long and synthetic division, finding zeros of polynomials, writing polynomials given their zeros, and graphing polynomials. It includes 79 problems to work through involving these topics.
This document provides an overview of topics covered in intermediate algebra revision including: collecting like terms, multiplying terms, indices, expanding single and double brackets, substitution, solving equations, finding nth terms of sequences, simultaneous equations, inequalities, factorizing common factors and quadratics, solving quadratic equations, rearranging formulas, and graphing curves and lines. The document contains examples and practice problems for each topic.
The document discusses the binomial theorem, which provides a method for expanding binomial expressions of the form (x + y)^n. It explains that each term of the expansion is of the form x^i y^(n-i), with coefficients given by the binomial coefficients. Pascal's triangle is introduced as a way to determine the coefficients. Examples are provided to demonstrate expanding binomial expressions and using the binomial coefficients and theorem.
This document contains solutions to problems from calculus and multivariable calculus courses. It begins with single variable calculus problems involving tangent lines, integrals, derivatives, and infinite series. The second part involves problems related to parametric equations, vectors, planes, cylinders, and graphing surfaces. The last part contains problems involving level curves, least squares regression, and using computer algebra systems to plot functions.
This module introduces polynomial functions of degree greater than 2. It covers identifying polynomial functions from relations, determining the degree of a polynomial, finding quotients of polynomials using division algorithm and synthetic division, and applying the remainder and factor theorems. The document provides examples and practice problems for each topic. It aims to teach students how to work with higher degree polynomial functions.
Polynomials with common monomial factors.pptxGeraldineArig2
This document contains a lesson on factoring polynomials. It begins with examples of factoring various polynomials and identifying the greatest common factor. It then discusses key terminology related to factoring like common monomial factor and factor. The document provides step-by-step worked examples of factoring polynomials. It concludes with a generalization section and seat work problems for students to practice factoring polynomials.
CAPE PURE MATHEMATICS UNIT 2 MODULE 1 PRACTICE QUESTIONSCarlon Baird
dy/dx = (x - 3y)/(6x - 4)
The stationary points on the curve C occur when tan(x) = 2.
The equation of the tangent to C at the point where x=0 is y = 2ex.
This document provides steps for solving radical equations:
1) Isolate the radical on one side of the equation by performing inverse operations
2) Raise both sides of the equation to a power equal to the index of the radical to eliminate the radical
3) Solve the remaining polynomial equation
It includes examples of solving simpler radical equations as well as more complex equations involving fractions and multiple radicals. Checking solutions is emphasized as extraneous solutions may occasionally occur. Graphing calculators can also help visualize and find solutions to radical equations.
A quadratic equation is a second-order polynomial with terms up to x^2. It has two roots, or solutions, which may be real or complex. The quadratic formula can be used to find the exact numeric values of the roots. A root is a value that makes the quadratic equation equal to zero. A double root occurs when the two roots are equal, making the quadratic a perfect square trinomial.
NTHU AI Reading Group: Improved Training of Wasserstein GANsMark Chang
This document summarizes an NTHU AI Reading Group presentation on improved training of Wasserstein GANs. The presentation covered Wasserstein GANs, the derivation of the Kantorovich-Rubinstein duality, difficulties with weight clipping in WGANs, and a proposed gradient penalty method. It also outlined experiments on architecture robustness using LSUN bedrooms and character-level language modeling.
This document provides an outline of topics in algebra including: indices, expanding single and double brackets, substitution, solving equations, solving equations from angle problems, finding the nth term of sequences, simultaneous equations, inequalities, factorizing using common factors, quadratics, grouping and the difference of two squares. It also includes examples and explanations for each topic.
This module covers quadratic functions and equations. Students will learn to determine the zeros of quadratic functions by relating them to the roots of quadratic equations. They will also learn to find the roots of quadratic equations using factoring, completing the square, and the quadratic formula. The module aims to help students derive quadratic functions given certain conditions like the zeros, a table of values, or a graph.
This document contains 3 short entries dated October 06, 2014 that are all labeled "6th october 2014". The document appears to be a log or record with multiple brief entries made on the same date.
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