In this task students investigate and ultimately prove the validity of the method of generating Pythagorean Triples that involves the polynomial identity (x 2 y 2) 2 =(x 2y 2) 2 (2xy) 2 Type ProblemSolving Task I am trying to solve the equation $$ (x^2y^2)y' 2xy = 0 $$ I have rearranged to get $$ y' = f(x,y) $$ where $$ f(x,y) = \frac{2xy}{x^2y^2} $$ From here I tried to use a trick that I learned Stack Exchange Network Stack Exchange network consists of 178 Q&A communities including Stack Overflow,Add − x 2 y 2 x 2 y 2 and x 2 y 2 x 2 y 2 Add x 2 x 2 x 2 x 2 and 0 0 Simplify each term Tap for more steps Multiply x 2 x 2 by x 2 x 2 by adding the exponents Tap for more steps Use the power rule a m a n = a m n a m a n = a m n to combine exponents Add 2 2 and 2 2
Pythagorean Identities Mathbitsnotebook Ccss Math
The identity (x^2 y^2)^2=(x^2-y^2)^2 (2xy)^2 can be used to generate pythagorean triples
The identity (x^2 y^2)^2=(x^2-y^2)^2 (2xy)^2 can be used to generate pythagorean triples-Write a(x)/ b(x) in the form q(x) r(x)/ b(x), where a(x), b(x), q(x), and r(x) areFor example, the polynomial identity (x 2 y 2) 2 = (x 2 – y 2) 2 (2xy) 2 can be used to generate Pythagorean triples Suggested Learning Targets Understand that polynomial identities include but are not limited to the product of the sum and difference of two terms, the difference of two squares, the sum and difference of two cubes, the square of a binomial, etc
Understanding Pythagorean Identities Studypug
For x^2y^2=2xy, we get (by differentiating implicitly), dy/dx =1 That's the same as the derivative of a linear function with slope, 1 Hmmmmm Let's see If we have x^2y^2=2xy The we must also have x^22xy y^2=0 Factoring gets us (xy)^2 = 0 And the only way for that to happen is to have xy=0 So y=x and dy/dx =1X2 y2 can be written as (xy)2 this is in the form of (a b)2 = a2 2ab b2 so the above can be written as x2 2xy y2 or there is another one too x2 y2 = (xy) (x yX^ {2}2yxy^ {2}=0 x 2 2 y x y 2 = 0 This equation is in standard form ax^ {2}bxc=0 Substitute 1 for a, 2y for b, and y^ {2} for c in the quadratic formula, \frac {b±\sqrt {b^ {2}4ac}} {2a} This equation is in standard form a x 2 b x c = 0
The reference for Conic Section General Cartesian form tells you how to determine what conic section it is, when given the General Cartesian form #Ax^2 Bxy Cy^2 Dx Ey F = 0# Here is the given equation in the general form #2x^2 4y^2 8 = 0# Please observe the value of #B^2 4AC = 0^2 4(2)(4) = 32#The reference says that this is an ellipseIdentity (x 2 y 2)2 = (x 2 – y2)2 (2xy) 2 can be used to generate Pythagorean triples NCN8 () Extend polynomial identities to the complex numbers For example, rewrite x 2 4 as (x 2i)(x – 2i) NCN9 () Know the Fundamental Theorem of Algebra;For example, the polynomial identity (x 2 y 2) 2 = (x 2 y 2) 2 (2xy) 2 can be used to generate Pythagorean triples CCSSMathContentHSAAPRC5 () Know and apply the Binomial Theorem for the expansion of ( x y ) n in powers of x and y for a positive integer n , where x and y are any numbers, with coefficients determined for example
The polynomial identity (x2 y2)2 = (x2 – y2)2 (2xy)2 can be used to generate Pythagorean triples D Rewrite rational expressions AAPRD6 Rewrite simple rational expressions in different forms;Generate Pythagorean Triples using an identity About this video In this lesson you will learn to generate a Pythagorean Triple by using the identity (x^2 y^2)^2 (2xy)^2 = (x^2 y^2)^2MGSE912AAPR4 Prove polynomial identities and use them to describe numerical relationships For example, the polynomial identity (x^2 y^2)^2 = (x^2 – y^2)^2 (2xy)^2 can be used to generate Pythagorean triples Video Lessons ( p1, p2a,
Pythagorean Triples
Aim How Do We Use Structure To Prove Find Pythagorean Triples Ppt Download
For example, the polynomial identity (x 2 y 2) 2 = (x 2 – y 2) 2 (2xy) 2 can be used to generate Pythagorean triples Interpret functions that arise in applications in terms of the context MGSE912FIF4 Using tables, graphs, and verbal descriptions, interpret the key characteristics of a function which models the relationship4 Prove polynomial identities and use them to describe numerical relationships For example, the polynomial identity (x 2 y 2) 2 = (x 2 – y 2) 2 (2xy) 2 can be used to generate Pythagorean triples With the increase in technology and this huge new thing called the Internet, identity theft has become a worldwide problemWrite a(x)/b(x) in the form q(x) r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the
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The Pythagorean Theorem Educational Outreach
For example, the polynomial identity (x 2 y 2) 2 = (x 2 – y 2) 2 (2xy) 2 can be used to generate Pythagorean triples Use complex numbers in polynomial identities and equations MGSE912NCN8 Extend polynomial identities to include factoring with complex numbersExample, the polynomial identity (x2 y2) 2 = (x2 – y 2) 2 (2xy)2 can be used to generate Pythagorean triples 8) AAPR6 Rewrite simple rational expressions in different forms;Students will prove the polynomial identity ( x^2 y^2 )^2 ( 2xy )^2 = ( x^2 y^2 )^2 and use it to generate Pythagorean triplesUse this activity as independent/partner practice or implement it as guided notes and practice for students in need of extra supportThis activity is in PDF formatPar
Generating Pythagorean Triples Chilimath
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CCSSMathContentHSAAPRC4 Prove polynomial identities and use them to describe numerical relationships For example, the polynomial identity (x 2 y 2) 2 = (x 2 y 2) 2 (2xy) 2 can be used to generate Pythagorean triples Authors National Governors Association Center for Best Practices, Council of Chief State School OfficersTap for more steps Add 1 1 to both sides of the equation x 2 y 2 2 x 2 y = 1 x 2 − y 2 − 2 x − 2 y = 1 Complete the square for x 2 2 x x 2 − 2 x Tap for more steps Use the form a x 2 b x c a x 2 b x c, to find the values of a a, b b, and c c a = 1, b = 2, c = 0 a = 1, b = − 2, c = 0Identity (x2 y2)2 = (x 2 – y 2) 2 (2xy) 2 can be used to generate Pythagorean triples Desired Student Performance A student should know • Number theory • Consecutive numbers forms A student should understand
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2 Using suitable identities find (1092)2 3 Using the identity (ab)2 = a2 2ab b2, find (5a 7b)2 4 Find 194 * 6 using suitable identity 5 Use a suitable identity to find the product of (3a 1/3)(3a 1/3) 6 The length and breadth of a rectangle are 3x2 2 and 2x 5 respectively Find its areaGet stepbystep solutions from expert tutors as fast as 1530 minutes Your first 5 questions are on us!Tangent of x^22xyy^2x=2, (1,2) \square!
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