Maths Teaching Guide: Algebraic Expressions

Maths Teaching Guide: Algebraic Expressions 6 Algebraic Expressions You know to write the terms, coefficients and factors of an algebraic expression. to classify an algebraic expression as monomial, binomial, trinomial. to identify like terms. to add and subtract algebraic expression. You will learn multiplication and division of given polynomials. the difference between an identity and an equation. algebraic identities and their applications. factorization of algebraic expression by regrouping , by taking common factors or using algebraic identities. Let us recall the basic definitions of algebra Constants and variables : A quantity having a fixed numerical value is called a constant whereas variables in algebra are letters such as x, y, z or any other letter that can be used to represent unknown numbers. Algebraic expression : An expression which has a combination of constants and variables connected to each other by one or more operation (+,-,X,à ·) is called an algebraic expression. Example are all algebraic expressions Term : The parts of an algebraic expression separated by an addition or a subtraction sign are called terms of the expression. In the expression the terms of the expression are are variable terms as their values will change with the value of x, while (-4) is a constant term. On the basis of the number of terms in an algebraic expression, they are classified as monomials, binomials, trinomials and polynomials. Monomials are algebraic expressions having one term . Binomials are algebraic expressions having two terms. Trinomials are algebraic expressions having three terms. Polynomials are algebraic expressions having one or more than one term. Remember – Only expressions with positive powers of variables are called polynomials. An expression of the type is not a polynomial as and the power of variable p is (- 1) which is not a whole number. Example 1 Classify the algebraic expressions as monomials, binomials or trinomials. Solution binomial monomial trinomial monomial binomial Like and Unlike terms : Terms having the same algebraic factors are called like terms . The numerical coefficients may be different. 2x2yz, 5x2yz, 8x2yz and 2x2yz are like terms 3p 3q2, 7p 3q2and 9p 3q2 are also like terms. Unlike terms : Terms having different algebraic factors are called unlike terms, , 3x2yz 3p 3q2 are unlike terms. Addition and Subtraction of Algebraic Expressions. In algebra, like terms can be added or subtracted. To add or subtract algebraic expressions we can use the horizontal method or the column method. The horizontal method All algebraic expressions are written in a horizontal line; the like terms are then grouped. The sum or difference of the numerical coefficients is then found. Example 2 Add the following Solution Example 3 Subtract Solution The column method In the column method, each expression is written in a separate row in such a way that like terms are arranged one below the other in a column. The sum or difference of the numerical coefficients is then found. Example 4 Add : Solution To add by horizontal method, collect the like terms and add coefficients. To add by column method, arrange the like terms in column and add Example 5 Subtract : Solution We know that the subtraction of two algebraic expressions or terms is addition of the additive inverse of the second term to the first term. Since the additive inverse of a term has opposite sign of the term, hence we can say that in subtraction of algebraic expressions change + to – and change – to + for the term to be subtracted and then add the two terms To subtract by column method, arrange the like terms in columns and change the sign of the subtrahend Example 6 What should be added to to get Solution The expression to be added will be Exercise 6.1 Classify the algebraic expressions as monomials, binomials or trinomials. Also write the terms of the expression Add the following algebraic expressions by the horizontal method Add the following algebraic expressions. Subtract the following expressions. Subtract the sum of from the sum of . Two adjacent sides of a rectangle are . What will be the perimeter of the rectangle. The perimeter of a triangle is and the measure of two sides is. What will be the measure of the third side? What should be added to to get . What should be subtracted from to get By how much is greater than . Multiplication of Algebraic Expressions Multiplication of a monomial by another monomial To multiply 2 monomials Multiply the numerical coefficients Multiply the literal coefficients and use laws of exponents if variables are same. The product of two monomials is always a monomial. Example 1 Find the product of Solution Geometrical interpretation of product of two monomials The area of a rectangle is given by the product of length and breadth. If we consider the length as l and breadth as b, then Area of rectangle = l x b Thus, it can be said that the area of a rectangle is product of two monomials. Let us consider a rectangle of length 4p and breadth 3p, Area of rectangle ABCD =AB x AD = 4p x 3p = 12p2 Multiplication of a monomial by a binomial To multiply a monomial by a binomial, we use the distributive law Multiply the monomial by the first term Multiply the monomial by the second term of the binomial. The result is the sum of the two terms The product of a monomial and a binomial is always a binomial. Example 2 Find the product Solution Example 3 Multiply Solution Geometrical interpretation of product of a monomial and a binomial Area of rectangle = l x b Let us draw a rectangle ABCD with length (p+q) and breadth k. Take a point P on AB such that AP = p and PB = q. Draw a line parallel to AD from the point P, PQà ¢Ã‚ «Ã‚ ½AD meeting DC at Q. Area of rectangle ABCD = area of rectangle APQD +area of rectangle PBCQ = k x p + k x q = k(p + q) Thus, the product k(p + q) represents the area of a rectangle with length as a binomial (p+q) and breadth as a monomial k. Multiplication of a monomial by a polynomial To multiply a monomial with a binomial, we can extend the distributive law further The product of a monomial and a polynomial is a polynomial. Example 3 Find the product of Solution We have multiplied horizontally in all the above examples We can also multiply vertically as shown below Multiply Geometrical interpretation of product of a monomial and a polynomial Let us consider a rectangle with length = (p +q + r) and breadth= k Take points M and N on AB such that AM = p and MN = q and NB = r .from the points M and N draw parallel to AD, MXà ¢Ã‚ «Ã‚ ½AD and NYà ¢Ã‚ «Ã‚ ½AD meeting DC at X and Y. Area of rectangle ABCD = area of rectangle AMXD +area of rectangle MNYX +area of rectangle NBCY Area of rectangle ABCD=pk + qk + rk = k(p + q+ r) Thus, the product of a monomial and a polynomial represents the area of a reactangle with length as a polynomial and breadth as a monomial. Example 4 Simplify Solution Multiplication of binomials To multiply two binomials (a + b) and (c + d) we will again use the distributive law of multiplication over addition twice Example 5 Multiply Solution We have multiplied horizontally in all the above examples We can also multiply vertically as shown below Multiplication of polynomial by a polynomial A polynomial is an algebraic expression having 1 or more than one term To multiply two polynomials, we will use the distributive property that is multiply each term of the first polynomial with each term of the second polynomial. Example 6 Multiply Solution We have multiplied horizontally in the above example, We can also multiply vertically as shown below Exercise 6.2 Multiply the following monomials 2a and 9b Find the following products and evaluate for x = 1, y = -1 Find the following products by horizontal method Find the following products using column method Find the area of the rectangle with the given measurements Length = 3p, breadth = 4p Length = (2a+4), breadth = 5a Multiply the following Simplify the following expressions Multiply . Simplify If the length of a rectangle is and breadth is 3abc,find the area of the rectangle. Algebraic identities An identity is a special type of equation in which the LHS and the RHS are equal for all values of the variables. The above equation is true for all possible values of a and b; so it is called an identity. An identity is different from equation as an equation is not true for all values of variables,;it has a unique solution. Example There are a number of identities which are used in mathematics to make calculations easy. We are going to study 4 basic identities Verification of identities in this identity a and b can be positive or negative Geometrical verification of identities Geometrical demonstration for. Draw a square with length as shown in the figure. Let the area of original square be X then, area of Square PQRS=(side)2 ∠´ , Mark a point M on PQ such that length of PM = a and length of MQ= b. Draw a line MC parallel to PS intersecting SR at C. Similarly, mark a point B on RQ such that RB = a and QB = b. Draw a line BD parallel to QP intersecting PS at D. The whole square is divided into 2 squares and 2 rectangles say A1, A4,A2and A3 Area of Square X1 = side2= a2 Area of rectangle X2= length x breadth = ab Area of rectangle X3= length x breadth = ab Area of Square X4 = side2= b2 area of Square PQRS = sum of inside area = area of X1+ area of X2+ area ofX3+ area ofX4 Geometrically demonstration for . We draw a square with length a as shown in the figure. Let the area of original square is A Then, area of Square PQRS=(side)2 ∠´ Mark a point M on PQ such that the length of PM = a-b and length of MQ= b. Draw a line MC parallel to PS intersecting SR at C. Similarly, mark a point B on RQ such that RB = a b and QB = b. Draw a line BD parallel to QP intersecting PS at D. The whole square

Eulogy: Love and Long Remember Desdemona

I would like to start by thanking Desdemona’s family who have travelled from Venice to be here today, and to all her loyal friends here in Cyprus for your unwavering support during this difficult time. Many of you may be offended by my appearance here today, but to that I will say, I was blinded by my intense love for Desdemona and that love is as strong now as it was the day I married her. We will all long remember Desdemona as a generous and devoted wife and friend. She had a childlike spirit that could shine light into the darkest day. As Desdemona’s husband I loved and wanted to show her how much she meant to me, and in that I failed. She was a very significant person to all of us here, along with so many others. She was such an intrinsic part in each of our lives that it will be long before she is ever forgotten. As her husband, we shared a relationship which was from the start one of loyalty and trust.. As a young and exquisitely beautiful woman she was also accused of many things that I now come to believe untrue. Desdemona was born on 22 April 1603 in Venice to senator, Brabantio, with whom she is now reunited. She was a humble woman who was hard-working, faithful and had a very strong belief in God. Today we mournfully remember the naive Desdemona and the sadistic death she suffered, at my hands. As we are saddened by Desdemona’s passing it is difficult for us to think of continuing life without her. Today we to reflect upon my darling Desdemona’s life and the virtues, achievements and qualities she showed us day by day. She was admired by both men and women for her trusting nature, her intelligence, loyalty to both myself and her father. Desdemona was one to always see the good in others and support them no matter what. Like many here today she was a woman, but through no fault of her own; she customarily received unfair treatment. I treated her unfairly a way no person should be treated i should have trusted her as she would trust me. A friend would tell me how women are manipulative and were seductresses. However Desdemona was the complete opposite She was pure, and kind hearted. Desdemona was devoted and truly submissive to her father. She only did defy him once in her life an that was when she said i do to me. And her husband as long as she lived. When she was nearing the end of her life, I am saddened to say that she was accused of many things, things which I should have known not to be true. Such a quiet and unassuming person, the sweet Desdemona deserved much better treatment than this and this is where I failed her. She never seemed to cease in amazing me, even though she had been through so much she still maintained a forgiving and positive attitude towards life. Desdemona has gone home now, guided by her faith and by the light of those whom she has loved and lost. She will be forever remembered for her kind and loving soul. As Desdemona leaves us broken-hearted with her passing, we can be thankful for the memories she has left in our hearts. My hope is that she forgives the role I had to play in her demise, as a man blinded by trust I listened when I should have felt with my heart, I believed the words of another when I always knew the truth within myself. I did not understand the ways of women i was rather accustomed only to the battlefield and the world of men. I will have to endure life without her because of my mistakes and my blindness. God Bless Desdemona and may she rest in perpetual peace.

Gear Design Software

White Paper on E-Yantra Gear Overview : E-Yantra Gear E-Yantra Gear is a software for design and optimization of gears. E-Yantra Gear gives engineers an opportunity to have several design alternatives and select the optimum design in a very short time, thereby reducing the time taken by design phase in the product life-cycle. In E-Yantra Gear, fatigue-based design is done so gears can be designed for finite or infinite life. E-Yantra Gear includes design (as per IS / AGMA / DIN standard) and analysis of following gear types: ? Spur Gear ? Helical Gear ? Bevel Gear ? Worm Gear Specifications : E-Yantra Gear E-Yantra Gear module has a very user friendly interface with which gear can be designed starting from minimum data like power, gear ratio and input speed. Initially an optimum module is calculated on the basis of which the gear is checked for acceptability. In addition, E-Yantra Gear includes other modules as given below: ? Gear Correction (as per IS/DIN/BIS standard) ? Gear Measurement (MOT/Base Tangent/Chordal Measurement) ? Magnification/Reduction Gear Box Design for Spur & Helical Gear (as per IS/DIN/BIS standard). ? Differential gear box ? Epicyclic gear box (Simple & Compound) ? Machine Tool Gear Box (Ray diagram) ? Tolerances E-Yantra Gear module can develop two dimensional sketches in the form of IGES and DXF formats which can be transferred to any standard modeling software such as AutoCAD or any analysis software. FEAST Software Pvt. Ltd. 1st Floor, Amartara Plastic Compound, Saki Vihar Road, Saki Naka, Andheri (E), Mumbai – 400 072, INDIA Phone: +91-22 6770 8753 Email : [email  protected]are. com www. feastsoftware. com  © 2010. FEAST Software Pvt. Ltd. All rights reserved. White Paper on E-Yantra Gear Input Screen Output Screen  © 2010. FEAST Software Pvt. Ltd. All rights reserved.

Duty Honor Country- Douglas Macarthur - 2130 Words

Duty, Honor, Country â€Å"And through all this welter of change and development, your mission remains fixed, determined, inviolable: it is to win our wars,† this statement embodies Douglas Macarthur’s Speech â€Å"Duty, Honor, Country†. It was given in 1962 in acceptance of the Thayer Award, â€Å"The Award given†¦ citizen of the United States, whose outstanding character, †¦ comparison to the qualities for which West Point strives, in keeping with its motto - Duty, Honor, Country.† (AOGUSMA) It has been presented to other distinguished leaders such as Former Presidents Dwight D. Eisenhower and Ronald Regan, all of who upheld American values and the pillars, Duty, Honor and Country of the US Army. General Douglas Macarthur was one of four Men to reach†¦show more content†¦This parallelism is reflected in the speech. The Army creed, Duty, Honor, Country is repeated eight times. This speech was to a hall of men and woman who also proudly associated themselve s with the US Army. Individually duty, honor, country are strong ideographs, but together they ring true for the corps. Duffy puts is well,† After all duty is what one ordinarily obliged to do; achieving honor for so doing exemplary performance; but to do this not for personal recognition but for country implies altruism on who’s behalf people might giver their lives. ( 115)† This is a great definition of what this creed means for the cadets. These guidelines are utilized for maximum resonance: â€Å"The probably of recalling a repeated word is about twice the probably a unique word. (Perelman 471)† However, in this case the memorability is greater because of previous habit. Duffy Writes, â€Å"advantage accrues†¦ whereby thresholds vary inversely with frequency of prior usage†¦Of all the words uttered at West Point, Duty, Honor, Country clearly had the advantage for emphasis and resultant memorability. (119)† Macarthur ends each of his main points with the army creed. For the audience, this ties every lesson, whether it is civilian versus army focus, the history of west point, or the pride in fighting for this nation into a strong physical application of what a soldier should do. Macarthur takes their training and puts it into action. Parallelism is also used in explaining the ArmyShow MoreRelatedGeneral Douglas Macarthur s Duty, Honor, Country ``1359 Words   |  6 PagesThe Speech â€Å"Duty, Honor, Country† was delivered by General Douglas MacArthur to the Cadets of the Military Academy in West Point, New York. General Douglas MacArthur was being awarded with the Sylvanus Thayer Award to commend the General for all his hard work and devotion to the Military. The Award is a memorable tribute to the ideals that inspired the late General Sylvanus Thayer, who was also known as â€Å"the Father of West Point†. In General Douglas MacArthur’s â€Å"Duty, Honor, Country† speech, he used Read MorePathos And Ethos In Speech1202 Words   |  5 PagesOn May 12, 1962, General Douglas MacArthur addressed the cadets at the West Point Military Academy in his speech Duty, Honor, Country, during the reception for the Sylvanus Thayer Award. This award is given to an â€Å"†¦outstanding citizen of the United States whose service and accomplishments in the national interest exemplify personal devotion to the ideals expressed in the West Point motto, duty, honor, country.† As the one receiving this prestigious award, General MacArthur truly exemplified theseRead MoreEssay about Rhetorical Analysis of MacArthur’s Duty Honor Country1283 Words   |  6 Pagesmust remind ourselves to persevere and continue to defend the country. 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He became known for his great strategic battles that ultimately won World War II for the United States and its allies.Read MoreEssay A Brief Biography of General MacArthur1916 Words   |  8 Pages General MacArthur was a dynamic person who always has been a leader fighting battles throughout the world. Since a young age MacArthur was trained in the military ways to be a leader. It was these skills he learned earl in life, which would lead him to success in the battlefield. He was able to attend West Point where excelled. Eventually, he was appointed to General overseeing the defeat of the Japanese in World War Two. Afterwards, he retired devoted him to staying away from politics as muchRead MoreCritical Analysis The Death Of The Ball Turret Gunner1047 Words   |  5 PagesA Doomed Fate A Critical Analysis of Three Messages in Randall Jarrell’s, Gunner   Ã‚  Ã‚   Douglas MacArthur, an American general during World War II, described those who fight in war as, â€Å"The soldier, above all others, prays for peace; for it is the soldier who must suffer and bear the deepest wounds and scars of war.† Throughout history, war has been a part of every nation. From medieval times to present day, there have been a countless number of wars fought and even more human lives lost. Many shortRead MoreEthics : Defining Police Ethics1455 Words   |  6 Pagesdefine police ethics. Defining Police Ethics Ethics is a word that can be defined as right behavior, but the word has so many other words that fit into it. What words fall into Ethics? 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