OPERATIONS INVOLVING POLYNOMIALS in .NET framework

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OPERATIONS INVOLVING POLYNOMIALS
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Addition of Polynomials
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EXAMPLE: Add the following polynomials: 3x 2xy + 4y2 , 5xy + y 2 2x , and 3y 2 + 7x + 10xy SOLUTION: Arrange the three polynomials for addition such that like terms are in the same column 3x 2x + 7x 8x 2xy 5xy 10xy 13xy 4y2 y2 3y2 2y2
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Thus, the sum of the three polynomials is 8x + 13xy + 2y2
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Subtraction of Polynomials
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EXAMPLE: Subtract 3a 2 2ab 4b2 from 9a 2 5ab + 6b2 SOLUTION: First, change the sign of each term in the polynomial being subtracted: 3a 2 2ab 4b2 3a 2 + 2ab + 4b2 Arrange the two polynomials for addition such that like terms are in the same column 9a2 5ab 6b2 3a2 6a2 2ab 3ab 4b2 10b2
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Thus, the difference of the two polynomials is 6a 2 3ab + 10b2
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Multiplication of a Monomial by Another Monomial
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EXAMPLE: Multiply 4a2b4c by 7a5b3 SOLUTION: Step 1: Write the product of the two monomials as: (4a2b4c)( 7a5b3) Step 2: Arrange the terms according to the commutative and associative properties: {(4)( 7)}{(a2)(a5)}{(b4)(b3)}{(c)} Step 3: Use the rules of signs and laws of exponents to obtain the product of the two monomials: 28a7b7c
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Multiplication of a Polynomial by a Monomial
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EXAMPLE: Multiply 5x 2 2xy + 7xy 2 + 8y2 by 4x3y5 SOLUTION: Step 1: Write the product of the polynomial and monomial as: (4x3y5)(5x 2 2xy + 7xy 2 + 8y2) Step 2: Multiply each term of the polynomial by the monomial: (4x3y5)(5x 2 ) (4x3y5)(2xy ) + (4x3y5)(7xy 2 ) + (4x3y5)(8y2) Step 3: Use the rules of signs and laws of exponents to obtain the product of each term: (20x5y 5 ) (8x4y 6 ) + (28x4y 7 ) + (32x3y7)
Multiplication of a Polynomial by Another Polynomial
EXAMPLE: Multiply 4x 7x 2 + 8 by x + 3 SOLUTION: Step 1: Arrange in descending powers of x and write the product of the two polynomials as: 7x2 +4x x +8 +3
Step 2: Perform the multiplication process: (1) Multiply 7x 2 + 4x + 8 by x which results in 7x 3 + 4x 2 + 8x (2) Multiply 7x 2 + 4x + 8 by 3 which results in 21x 2 + 12x + 24 Step 3: Add the results making sure to align similar terms in the same column: 7x3 + 7x3 + 4x2 21x2 17x2 + 8x + 12x + 20x + 24 + 24
Thus, the product of the two polynomials is 7x 3 17x 2 + 20x + 24
Division of a Monomial by Another Monomial
EXAMPLE: Divide 36x5y3z2 by 9x4yz4 SOLUTION: Step 1: Write the two monomials with the rst monomial in the numerator and the second monomial in the denominator
36x5y3z2 9x 4 yz 4
Step 2: Arrange the expressions so that you can calculate the quotients of the coef cients and of the variables 36 x 5 y 3 z 2 36 x 5 y 3 z 2 = 9 x 4 y z 4 9 x 4 yz 4 Step 3: Calculate the individual quotients: 36 x 5 y 3 z 2 5 4 y 3 1 z 2 4 = 4 x y 2 z 2 4 4 = 4 x 9 x y z
( )(
) ( )( )( )( )
4xy2 z2
Step 4: Report the answer:
4xy2z 2 =
Division of a Polynomial by Another Polynomial
EXAMPLE: Divide x 2 + 2x 4 3x 3 + x 2 by x 2 3x + 2
SOLUTION: Arrange polynomials in descending powers of x and divide: 2x2 + 3x+ 6 x 3x + 2)2x4 3x3 + x2 + x 2 2x4 6x3 + 4x2 3x3 3x2 + x 2 3x3 9x2 + 6x 6x2 5x 2
Factoring is the mathematical process of simplifying an algebraic expression in terms of two or more algebraic expressions which, when multiplied together, produce the original expression A Common monomial factor B Difference of two squares C Perfect square trinomials ac a2 a2 a2 D Grouping of terms ac (a ad b2 2ab 2ab bc a(c d) (a b)(a b2 b2 ad (a (a bd d) b) b)2 b)2 c(a b) d(a b)
b)(c
Solving Equations for Unknowns involves the use of mathematical relationships and operations applied to both sides of an equation to isolate and then solve for the unknown variable
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