binomials polynomials many terms are algebraic expressions formed by adding or subtracting monomials single terms with positive exponents or constants a binomial is a polynomial containing two terms bi meaning two in terms of nomenclature bi means 2 hence binomial means two terms so in general a binomial is a polynomial containing two terms the general form can be written as axn ± bym a different and simpler way of defining a binomial is that it is an algebraic expression containing two terms connected by a sum of a difference sign example 3x+5y,a+3x x2-3x etc what are binomials an algebraic expression which contains two terms is called a binomial examples :2x -3 this binomial is formed by subtracting 3 from 2x x2 4y2 the term 4y2 is added to x2 to give a binomial expression the product of many variables yield only monomials 3xyz has three variables namely x y and z but it is a single term or a monomial the constant 3 is the coefficient of the term xyz.
more examples for binomials 5abc 6xy ax by x33 y22 the verbal expressions are often expressed as binomials verbal expressions algebraic expression x+3 z2 5 4 less 2w 4 x2 10 a2 b2 three more than a number five added to square of a number the length of a rectangle is than twice the width dan s age is ten less than half his father s age sum of the squares on the legs of a right triangle adding binomials when two binomials are added the like terms are combined together so in such cases addition can be done between those terms that have the same variable with the same exponent example 3a+5y a+4y 3a+a 5y+4y =4a+9y how ever 5y+y2 cannot be added so we write it as it is 5y+y2 this is because its not possible to add unlike terms when two binomials are being added try to gather the like terms by doing this the like terms come together and simplification is made simple.
in case there are any like terms missing we can complete it by using 0 for example x+2y y x+2y 0x+y x+0x 2y+y 1+0 x 2+1 y =x+3y subtracting binomials when two binomials are subtracted the like terms are combined together example x+5y 2x+7y x-2x 5y-7y x 2y x-2y this operation can also be done in another way that is reversing the sign and then the terms can be added x+5y 2x+7y x+5y 2x-7y x-2x 5y-7y x 2y x-2y multiplying binomials the product of two binomial forms a trinomial product of two binomials can be found using a method called foil f-front o outer i-inner l-last
consider for example 2x+y × 6y+x f-here the front terms of both the binomials are taken and multiplication is performed here front terms are 2x and 6y multiplying them=2x × 6y=12xy o-here both the outer terms of the binomials are taken and multiplied here outer terms are 2x and x multiplying them=2x × x=2x2 i-here both inner terms are taken and multiplied here inner terms are y and 6y multiplying them=y times 6y=6y2 l-here the last terms of the binomials are taken and multiplied here last terms are y and x multiplying them=y times x=xy so 2x+y times 6y+x =12xy+2x2+6y2+xy now like terms are combined so 13xy+2x2+6y2 example y 3 y+4 y y y2 y 4 4y 3 x 3y 3 4 12 y 3 y 4 y2+4y+3y+12 y2+7y+12 when dealing with the negative terms in a binomial each step must be taken carefully two methods can be employed for doing this.
method i example x+5 3x-2 here first of all insert the parenthesis so x+5 3x-2 then use foil method [3x2-2x+15x-10 simplify [3x2+13x-10 remove and change the sign -3x2-13x+10 so x+5 3x-2 -3x2-13x+10 method ii example x+5 3x-2 first distribute the sign for first binomial x-5 3x-2 expand using foil method -3x2+2x-15x+10 simplify -3x2-13x+10 dividing binomials when binomials are divided it is first important to arrange the divisor and the dividend in order ascending or descending order of some variable used in it there are mainly two ways of dividing method i take the dividend s first term and divide it by the divisor s first term and write the value as the quotient in the next step take the quotient obtained and multiply the divisor by it and place the.
respective value below the dividend keeping the like terms under each other subtract this from the dividend take the reminder and any remaining terms from the dividend as the new dividend and the process once again example x2-4 x-2 here dividend =x2-4 divisor x-2 the following process is performed x-2 x2+0x-4 x+2 x2-2x 2x 4 2x 4 0 answer x+2 method ii :the binomial can be factored using the algebraic formulas mentioned above then the common factors can be cancelled example x2-4 x-2 x2-22 x-2 x-2 x+2 x-2 x+2 factoring binomials factoring can be done by taking the gcf of the two terms of the binomial and factoring that out.
for example 4x 6x2 here the gcf 2x so 4x-6x2 2x 2-3x another way is to use special algebraic formulas so general used form are sum and difference binomial squared binomial cube x+y x-y x2 y2 x+y2 x2 2xy y2 x+y3 =x3+3x3y+3xy3+y3
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