Multiplying Polynomial

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```					                          Multiplying Polynomial
Multiplying Polynomial

A polynomial is the expression which is written in the combination of the different terms
joined together with the + or - sign.

There can be one or more terms in a polynomial. A polynomial with the one term is called a
monomial, a term with two terms is called a binomial and the term with three terms is called a
trinomial.

A degree of the polynomial is the highest power of the variable among the all given terms in
the polynomial.

Let’s learn how to perform Multiplying Polynomials. For Multiplying Polynomials, we will first
learn to perform multiplication of a monomial with a monomial.

In this case we will multiply the numerical value of both the monomials and then we will add
the powers of the variables in the two monomials. For this let us take an example: 3x^2 * 2x^4

= ( 3 * 2 ) x^6 = 6 x^6

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Now if we have a monomial to be multiplied by a binomial, we say that one term of the
monomial will be multiplied by the two terms of the binomial separately. It can be learned as
follows : if we have to multiply 5x * ( 4x^2 + 2x^3)

Here when we multiply, we will first multiply 5x by 4x^2   and then the same term 5x will be
multiplied by 2x^3. So we get the result as follows:

( 5 * 4 ) x^3 + ( 5 * 2 ) x^4

= 20x^3 + 10x^4

In the same way if we have a trinomial to be multiplied with a monomial, then we will multiply
all the terms of the monomial with all the three terms of the trinomial separately and thus we
get the product of a monomial with the trinomial.

It will be clear with the following example:

2x * ( 3x + 5y + 7z )

Here the term of a monomial 2x will be multiplied with each term of the trinomial and then we
get :

= 2x * 3x + 2x * 5y + 2x * 7z

= 6x^2 + 10xy + 14 xz

All the terms of the above resultant expressions are unlike so they can not be summed up
and thus we get the above given expression as the product of the given trinomial a with the
monomial term.

Let us take an example of multiplying the binomial with a binomial

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( 2x + 3y ) * ( 3x + 5y )

Here we have two binomials ( 2x + 3y ) and ( 3x + 5y ) to be multiplied. For this we need to
multiply the term ( 3x + 5y ) first by 2x and then we will multiply ( 3x + 5y ) by 5y

So we say that we get 2x * ( 2x + 3y ) + 5y * ( 3x + 5y )

So we get 4 x^2 + 6 xy + 15 xy + 25 y ^2

Now we observe that 6xy and 15 xy are like terms so they can be added together and we get
the result as follows :

4 x^2 + 21 xy + 25 y^2

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