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Multiplication of Polynomials

Multiplication of Polynomials

   Definition and Notation

The product of two natural numbers, 3and 4, is defined by

          3 x 4 = 4+4+4 three terms of 4
Similarly   5 a = 5·a = a+a+a+a+a five terms of a
          4 a b = a b+a b+a b+a b four terms of a b
          a b=a x b = b+b +....+ b a terms of b

The following are some of the laws from multiplication of real numbers

1. The commutative law of multiplication: a b = b a

2. The associative law of multiplication: a (b c) = (a b) c

3. The distributive law of multiplication: a (b+c) = (b+c) a

=a b+a c

4. Multiplication of signed numbers:

            (+ a) (+ b) = + a b; (+ a) ( b) = a b

            ( a) (+ b) = a b; ( a) ( b) = + a b

When we have 2·2·2·2. that is, four factors of 2, the notation 24 is used, which reads. “two to the power four," or “two to the fourth power."

Similarly, a·a·a·a·a = a5 means five factors of a. The a is called the base, and the 5 is called the exponent. When there is no exponent, as in x, we always mean x to the power 1.

DEFINITION

                  Notation of Polynomial

Note the difference between

          ( 24) = ( 2) ( 2) ( 2) ( 2) = +16

and 24 = (24) = -(2·2·2·2) = 16

Note also 2 a3 = 2 (a·a·a)

While (2 a)3 = (2 a) (2 a) (2 a)

=(2·2·2) (a·a·a)

=23 a3 = 8 a3

Remark a , a2 , a3,.... are not like terms.

EXAMPLES 1. 7 a·a·a·a = 7 a4

2. ( 3) ( 3) ( 3) ( 3) = ( 3)4

3. (x-1)3 = (x-1) (x-1) (x-1)

4. 22·33 = (2·2)·(3·3·3) = 4·27 = 108

5. 22+23 = 2·2+2·2·2 = 4+8 = 12

6. 23-2 = 2·2·2-2=8-2 = 6

 

Multiplication of Monomials    

We will discuss the multiplication of monomials, then the multiplication of a monomial and a polynomial, and finally the multiplication of two polynomials.

From the definition of exponents we have

    a3·a5 = (a·a·a) (a·a·a·a·a)

  =a·a·a·a·a·a·a·a

  =a8

  =a3+5

THEOREM 1

  Multiplication of Monomial Theorem 1

Proof

     Explaination for Theorem 1 Multiplication of Monomial

EXAMPLES 1. 23·25 = 23+5 = 28

2. a2·a4 = a2+4 = a6

3. 24·23 = 24+3 = 27

4. 3 x3·x2 = 3 x3+2 = 3 x5

5. x5·x = x5+1 = x6

6. (a+1)2·(a+1)3 = (a+1)2+3 = (a+1)5

 

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Remark 23·27 = 23+7 = 210 , and not 410

Remark 24·35 = 24·35 , to find the product, multiply 24 =16 by 35 =243 ; that is 24·35 = (16) (243) = 3888

Since the commutative and associative laws for multiplication hold for numbers, specific or general. we have

 

EXAMPLES 1. (2 a b2) (3 a4 b c2)= (2·3) (a1·a4) (b2·b1) (c2)

= 6 a5 b3 c2

2. ( 3 b2 c3) (8 a b3 c) = ( 3·8) (b2·b3) (c3·c) (a)

= 24 b5 c4 a

3. ( 32 x y2) ( 5 x2 y3) = ( 9 x y2) ( 5 x2 y3)

=( 9) ( 5) (x·x2) (y2·y3)=45 x3 y5 

       

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From the definition of exponents we have

        (a2)3 = (a2) (a2) (a2)

  =(a2·a2) (a2)

  =a2+2·a2

  =a2+2+2

  =a3·2 = a2·3

  =a6

 

THEOREM 2

      Multiplication of Monomial Theorem 2

Proof

      Explaination for Theorem 2 Multiplication of Monomial

EXAMPLES 1. (32)4 = 32·4 = 38

2. (a3)5 = a3·5 = a15

3. ( 32)3 = 32·3 = 36

4. ( a3)2 = a3·2 = a6

  

Note 23·24 = 23+4=27, While (23)4 = 23·4=212

From the definition of exponents we have

  64 = (2·3)4 = (2·3) (2·3) (2·3) (2·3)

  = (2·2·2·2) (3·3·3·3)

  =24·34

THEOREM 3

   Multiplication of Monomial Theorem 3

Proof

    Explaination for Theorem 3 Multiplication of Monomial

  Note a and b are factors. If a=3, b=x, and m=5,(3 x)5=35 x5

Do not forget to raise the number 3 to the power 5

Applying Theorem 3 repeatedly, we obtain

        (a b c d)m=[(a b) (c d)]m

  =(a b)m (c d)m

  =ambmcmdm

    Remark 212 = (3·7)2 = 32·72 = 9 x49 = 441

    Remark The quantity Condition to simplify the multiplication theorem problem

          (5+3)2 = (8)2 = 64, but 52+32 = 25+9 = 34

If we consider (a+b) as one quantity, then

          (a+b)5 = (a+b) (a+b) (a+b) (a+b) (a+b)

The method of calculating the product will be explained later

COROLLARY Applying theorems with conditions

          (am) (bn)k = [(am) (bn)]k

=(am)k (bn)k

=am kbn k

EXAMPLE Perform the following multiplication: (2 x2 y z3)( 4 x3 y2)

Solution (2 x2 y z3)( 4 x3 y2) = (2) ( 4) (x2·x3) (y·y2) (z3)

= 8 x5 y3 z3

 

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EXAMPLE Perform the following multiplication: (5 a2 b)3

Solution (5 a2 b)3 = (5)3 (a2)3 (b)3 = 53 a6 b3 = 125 a6 b3

 

EXAMPLE Perform the following multiplication: 22 a3 (a b3)2

Solution 22 a3 (a b3)2 = 4 a3 (a2 b6) = 4 (a3·a2) (b6) = 4 a5 b6

 

EXAMPLES Perform the following multiplication: (3 x2 y)2 (2 x y3)3

Solution (3 x2 y)2 (2 x y3)3 = (32 x4 y2) (23 x3 y9) = (32·23) (x4·x3) (y2·y9)

=(9·8) x7 y11 = 72 x7 y11

Remark Perform the outside exponents first

 

EXAMPLE Perform the following multiplication:

      ( 2 a b2)2 ( 3 a2 b)3 ( b c2)4

Solution ( 2 a b2)2 ( 3 a2 b)3 ( b c2)4 = ( 2)2 a2 b2·( 3)3 a6 b3·( 1)4 b4 c8

=( 2)2 ( 3)3 ( 1)4 (a2·a6) (b4·b3·b4) (c8)

=(4) ( 27) (+ 1) a8 b11 c8

= 108 a8 b11 c8

 

EXAMPLE Perform the indicated operations and simplify:

       (2 a b)4 ( a3 b)2-( 3 a2)3 (a2 b3)2

Solution (2 a b)4 ( a3 b)2-( 3 a2)3 (a2 b3)2 = (16 a4 b4) (a6 b2)-( 27 a6) (a4 b6)

=16 a10 b6+27 a10 b6

=43 a10 b6

 

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Note: To evaluate expressions involving exponents. first replace each letter by its indicated specific value. Use grouping symbols where necessary so as not to confuse operation signs with number signs.

 

Evaluate a2 b3, give that a = 3 and b = 2

Solution a2 b3 = ( 3)2 (2)3 = (9) (8) = 72

 

EXAMPLE Evaluate the expression b2-a2 (c3-b3), give that a = 2 , b = 3, and c = 1

Solution b2-a2 (c3-b3) = (3)2-( 2)2 [(( 1)3-(3)3)]

=9-(+ 4) [( 1)-(27)]

=9-4 ( 1-27)

=9-4 ( 28)

=9+112 = 121

Multiplication of a Polynomial by a Monomial    

Use of subscript

How to read subscript

The extended distributive law of multiplication

  extended distributive law of multiplication

is used to multiply a monomial by a polynomial

EXAMPLE Multiply 3 x2+x-2 by x

Solution x (3 x2+x-2) = x (3 x2)+x (x)+x ( 2)

=3 x2+x2-2 x

EXAMPLE Multiply x2-x+4 by 2 x2

Solution ( 2 x2) (x2-x+4) = ( 2 x2) (x2) + ( 2 x2) ( x) + ( 2 x2) (4)

= 2 x4+2 x3-8 x2

 

EXAMPLE Multiply a2 b-2 b2 c+5 c2 a by 3 a2 b

Solution 3 a2 b (a2 b-2 b2 c+5 c2 a)

=3 a2 b (a2 b)+3 a2 b ( 2 b2 c)+3 a2 b (5 c2 a)

=3 a4 b2-6 a2 b3 c+15 a3 b c2

EXAMPLE Multiply 3 x-24 - 2 x-16 by 12

Solution 121[3 x-24-2 x-16] = 121[3 x-24] - 121 [2 x-16]

=3 (3 x-2)-2 (2 x-1)

=9 x-6-4 x+2

=5 x-4

 

Multiplication of Polynomials

Multiplication of two polynomials is the same as multiplication of a monomial and a polynomial where the first polynomial is considered as one quantity.

To multiply (x+2) by (x-3), consider (x+2) as one quantity and apply the distributive law:

    multiplication of polynomials - distributive law

Then reapply the distributive law

=x2+2 x-3 x-6

=x2-x-6

Notice that each term of the second polynomial has been multiplied by each term of the first polynomial.

The same result can be obtained by arranging the polynomials in two rows and multiplying the upper polynomial by each term of the lower polynomial. Arrange like terms of the product in the same column so that addition is easier.+

      rearrange problem for multiplication of polynomial

applicable condition for distributive law problem

 

EXAMPLE Multiply (3 x-4)2

Solution (3 x-4)2 = (3 x-4) (3 x-4)

        rearrange polynomal expression for simplification

Condition for Polynomial problem Solution

Notes 1. (a+b)2 = a2+2 a b+b2

2. (a-b)2 = a2-2 a b+b2

3. (a+b) (a-b) = a2-b2

EXAMPLE Multiply (x2-2 x+1) by (2 x-3)

Solution  

      Another way of writing the polynomial multification expression

Therefore (x2-2 x+1) (2 x-3) = 2 x3-7 x2+8 x-3

EXAMPLE Perform the indicated operations and simplify

   (2 x-3) (x+4)-(x+2) (x-6)

Solution (2 x-3) (x+4)-(x+2) (x-6)

=(2 x2-+ 5 x-12)-(x2-4 x-12)

=2 x2+5 x-12-x2+4 x+12

=x2+9 x

 

Grouping Symbols

Grouping symbols, such as parentheses ( ), braces { }, and brackets [ ], are used to designate, in a simple manner, more than one operation.

When we write the binomial 3 a+5 b as (3 a+5 b), we are considering the sum of 3 a and 5 b as one quantity. The expression a-(b+c) means the sum of b and c is to be subtracted from a.

The statement, three times x minus four times the sum of y and z, can be written in algebraic notation as

3 x-4 (y+z)

Removal of the grouping symbols means performing the operations that these symbols indicate. Remove the symbols one at a time, starting with the innermost, following the proper order of operations to be performed.

EXAMPLE Remove the grouping symbols and combine like terms

        2 x-(5 x-2 y)+(x-6 y)

Solution 2 x-(5 x-2 y)+(x-6 y) = 2 x-5 x+2 y+x-6 y

  =(2 x-5 x+x)+(2 y-6 y)

  = 2 x-4 y

 

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EXAMPLE Remove the grouping symbols and combine like terms

        7 a+2 [2 b-3 (3 a-5 b)]

Solution 7 a+2 [2 b-3 (3 a-5 b)] = 7 a+2 [2 b-9 a+15 b]

  =7 a+4 b-18 a+30 b

  =34 b-11 a

 

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EXAMPLE Remove the grouping symbols and combine like terms

        6 a-{2 b+[3-(a+b)+(5 a-2)]}

Solution 6 a-{2 b+[3-(a+b)+(5 a-2)]}

=6 a-{2 b+[3-a-b+5 a-2]}

=6 a-{2 b+3-a-b+5 a-2}

=6 a-2 b-3+a+b-5 a+2

=(6 a+a-5 a)+( 2 b+b)+( 3+2)

=2 a-b-1

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It is sometimes necessary to group some of the terms of an expression. This can be accomplished by use of a set of parentheses.

When the grouping symbol is preceded by a plus sign. we keep the signs of the terms the same when it is preceded by a minus sign, we use the additive inverses (negatives) of the terms.

 

EXAMPLE Group the last three terms of the polynomial 3a -  5b + c -  2 with a grouping symbol in two ways, one preceded by a plus sign, the second preceded by a minus sign.

Solution

              How to use grouping symbols while using parentheses

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