Example: 2x-1=y,2y+3=x

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# Multiplication of Polynomials

**Multiplication of Polynomials**

**Deﬁnition and Notation**

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

3 x 4 = 4 + 4 + 4 three terms of 4

Similarly 5a = 5 * a = a + a + a + a + a ﬁve terms of a

4ab = ab + ab + ab + ab four terms of ab

ab=a x b = b + b +....+ b a terms of b

The following are some of the laws from

1. The commutative law of multiplication: ab = ba

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

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

=ab + ac

4. Multiplication of signed numbers:

+a)(+b) = +ab; (+a)(-b) = -ab

(-a)(+b) = -ab; (-a)(-b) = +ab

When we have 2 * 2 * 2 * 2. that is, four factors of 2, the notation 2^4 is used, which reads. “two to the power four," or “two to the fourth power."

Similarly, a * a * a * a * a = a^5 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 **

Note the difference between

(-2^4) = (-2)(-2)(-2)(-2) = +16

and -2^4 = -(2^4) = -(2 * 2 *2 *2) = -16

Note also 2a^3 = 2(a*a*a)

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

=(2*2*2)(a*a*a)

=2^3a^3 = 8a^3

Remark a , a^2 , a^3,.... are not like terms.

**EXAMPLES **1. 7a*a*a*a = 7a^4

2. -(-3)(-3)(-3)(-3) = -(-3)^4

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

4. -2^2*3^3 = -(2*2)*(3*3*3) = -4*27 = -108

5. 2^2 + 2^3 = 2*2 + 2*2*2 = 4 + 8 = 12

6. 2^3 - 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

a^3*a^5 = (a*a*a)(a*a*a*a*a)

=a*a*a*a*a*a*a*a

=a^8

=a^(3+5)

**THEOREM 1**

**Proof**

**EXAMPLES **1. 2^3 * 2^5 = 2^(3+5) = 2^8

2. a^2 * a^4 = a^(2+4) = a^6

3. -2^4 * 2^3 = -2^(4+3) = -2^7

4. -3x^3*x^2 = -3x^(3+2) = -3x^5

5. x^5*x = x^(5+1) = x^6

6. (a + 1)^2*(a + 1)^3 = (a + 1)^(2+3) = (a + 1)^5

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**Remark **2^3*2^7 = 2^(3+7) = 2^10 , and not 4^10

**Remark **2^4*3^5 = 2^4*3^5 , to find the product, multiply 2^4 =16 by 3^5 =243 ; that is 2^4*3^5 = (16)(243) = 3888

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

**EXAMPLES **1. (2ab^2)(3a^4bc^2)= (2*3)(a^1*a^4)(b^2*b^1)(c^2)

= 6a^5b^3c^2

2. -3b^2c^3)(8ab^3c) = (-3*8)(b^2*b^3)(c^3*c)(a)

=-24b^5c^4a

3. -3^2xy^2)(-5x^2y^3) = (-9xy^2)(-5x^2y^3)

=(-9)(-5)(x*x^2)(y^2*y^3)=45x^3y^5

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From the deﬁnition of exponents we have

(a^2)^3 = (a^2)(a^2)(a^2)

=(a^2*a^2)(a^2)

=a^(2+2)*a^2

=a^(2+2+2)

=a^(3*2) = a^(2*3)

=a^6

**THEOREM 2**

**Proof**

**EXAMPLES **1. (3^2)^4 = 3^(2*4) = 3^8

2. (a^3)^5 = a^(3*5) = a^15

3. (-3^2)^3 = -3^(2*3) = -3^6

4. (-a^3)^2 = a^(3*2) = a^6

**Note **2^3*2^4 = 2^(3+4)=2^7, While (2^3)^4 = 2^(3*4)=2^12

From the deﬁnition of exponents we have

6^4 = (2*3)^4 = (2*3)(2*3)(2*3)(2*3)

= (2*2*2*2)(3*3*3*3)

=2^4*3^4

**THEOREM 3**

**Proof**

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

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

Applying Theorem 3 repeatedly, we obtain

(abcd)^m=[(ab)(cd)]^m

=(ab)^m(cd)^m

=a^mb^mc^md^m

**Remark **21^2 = (3*7)^2 = 3^2*7^2 = 9 x49 = 441

**Remark **The quantity

(5 + 3)^2 = (8)^2 = 64, but 5^2 + 3^2 = 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 **

**(a^m)(b^n)^k** = [(a^m)(b^n)]^k

=(a^m)^k(b^n)^k

=a^mkb^nk

**EXAMPLE **Perform the following multiplication: (2x^2yz^3)(-4x^3y^2)

**Solution **(2x^2yz^3)(-4x^3y^2) = (2)(-4)(x^2*x^3)(y*y^2)(z^3)

= -8x^5y^3z^3

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**EXAMPLE **Perform the following multiplication: (5a^2b)^3

**Solution **(5a^2b)^3 = (5)^3(a^2)^3(b)^3 = 5^3a^6b^3 = 125a^6b^3

**EXAMPLE **Perform the following multiplication: -2^2a^3(ab^3)^2

**Solution **-2^2a^3(ab^3)^2 = -4a^3(a^2b^6) = -4(a^3*a^2)(b^6) = -4a^5b^6

**EXAMPLES **Perform the following multiplication: (3x^2y)^2(2xy^3)^3

**Solution **(3x^2y)^2(2xy^3)^3 = (3^2x^4y^2)(2^3x^3y^9) = (3^2*2^3)(x^4*x^3)(y^2*y^9)

=(9*8)x^7y^11 = 72x^7y^11

**Remark **Perform the outside exponents first

**EXAMPLE **Perform the following multiplication:

(-2ab^2)^2(-3a^2b)^3(-bc^2)^4

**Solution **(-2ab^2)^2(-3a^2b)^3(-bc^2)^4 = (-2)^2a^2b^2*(-3)^3a^6b^3*(-1)^4b^4c^8

=(-2)^2(-3)^3(-1)^4(a^2*a^6)(b^4*b^3*b^4)(c^8)

=(4)(-27)(+1)a^8b^11c^8

=-108a^8b^11c^8

**EXAMPLE **Perform the indicated operations and simplify:

(2ab)^4(-a^3b)^2 - (-3a^2)^3(a^2b^3)^2

**Solution **(2ab)^4(-a^3b)^2 - (-3a^2)^3(a^2b^3)^2 = (16a^4b^4)(a^6b^2) - (-27a^6)(a^4b^6)

=16a^10b^6 + 27a^10b^6

=43a^10b^6

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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** -a^2b^3, give that a = -3 and b = 2

**Solution **-a^2b^3 = -(-3)^2(2)^3 = -(9)(8) = -72

**EXAMPLE **Evaluate the expression b^2 - a^2(c^3 - b^3), give that a = -2 , b = 3, and c = -1

**Solution **b^2 -a^2(c^3 - b^3) = (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**

The extended distributive law of multiplication

is used to multiply a monomial by a polynomial

**EXAMPLE **Multiply 3x^2 + x - 2 by x

**Solution **x(3x^2 + x - 2) = x(3x^2) + x(x) + x(-2)

=3x^2 + x^2 - 2x

**EXAMPLE **Multiply x^2 - x + 4 by -2x^2

**Solution **(-2x^2)(x^2 - x + 4) = (-2x^2)(x^2) + (-2x^2)(-x) + (-2x^2)(4)

=-2x^4 + 2x^3 - 8x^2

**EXAMPLE **Multiply a^2b - 2b^2c + 5c^2a by 3a^2b

**Solution **3a^2b(a^2b - 2b^2c + 5c^2a)

=3a^2b(a^2b) + 3a^2b(-2b^2c) + 3a^2b(5c^2a)

=3a^4b^2 - 6a^2b^3c + 15a^3bc^2

**EXAMPLE **Multiply (3x - 2)/4 - (2x - 1)/6 by 12

**Solution **12/1[(3x - 2)/4 - (2x - 1)/6 = 12/1[(3x - 2)/4 - 12/1[(2x - 1)/6

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

=9x - 6 - 4x + 2

=5x - 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:

Then reapply the distributive law

=x^2 + 2x - 3x - 6

=x^2 - 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.+

**EXAMPLE** Multiply (3x - 4)^2

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

**Notes** 1. (a + b)^2 = a^2 + 2ab + b^2

2. (a - b)^2 = a^2 - 2ab + b^2

3. (a + b)(a - b) = a^2 - b^2

**EXAMPLE** Multiply (x^2 - 2x +1) by (2x - 3)

**Solution**

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

**EXAMPLE** Perform the indicated operations and simplify

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

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

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

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

=x^2 + 9x

**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 3a + 5b as (3a + 5b), we are considering the sum of 3a and 5b 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

3x - 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

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

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

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

=-2x - 4y

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**EXAMPLE **Remove the grouping symbols and combine like terms

7a + 2[2b - 3(3a - 5b)

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

=7a + 4b -18a +30b

=34b -11a

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**EXAMPLE **Remove the grouping symbols and combine like terms

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

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

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

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

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

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

=2a - 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 **