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The Four Operations on Polynomials
Unit 1
MTH-3001-2
William Low
Value of the Coefficient and the Exponent of a Given Base
A variable is a mathematical symbol which can be assigned any one of a set of values.
A monomial is an algebraic expression composed of the product of a numerical coefficient and variables.
Examples :
1.
2.
2t3 is a monomial in t ; t is the variable , 2 is the numerical coefficient and 3 is the exponent.
1 2 3
1
x y is a monomial in x and y ; x and y are variables,  is the numerical coefficient
2
2
2
3
and and are exponents.

3.
What is the numerical coefficient of mc 2 ?
4.
What is the numerical coefficient of the monomial  y 4 ?
5.
What is the numerical coefficient of the monomial 
3 3 4
x y ?
4
The set of rational numbers, designated Q , is composed of the natural numbers N , the integers Z , and
the fractions.
A numerical coefficient is a number which multiplies the letters (variables and constants) in an algebraic
expression.
In the algebraic expression y = 5x3 , x is the independent variable and y is the dependent variable , the
number 3 attached to the variable x in a raised position ( 3 ) is called an exponent and in this respect the
variable x is a base raised to the third power ; x3 means x × x × x ; that is , the exponent attached to a
quantity indicates how many times this quantity is taken as a factor.
Examples :
2
−2 × (−2) × (−2) = −2
3
3 3 3
32
    2
5 5 5
5
7  7  7  7  7 7 5
 3
10  10  10
10
3 2 4 3
3
x y z ,  is the numerical coefficient, x , y , and z are the bases, the exponent of
2
2
the base x is 2, the exponent of y is 4, and the exponent of z is 3.
In the monomial 
Two or more monomials are similar if they are composed of the same variables , respectively affixed
with the same exponents , regardless of their numerical coefficients.
Consider the monomials
12 x 3 y 2 z 3 ,  x 3 y 2 z 3 , and 5 x 3 y 2 z 3 . These monomials are composed of
the same variables x , y , and z respectively raised to the same power, that is, x3 , y2 , and z3. These
monomials are therefore similar monomials. Only the numerical coefficients are different.
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The Four Operations on Polynomials
Unit 2
MTH-3001-2
William Low
Monomials , Binomials , Trinomials and Polynomials
The product of a number and one or more variables affixed with an exponent is called a term.
3 3 4 2
x y z is a term. The binomial 3xyz  5x 3 z 2 consists of two terms, 3xyz is one term
5
and 5 x 3 z 2 is the other. The trinomial 2 x 2  3 x  7 consists of three terms, 2x 2 is one term, 3x is the
second term and 7 is the third term.
The monomial
A binomial is an algebraic expression containing two terms.
A trinomial is an algebraic expression containing three terms.
A polynomial is an algebraic expression containing many terms.
A polynomial in one or more variables is an algebraic expression composed of a term or a sum of terms.
It is standard practice to order the terms of a polynomial according to the increasing or decreasing
powers of a chosen variable.
Unit 3
Adding and Subtracting Two Polynomials
The sum of two similar monomials always yields a similar monomial.
7 x 3 y 4 z 2 + 13x 3 y 4 z 2 = 20 x 3 y 4 z 2
Addition rules for two integers :
1.
When two numbers have the same sign, they are added and the result keeps the sign of these two
numbers.
2.
When two numbers have opposite signs, the difference between them is calculated and the sign of
the number that is further from point 0 on the number line is kept i.e. the sign of the larger number
is kept.
To add two or more similar monomials , calculate the sum of the numerical coefficients of the given
monomials. This sum becomes the new numerical coefficient of the similar monomial.
3xy 2 z  5xy 2 z  9 xy 2 z  17 xy 2 z
The difference between two similar monomials always yields a similar monomial.
21x 3 y 2 z 5  5x 3 y 2 z 5  16 xy 2 z
Find the difference between – 5x2 and 3x2.
– 5x2 − (+ 3x2) = – 5x2 − 3x2 = (−5 −3) x2 = −8x2
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The Four Operations on Polynomials
MTH-3001-2
William Low
Calculate the difference between – 3a3 b2 and – 4a3 b2
– 3a3 b2 − (– 4a3 b2)
i.e.
1.
Change the coefficient of the monomial to be subtracted to its opposite :
– 4a3 b2 becomes 4a3 b2.
2.
Then perform an addition :
– 3a3 b2 − (– 4a3 b2) = – 3a3 b2 + 4a3 b2) = (– 3 + 4)a3 b2 = a3 b2
To subtract two numbers :
1.
Find the opposite of the number to be subtracted (the one located immediately to the right of the
subtraction sign) and change the subtraction to addition.
2.
Calculate the sum of the two numbers by following the rules for addition.
To subtract two or more similar monomials , calculate the difference between the numerical
coefficients of the given monomials. This difference gives the new numerical coefficient of the similar
monomial.
13xy 2 z  5xy 2 z  4 xy 2 z  3xy 2 z  7 xy 2 z
Unit 4
Multiplying Two Polynomials
An exponent is a mathematical symbol which indicates the power of a number, that is, the number of
times a number is multiplied by itself.
35 = 3 × 3 × 3 × 3 × 3
x7 = x × x × x × x × x ×x × x
y × y ×y × y = y4
The product of several powers of the same algebraic quantity is calculated by finding the sum of its
exponents.
35 × 37 = 35+7 = 312
x3 × x2 × x5 = x3+2+5 = x10
am × an = x m+n
5a3 × 7a2 = 5 × 7 × a3 × a2 = 35 × a3 × a2 = 35 × a3+2 = 35a5
Remember the commutative property of the multiplication numbers is that the product of these two or
more numbers is independent of the order of the factors.
To calculate the product of two or more monomials :
1.
Calculate the value of the product of the numerical coefficients of these monomials.
2.
Write the different variables by assigning them an exponent equal to the sum of their respective
exponents.
Calculate the product of 3x2y3 × 2x3y5z
3x2y3 × 2x3y5z = 3 × 2 × x3 × x2 × y3 × y5 × z = 6 × x3+2 × y3+5 × z = 6 × x5 × y8 × z = 6x5y8z
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The Four Operations on Polynomials
The law of signs :
+ times +
+ times −
− times +
− times −
gives
gives
gives
gives
MTH-3001-2
William Low
+
−
−
+
The product of two or more monomials is a monomial whose numerical coefficient is equal to the
product of the numerical coefficients of these monomials and whose different variables have as an
exponent the sum of their respective exponents.
The distributive property of multiplication over addition is the property whereby the number or the
variable located in front of a set of parenthesis can multiply each of the terms in this set of parenthesis
without changing the value of the expression. Regardless of how the multiplication is performed, the
distributive property applies.
5(3 + 7) = 3 × (2 + 7) = (3 × 2) + (3 × 7) = 6 + 21 = 27
2x(6x + 2y – 5) = (2x × 6x) + ( 2x × 2y) + (2x × (–5)) = 12x1+1 + (4 × x × y) + (–10x) = 12x2 + 4xy – 10x
To multiply a polynomial by a monomial , multiply each term of the polynomial by the monomial.
1 3 2 2 3
1
x  x  x 
3 7
5
4
 1 3 2 2   1 3 3   1 3  1   2 5   1 4   1 3  2 5 1 4 1 3
x  x  x
 x  x    x  x    x        x    x     x  
7  3
5  3
5
12
3
 4    21   5   12  21
The multiplication of a binomial by a binomial is performed by applying the distributive property
twice, that is, by multiplying the terms in the first polynomial by each of the terms in the second
polynomial.
Example :
(2x + 3)(3x + 2) = (2x + 3) × (3x + 2) =
2x × (3x + 2) + 3 × (3x + 2) = 2x × 3x + 2x ×2 + 3 × 3x + 3 × 2 = 6x2 + 4x + 9x + 6 = 6x2 + 13x + 6
Two terms are similar if they are composed of the same variables, raised respectively to the same
exponent regardless of the value of the numerical coefficient.
Example :
(4x – 2)(2x + 4) = 4x × (2x + 4) + (−2) × (2x + 4) =
4x × 2x + 4x × 4 + (−2) × 2x + (−2) × 4 = 8x2 + 16x – 4x – 8 = 8x2 + 12x – 8
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The Four Operations on Polynomials
MTH-3001-2
William Low
Example :
(4x2 – y2)(3x3 + 3y) = 4x2 ×3x3 + 4x2 × 3y + (– y2) × 3x3 + (– y2) × 3y = 12x5 + 12x2y − 3x3y2 – 3y3
Other methods of calculating the product of two binomials are by the method cross-multiplication (the
binomials are place under each other , then the two terms on the left are multiplied followed by the two
terms on the right , then the extremes of the two binomials are cross-multiplied and lastly the four
resulting terms are ordered according to decreasing powers of the variable) or by the method of foil.
Multiply the binomials
(4x2 + 8)(3x – 7)
4x2
+
8
3x
–
7
2. Multiply the two terms on the left and the two terms on the right
12x3
and
56
3. Then cross-multiply the two extremes of the two binomials
4x2
+
8
3x
–
7
24x
and
28x2
1. Place the binomials under each other
×
×
4, You obtain
12x3 − 56 + 24x – 28x2
5. Lastly, order the terms according to increasing powers of x
12x3– 28x2 + 24x − 56
Multiply the binomials
(3x3 + 2x)(2x2 + 3)
3x3
+
2x
2x2
+
3
2. Multiply the two terms on the left and the two terms on the right
6x5
and
6x
3. Then cross-multiply the two extremes of the two binomials
3x3
+
2x
2x2
+
3
4x3
and
9x3
1. Place the binomials under each other
×
×
4, You obtain
6x5 + 6x + 4x3 + 9x3
5. Lastly, order the terms according to increasing powers of x
6x5 +13x3 + 6x
Multiply the binomials
(4y3 + 2)(2y2 + 3)
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The Four Operations on Polynomials
MTH-3001-2
William Low
The square of a binomial is equal to the sum of the squares of each of its two terms increased or
decreased by double the product of its terms.
(a ± b)2 = a2 ± 2ab + b2
Unit 5
Dividing a Polynomial by a Monomial
Note : To divide one number by another , you can multiply the first number by the multiplicative inverse
of the second number.
12  4  12 
1
4
6y3  2y2  6y3 
1
2y2
To determine the quotient of several powers of the same algebraic number , simply calculate the
difference between these exponents.
57 ÷ 53 = 57 – 3 = 54
y9 ÷ y5 = y9 – 5 = y4
am
 a mn if m  n and a  0
n
a
Laws of exponents :
1.
To multiply monomials , add the exponents of the same variables.
2.
To divide monomials , subtract the exponents of the same variables.
3.
Note that x0 = 1 and that x1 = x
Examples :
(1).
1
1 4 1 18 a 4
18a  3a  18a  2  18   a  2   2  6a 42  6a 2
3
3 a
3a
a
(2).
1 3
x
 1  1 
 1  3 
1 3  1  2
3
x   x 
       x 31        x 2   x 2
2
2
 3   1 x  2  3 
 2  1 
3
(3).
14 x 2 y 3 z 4  7 xy 2 z 3 
(4).
 28a 4 b 3 c  4ab 
4
2
4
14 x 2 y 3 z 4 14 x 2 y 3 z 4
 
 2 x 21 y 32 z 43  2 xyz
7
7 xy 2 z 3
xy 2 z 3
 28a 4 b 3 c  28 a 4 b 3 c


 7a 41b 31c  7a 3b 2 c
4ab
4
ab
The quotient of two monomials is a monomial whose numerical coefficient is equal to the quotient of the
numerical coefficients of these monomials and whose different variables have as their exponents the
difference between their respective exponents.
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The Four Operations on Polynomials
MTH-3001-2
William Low
The division of a polynomial by a monomial is performed by dividing each term of the polynomial by
the given monomial.
Examples :
 8 x 4   16 x 3   8
  16

(1). 8 x 4  16 x 3  8x 2   2    2    x 42    x 32   1x 2  2 x1  x 2  2 x
 8

 8x   8x   8


   
(2).
9x
2
 9 x 2 y 2   36 x 2 y   9
  36

y 2  36 x 2 y  3x 2 y   2    2    x 22 y 21    x 22 y 11   3x 0 y  12 x 0 y 0  3 y  12
  3

 3x y   3x y   3
Unit 6

The Four Operations on Polynomials
The property of the order of operations :
1.
Operations in parenthesis or brackets in a problem must be performed first.
2.
Multiplication and division have priority over (are done before) addition or subtraction and they are
performed from left to right.
3.
Additions and subtractions are done in the order in which they appear.
Note : We can add or subtract two terms of a polynomial if and only if these two terms are similar , that
is, when they are composed of the same variables respectively raised to the same exponents. Regardless of
the value of their numerical coefficient.
An irreducible ordered polynomial is a polynomial which is expressed according to the decreasing
powers of a selected variable (ordered) and which does not contain any similar terms (irreducible).
As a general rule , polynomials are ordered according to the decreasing powers of the selected variable.
Arbitrarily , this variable is the one whose letter-symbol occurs first in the alphabet.
To simplify an arithmetic expression containing polynomials :
1.
Simplify expressions in parenthesis , brackets or both , if similar terms are present.
2.
Perform the multiplications and divisions of the polynomials in the order in which they appear.
3.
Regroup similar terms and add or subtract them as indicated.
4.
Order the polynomial obtained according to the decreasing powers of the variable whose lettersymbol occurs first in the alphabet.
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The Four Operations on Polynomials
MTH-3001-2
William Low
Examples :
(1).

25 x 3 y 5
 3 y 2 6 xy 2  2 x 2  2 xy 2
2
5x y




25 x 3 y 5
25 x 3 y 5
2
2
2

3
y
8
xy

2
x

 24 xy 4  6 x 2 y 2  5 xy 4  24 xy 4  6 x 2 y 2  6 x 2 y 2  19 xy 4
2
2
5x y
5x y



2
(2). 2 x 3  2  3x 2 4 x 2  4






 2 x 3  2 2 x 3  2  3x 2 4 x 2  4  4 x 6  8x 3  4  12 x 4  12 x 2  4 x 6  12 x 4  8x 3  12 x 2  4
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