Chapter P

Prerequisites: Fundamental Concepts of Algebra

P.4: Polynomials

ObjectivesAt the end of this session, you will be able to:

Use the terminology of polynomials.Add and subtract polynomials.Multiply polynomials.Use FOIL method in polynomial multiplication.Use special products in polynomial multiplication.

Index1. Polynomials2. Adding and Subtracting Polynomials3. Multiplying Polynomials4. Summary

1. PolynomialsIn this section, we will first understand the different components of polynomials. Then, we will move on to adding, subtracting and multiplying the polynomials.Let us start by recalling a few definitions before we get to the concept of polynomials:Recall: Constants: A symbol having a fixed numerical value is called a constant.

Variable: A symbol that can take different numerical values is known as a variable. For example: The circumference of a circle is given by the formula, C = 2 r,

where r is the radius of the circle. Here 2, and are constants, whereas C and r are variables. Algebraic Expressions: A combination of variables and constants using the

symbols of addition, subtraction, multiplication or division, is called an algebraic expression.

Example: x – 6y, 6xy, x + 5,

Polynomials: An algebraic expression in which the variables have only non-negative integral powers, is called a polynomial.

In other words, a polynomial is a finite sum of terms where the exponents on the variables are non-negative integers. Example: 3x3 – 7x2 + 4x is a polynomial having the exponents for the variable x as 3, 2 and 1, which are non-negative integers.

5x2 – 6x2y + 8y – 2xy2 is also a polynomial where the exponents of the variables x and

y are non-negative integers.

x4

1. Polynomials (Cont…) Polynomial terms have variables with only whole number exponents. A polynomial term

cannot have variables with square roots of exponents or fractional exponents.

Here are few examples of algebraic expressions which are not polynomials:

To understand the polynomials better, let us look at a typical polynomial in one variable, x:

Observe the exponents on the terms. The first term has the exponent 2, the second term has the exponent 1 (x1= x), and the last term is a constant term.As shown above, the polynomials are usually written in a descending order, with the term having the highest exponent appearing as first. Then, the terms with the next highest exponents are written until we get the constant term.

4x2 + 3x - 7

Terms

6x-2 Not a polynomial term.

This has a negative exponent.

1/x2 Not a polynomial term.

This has a variable in the denominator.

x Not a polynomial term.

This has a variable inside the radical.

4x2 Polynomial term.

5x2 + 9x3/2 + 7

This is an algebraic expression but not a polynomial as it

contains a term 9x3/2, where 3/2 is not a non-negative integer.

Having understood the concept of polynomials, now we can state a formal definition of a polynomial:

Standard form of a Polynomial in one variable, x:

A polynomial in x is an algebraic expression of the form

anxn + an-1xn-1 + an-2xn-2 + … + a1x + a0

where an, an-1, an-2, … , a1 and a0 are real numbers, an 0.

n is a non-negative integer, an is called the leading coefficient, a0 is called the constant term.

NOTE: In this definition, the coefficients of the terms are represented by an (read as “a sub n”), an-1

(read as “a sub n minus 1”), an-2 and so on. The notations appearing on the lower right of each

coefficient a are called subscripts and are not exponents. Subscripts are used to distinguish one constant from another when a large and undetermined number of such constants are needed.

Degree of a Term: The degree of a term is the sum of the exponents of the variables included in the term. 

Examples: In the term 2xy, the exponents of both the variables x and y is 1. Therefore, the degree of the

term 2xy will be 2, which is the sum of the exponents of the variables x and y. The degree of the term 5y3 will be 3 because the exponent of the only appearing variable y is

3.

NOTE: The degree of a non-zero constant is 0. For example, in the polynomial 5x4 – 3x2 + 2x – 4, the degree of the non-zero constant term 4 is zero because 4 = 4 . 1 = 4 . x0

1. Polynomials (Cont…)

Degree of a Polynomial in One Variable: The degree of a polynomial in one variable is the highest power of the variable.

For example, 3x – 5 is a polynomial in x, of degree 1; 4y2 + 6y – 3 is a polynomial in y, of degree 2; 5x4 – 7x2 + 4 is a polynomial in x, of degree 4.

Degree of a Polynomial in Two Variables: For polynomials in more than one variable, the powers of the variables in each term are added and the highest sum so obtained is taken as the degree of the polynomial.

For example, 7x2y2 + 3xy + 2 is a polynomial in x and y, of degree 4. (Reason: In the term 7x2y2, the degree of x is 2 and the degree of y is 2, so their sum is 4; the degree to term 3xy is 2; degree of non-zero constant term 2 is 0. The degree of a polynomial is the highest degree of all the terms, hence the degree of the given polynomial is the degree of the term 7x2y2 , which is 4.)

Standard form of Polynomials: In standard form, a polynomial is written in the descending order as anxn + an-1xn-1 + an-2xn-2 + … + a1x + a0 .  In other words, the term having the highest degree is written first, the term with the next highest degree is written next, and so forth. 

A polynomial can have “missing” terms.  For example, the polynomial 3x5 – 5x3 + x – 10 starts with a degree of 5, but notice that there is no term with an exponent of 4.

Types of Polynomials:

1. Polynomials (Cont…)

Type Definition Example

Monomial

A polynomial with one term. (‘Mono’ implies one)

4x2

Binomial A polynomial with two terms (‘Bi’ implies two)

4x2 – 5y

Trinomial

A polynomial with three terms. (‘Tri’ implies three)

8x4 + 5x2 – 10

1. Polynomials (Cont…)Let us find the degree of the following polynomials and indicate whether the polynomial is a monomial, binomial, trinomial, or none of these: 

i. 10As the degree of the polynomial is the highest degree of all the terms, the degree of this polynomial is 0. In addition, there is only one term, therefore, this is a monomial.

ii. 8xy – 5x2yAs the degree of the polynomial is the highest degree of all the terms, the degree of this polynomial is 3. In addition, there are two terms in the polynomial, therefore, it is a binomial. Note: This polynomial is not written in the standard form (descending order). Therefore, we had to move to the second term to get the highest degree. You should be careful that you do not fall into the trap of thinking that the degree of the polynomial is always the degree of the first term.

7x3 – 2x + 4 In this polynomial, the term with the highest degree is 7x3, therefore the degree of the polynomial is 3. In addition, as there are three terms in the polynomial, so it is a trinomial.

Polynomials are sometimes named in terms of their degree: A polynomial of degree 1, such as 3x – 4 is called a “Linear Polynomial”. A polynomial of degree 2, such as x2 – 3x + 5 or ax2 + bx + c is also called

“Quadratic Polynomial”. A polynomial of degree 3, such as 3x3 – 4x + 2 is also called “Cubic Polynomial”.

2. Adding and Subtracting Polynomials

Polynomials are added or subtracted by combining like terms. Like terms are terms that have the same variables with exactly same exponents.

For example, 4x2 and 5x2 are like terms because each term has x² as a common variable with same exponent. As 4x2 and 5x2 are like terms, they can be combined by addition or subtraction by simply adding or subtracting their numerical coefficients. 

4x² + 5x² = 9x² Another example of like terms is 5a2b and 9a2b. For addition and subtraction,

you can only combine like terms in a polynomial.

Steps to be followed for adding or subtracting polynomials: STEP 1: Remove the parentheses, ().

If there is only a ‘+’ sign in front of ( ), then the terms inside ( ) remain the same when we remove the ( ).

If there is a ‘-’ sign in front of the ( ), then we distribute it by multiplying every term in the ( ) by a –1.

STEP 2: Group the like terms together. STEP 3: Combine like terms for addition or subtraction.

2. Adding and Subtracting Polynomials (Cont…)

Let us understand these steps with the help of an example.

Example: Perform the indicated operation and simplify:

1. (13x3 – 9x2 – 7x + 1) – (-7x3 + 2x2 – 5x + 9)

= 13x3 – 9x2 – 7x + 1 ++ 7x3 -- 2x2 ++ 5x –– 9 (Step 1: We distribute the ‘-’ sign by multiplying every term inside the second bracket by –1. Also minus . minus = plus)

NOTE: Be sure to change the sign of each term inside the parenthesis preceded by the negative sign.

= (13x3 + 7x3) + (– 9x2 - 2x2 ) + (– 7x + 5x)+ (1 – 9) (Step 2: Group the like terms together)

= 20x3 + (– 11x2) + (– 2x) + (- 8) (Step 3: Combine like terms)

= 20x3 – 11x2 – 2x – 8 (Simplify)

2. (3x5y + 7x3y – 10xy) - (-5x5y + 10x3y + 10xy)

= 3x5y + 7x3y – 10xy ++ 5x5y -- 10x3y -- 10xy (Step 1: We distribute the ‘-’ sign by multiplying every term inside the second bracket by –1. Also minus . minus = plus)

NOTE: Be sure to change the sign of each term inside the parenthesis preceded by the negative sign.

= (3x5y + 5x5y) + (7x3y - 10x3y) + (– 10xy -- 10xy) (Step 2: Group the like terms together)

= 8x5y + (- 3x3y) + (-20xy) (Step 3: Combine like terms)

= 8x5y - 3x3y – 20xy (Simplify)

In general, when multiplying two polynomials together, we use the distributive property until each term of one polynomial is multiplied with every term of the other polynomial. Then, we simplify the answer by combining all like terms.Recall: Distributive Property a(b + c) = ab + ac; a(b - c) = ab - ac Now, we will look at multiplication of some common types of polynomials to illustrate this concept.

1. Multiplication of a Monomial with a MonomialIn this case, there is only one term in each polynomial. In such multiplications, we simply multiply the two terms together using properties of exponents.   Example: (3x3) . (-2x4) = (3 . -2) . (x3 . x4) (Multiplying the coefficients and

using xm. xn = xm+n)   = -6x3+4 = -6x7 

2. Multiplication of a Monomial with a PolynomialIn this case, there is only one term in one polynomial and more than one term in the other. In such multiplications, we multiply the monomial with every term of the other polynomial by using the distributive property and the properties of exponents.

Example: (-5b) . (2b5 + 7b3 – 9b)= (-5b) . 2b5 + (-5b) . 7b3 - (-5b) . 9b (Using distributive property)

= -10 b5 + 1 - 35b3 + 1 + 45b1 + 1 (Using xm.xn = xm+n) = -10 b6 - 35b4 + 45b2

3. Multiplying Polynomials

3. Multiplication of a Binomial with a Binomial

In this case, both polynomials have two terms each. In such multiplications, we distribute both terms of one polynomial over both terms of the other polynomial. One way to keep track of the distributive property is to use the FOILFOIL MethodMethod, where the letters of the word FOIL represent the following:

FF represents the product of the first terms in each binomial.

OO represents the product of the outside terms.

II represents the product of the inside terms.

LL represents the product of the last terms in each binomial.

In other words, use the distributive  property for every term in the first binomial.

Example: (x + 7) . (x + 3)

F O I LF O I L

(x + 7) . (x + 3) = (x).(x) + (x).(3) + (7).(x) + (7).(3) (Using FOIL method)

= x1+1 + 3x + 7x + 21 (Simplifying)

= x2 + (3x + 7x) + 21 (Grouping like terms)

= x2 + 10x + 21

NOTE: The FOIL method only works for the multiplication involving two binomials.

3. Multiplying Polynomials (Cont…)

FF First Terms

OO Outside terms

II Inside terms

LL Last Terms

Outside Terms

Inside Terms

First TermsLast Terms

Special Cases in Multiplication of Two Binomials: i. Square of a Binomial Sum and Square of a Binomial Difference:

Let us find (a + b)2, the square of a binomial sum. We begin by using the FOIL method and then arrive at a general rule. (a + b)2 = (a + b)(a + b) F O I L

= (a)(a) + (a)(b) + (b)(a) + (b)(b) (Using FOIL method) = a2 + ab + ab + b2 (Simplifying) = a2 + 2ab + b2 (Combining like terms)

This gives us the following special product rule for the square of a binomial sum. Similarly, we can have the special product rule for square of a binomial difference:

Square of a Binomial Sum:(a + b)2 = a2 + 2ab + b2

Square of a Binomial Difference:(a - b)2 = a2 - 2ab + b2

Whenever you have a binomial squared, you can use this shortcut method to find the product.

Let us look at some examples solved using the special product rule:Example 1: (x + 10)2 = x2 + 2 . x . 10 + (10)2 (Using (a + b)2 = a2 +

2ab + b2) = x2 + 20x + 100 (Simplifying)Example 2: (x – 4y)2 = x2 - 2 . x . 4y + (4y)2 (Using (a - b)2 = a2 - 2ab + b2) = x2 - 8xy + 16y2 (Simplifying)

3. Multiplying Polynomials (Cont…)

NOTE: The square of a sum is not equal to the sum of squares. That is, (a + b)2 a2 + b2

As per the special products rule (a + b)2 = a2 + 2ab + b2 , the middle term 2ab is missing.

ii. Multiplying the Sum and Difference of Two Terms:We can use the FOIL method to multiply (a + b) and (a – b) as follows:

F O I L (a + b)(a - b) = (a)(a) + (a)(-b) + (b)(a) + (b)(-b) (Using the FOIL method)

= a2 - ab + ab - b2 (Simplifying) = a2 - b2 (Combining like terms)

This FOIL multiplication provides us with a quick rule for multiplying the sum and difference of two terms. This is another special products rule. 

The Product of the Sum and Difference of Two Terms: (a + b)(a - b) = a2 - b2

Let us solve an example using the above special product rule:Example : (10x + 3)(10x - 3) = (10x)2 – (3)2 (Using (a + b)(a - b) = a2 - b2)

= 100x2 – 9 (Simplifying)

3. Multiplying Polynomials (Cont…)

4. Multiplication of a Polynomial with a Polynomial

As mentioned before, use the distributive property until every term of one polynomial is multiplied by every term of the other polynomial.  Then, simplify the answer by combining any like terms.

Example: (5x - 2)(3x2 - 5x + 4)

= (5x - 2)(3x2 - 5x + 4)

= 5x . (3x2 - 5x + 4) + (-2) . (3x2 - 5x + 4) (Multiplying the trinomial with each term of the binomial)

= 5x . 3x2 - 5x . 5x + 5x . 4 + (-2) . 3x2 - (-2) . 5x + (-2) . 4 (Using distributive property)

= 15x2+1 – 25 x1+1 + 20x – 6x2 + 10x – 8 (Multiplying monomials; Using (Minus. Minus) = Plus)

= 15x3 – 25x2 + 20x – 6x2 + 10x – 8 (Simplifying)

= 15x3 – 25x2 – 6x2 + 20x + 10x – 8 (Combining like terms)

= 15x3 – 31x2 + 30x – 8

     

3. Multiplying Polynomials (Cont…)

Special Product Rule for Cubing a binomial :Now let us find the special product rule for cubing a binomial:(a + b)3 = (a + b)2 + 1 (Using xm . xn = xm+n)

= (a + b)2 . (a + b) = (a2 + 2ab + b2) . (a + b) (Using (a + b)2 = a2 + 2ab +

b2) = (a + b) . (a2 + 2ab + b2) = a . (a2 + 2ab + b2) + b . (a2 + 2ab + b2) (Using distributive property) = a . a2 + a . 2ab + a . b2 + b . a2 + b . 2ab + b . b2 (Using distributive property) = a3 + 2a2b + ab2 + ba2 + 2ab2 + b3

= a3 + (2a2b + ba2 )+ (ab2 + 2ab2) + b3 (Grouping like terms) = a3 + 3a2b + 3ab2 + b3 (Combining like terms)

Thus, we have the following special product rule for cubing a binomial:

Special Product Rule for Cubing a Binomial: (a + b)3 = a3 + 3a2b + 3ab2 + b3

(a - b)3 = a3 - 3a2b + 3ab2 - b3

Whenever you have a binomial cubed, you can use this shortcut method to find the product.

3. Multiplying Polynomials (Cont…)

3. Multiplying Polynomials (Cont…)Now let us solve some examples using the special product rule for cubing a binomial. Example: (2x + 3)3

(2x + 3)3 = (2x)3 + 3(2x)2(3) + 3(2x)(3)2 + (3)3 (Using (a + b)3 = a3 + 3a2b + 3ab2 + b3)

= 23 x3 + 3(22 x2)(3) + 3(2x)(3)2 + (3)3 (Using (xy)n = xn . yn)

= 8x3 + 3(4x2)(3) + 3(2x)(9) + (27) (23 =8, (3)2=9, (3)3=27 )

= 8x3 +36x2 + 54x + 27

Example: (3x - 4)3

(3x - 4)3 = (3x)3 - 3(3x)2(4) + 3(3x)(4)2 - (4)3 ( Using (a - b)3 = a3 - 3a2b + 3ab2 - b3)

= 33 x3 - 3(32 x2)(4) + 3(3x)(4)2 - (4)3 (Using (xy)n = xn . yn)

= 27x3 - 3(9x2)(4) + 3(3x)(16) - (64) (33 =27, (4)2=16, (4)3=64 )

= 27x3 - 108x2 + 144x – 64

4. SummaryLet us recall what we have learnt so far:

Polynomials: A polynomial is an algebraic expression in which the variables involved have only non-negative integral powers.

Degree of a Term: The degree of a term is the sum of the exponents on the variables contained in the term.

Degree of a Polynomial: The degree of the polynomial is the largest degree of all its terms.

Some types of polynomials: Monomial – A polynomial with one term. Binomial – A polynomial with two terms. Trinomial – A polynomial with three terms.

FOIL Method:

Outside Terms

Inside Terms

First TermsLast Terms

(ax + b) . (cx + d) = ax . cx + ax . d + b . cx + b . d

4. Summary (Cont…)Special Products:

Sum and Difference of Two Terms: (a + b) . (a - b) = a2 – b2

Squaring a binomial: (a + b)2 = a2 + 2ab + b2

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

Cubing a binomial: (a + b)3 = a3 + 3a2b + 3ab2 + b3

(a - b)3 = a3 - 3a2b + 3ab2 - b3