PolynomialsBy

Dr. Julia Arnold Tidewater Community College

Copyright 10/19/2002

Introduction

What are Polynomials?

A polynomial in x consists of a finite number of terms of the form axn where a can be any Real number but n must be a whole number. (Recall a term is any algebraic expression separated from another algebraic expression by “+” or “-” signs. Whole numbers are {0,1,2,3,4,...})

The following are examples of polynomials:

2x4 A one term polynomial is called a monomial.

-5x6 + 7.9x2 A two term polynomial is called a binomial.

2x2 + 3x - 1 A three term polynomial is called a trinomial. The following are not polynomials:3x-2- 4x-1 + 2 is not a polynomial because the exponents on the variables are not whole numbers.

19y1/2 + 5 is not a polynomial because the exponent on the variable is not a whole number.

The following are examples of polynomials in more than one variable:

2x4 y2 is a monomial in x and y

-5x6yz + 7.9x2yz2 is a binomial in x, y & z2x2w + 3xw2 - w3 is a trinomial in x and w

The polynomial 7x4 - 3x2 + 4x3 - 9x +5 has 5 terms. 1 2 3 4 5

How many terms does the following polynomial have?

7x4 - 3x2 + 4x3 - 9x +5

Descending PowersWriting a polynomial in descending

powers means to begin with the term having the largest exponent on the variable and then proceeding to the lowest.

For example: - 3x2 + 4x3 - 9x + x4 +5 would

be written x4 + 4x3 - 3x2 - 9x + 5

Degree of a TermThe degree of a term is the sum of the exponents on all variables.

For example: the degree of 5x2y3z is (2 + 3 + 1) or 6

For the polynomial x4 + 4x3 - 3x2 - 9x + 5 the degree of each term from left to right is4, 3, 2, 1, and 0.

The constant 5 is equal to 5x0, thus it has degree 0.

Degree of a Polynomial

The degree of a polynomial is the largest degree of any one term. Thus in the preceding polynomial, x4 + 4x3 - 3x2 - 9x + 5, the degree would be 4.

What is the degree of

x7 + 4x8 - 3x9 - 9x3 + 5 ?

The degree of x7 + 4x8 - 3x9 - 9x3 + 5 is 9

the highest degreed term.

What are Like Terms?

Like terms are terms with the same variables raised to the same powers.For example:5x2y3 is like -4y3 x2 but is not like 5y3x

x is like .35x but is not like x2

Which of the following pairs are pairs of like terms?

(A) 3xy and 2yz (B) -2xyz and 5xyz (C) 3x2 and 4 x3

The answer is (B) -2xyz is like 5xyz because both have the same variables raised to the same powers or exponents.

To add polynomials(1) remove the grouping symbols, (2) find the like terms of the polynomial, and

then(3) add the numerical coefficients of the like

terms.[Note: the numerical coefficient is the number

with the variables; i.e. 3xyz has numerical coefficient 3, -5x2 has numerical coefficient -5, and x has numerical coefficient 1]

Adding Polynomials

Example 1: (2x +3) + (5x - 6)First remove grouping symbols 2x + 3 + 5x - 6

Next find the like terms 2x + 5x + 3 - 6

Add numerical coefficients (2+5)x + (3 - 6) = 7x - 3

[Question: Does 2x + 5x = (2 + 5)x remind you of a property stressed earlier in the course? ]

The distributive property:

a(b + c) = ab + ac

Example 2: (5x2 + 8x - 7) + (-9x3 - 8x2 - 7x + 3) [Remember to remove the grouping symbols, multiply by whatever number is in front of the grouping symbols, using the distributive property. That number in this problem is 1 for both polynomials. 1(5x2 + 8x - 7) + 1(-9x3 - 8x2 - 7x + 3)]5x2 + 8x - 7 - 9x3 - 8x2 - 7x + 3Combining like terms-9x3 - 3x2 + x - 4

Adding in ColumnsIf you prefer, you can use the column method of adding polynomials. Like terms are placed under each other.Example 3: Add (-10x4 + 8x2 - 1) and ( 2x4 - 5x2 + 4x + 3)Write -10x4 + 8x2 - 1 2x4 - 5x2 + 3 + 4x like terms under each other

-8x4 + 3x2 + 2 + 4x add columns

Subtracting Polynomials

To subtract polynomials you must remove thegrouping symbols by multiplying the firstexpression by 1 and the second expression by -1.Example 1: (2x +3) - (5x - 6)

1(2x + 3) -1 ( 5x - 6) First remove grouping symbols 2x + 3 - 5x + 6 (Note: Multiplying by -1 causes the signs to change in your expression.)

2x - 5x and 3 + 6 Add the like terms -3x + 9 is the result.

Example 2: (8x2 - 2x - 5) - (x3 - 9x2 - 2x + 5)1(8x2 - 2x - 5) - 1(x3 - 9x2 - 2x + 5)Remove grouping symbols8x2 - 2x - 5 - x3 + 9x2 + 2x - 5Add like terms- x3 + 17x2 -10 (the answer)

Subtracting in ColumnsBe very careful when using this method. You must make sure you change all the signs of the polynomial being subtracted.Example 3: Subtract 5x3 - 3x -10 from 8x3 - 2x 8x3 - 2x

8x3 - 2x -5x3 + 3x + 10

3x3 + x + 10

-1(5x3 - 3x -10) becomes

Multiplication of Polynomials

Multiplying a Monomial by a Monomial

Example 1: (-2x6)(3x4) =-2 . 3. x6 . x4 = -6x(6+4)

= -6x10

Example 2: (10x2y)(3x9y2)Write the answer before you click your mouse.

10 . 3 . x2 . x9 . y1 . y2

= 30x(2+9)y(1+2) = 30x11y3

Example 3: (-4x7y0)(-9x0yz3) =

-4 . -9 . x7 . x0 . y0 . y1 . z3

36x(7+0)y(0+1)z3

36x7yz3

Write the answer before you click your mouse.

Multiplying a Monomial by a Polynomial

Example 1: Distribute -2x thru parenthesis to each term -2x ( x2 - 3x + 9) = -2x(x2) - (-2x)3x + (-2x)9 =

-2x3 + 6x2 - 18x

Example 2: 3a2(-2a3 + 8a - 10) =

3a2(-2a3) + 3a2(8a) + 3a2(-10) = -6a5 + 24a3 - 30a2

Write your answer before you click your mouse.

Example 3: (5x2 - 4x + 6) (3x) =

5x2 (3x) - 4x (3x) + 6 (3x) = 15x3 - 12x2 +18x

Write your answer before you click your mouse.

Example 4: (-3x4 - 5x2 + 1) (-3x2)

= -3x4 (-3x2) - 5x2 (-3x2) + 1 (-3x2) 9x6 + 15x4 - 3x2

Example 5: -2a( a3 + a2 - a + 4) =

-2a( a3) + -2a (a2 ) -2a( - a) + -2a(4) = -2a4 - 2a3 + 2a2 - 8a

Write your answer before you click your mouse.

Write your answer before you click your mouse.

Multiplying a Binomial by a Binomial

To multiply two binomials together we use an acronymcalled FOIL to help us remember the products.

F stands for first. In the problem (x + 4)(2x -5) The first terms are x and 2x Their product is 2x2

O stands for outside. (x + 4)(2x -5) The outside terms are x and -5

More on the next slide.

Multiplying a Binomial by a Binomial Continued

FOIL stands for First, Outside, Inside, Last.

I stands for inside. (x + 4)(2x -5) The inside terms are 4 and 2x Their product is 8x

L stands for last. (x + 4)(2x -5) The last terms are 4 and -5Their product is -20

More on the next slide.

Putting all the products together we get:(x + 4)( 2x - 5) = 2x2 - 5x + 8x - 20 F O I LCombining like terms the final answer is 2x2 + 3x - 20

Example 1: Multiply (3y - 7)(5y - 6)First 3y(5y) = 15y2

Outside 3y(-6) = - 18yInside -7 (5y) = - 35yLast -7 ( -6) = +42Answer is 15y2 - 18y - 35y + 42Final Answer is 15y2 - 53y + 42

Do you see that each term of the first polynomial is multiplied by each term in the second polynomial?

Example 2: Multiply (a + b)(c + d)Distribute a thru (c + d) a(c + d) = ac

Final Answer is ac + ad + bc + bd

First

+ ad

OutsideThen distribute b b(c + d) =bc +

bdInside Last

If you understand this basic premise: that each term of the first polynomial is multiplied by each term in the second polynomial, then it will be an easy transition to multiply polynomials containing more than two terms.

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

Because the second polynomial is not a binomialwe cannot use FOIL. Instead multiply 2x by ( 4x2 - 3x -2) and then multiply 3 by ( 4x2 - 3x -2).The result is 2x( 4x2 - 3x - 2) = 8x3 - 6x2 - 4x then 3( 4x2 - 3x -2) = 12x2 - 9x -6

Now add down: 8x3 +6x2 - 13x -6

Division of Polynomials

There are two types of division techniques. The first kind that will be illustrated is division by a monomial.The second kind is for division by any other type of polynomial.

Occasionally, monomial division produces some unexpected answers. If you try to use the “second method” for dividing by a monomial, you may find yourself unable to complete the task.

Division by a Monomial Divisor

Example 1: Divide (3x4 - 5x3 +7x - 8) by 5x2

Write each term of the dividend as a fraction with adenominator of 5x2.Simplify each fraction to...3x4 - 5x3 + 7x - 8 = 3x2 - x + 7 - 8 5x2 5x2 5x2 5x2 5 5x 5x2

Example 2: 9x3 - 4x2 + 8x - 6 3xWrite 9x3 - 4x2 + 18x - 6 = 3x2 - 4x + 6 - 2 3x 3x 3x 3x 3 3x

Divisor Dividend

X+3 X2 +0X + 12

Divide (12 + X2 ) by (X + 3)In a long division problem you must follow two set-up rules. 1) The dividend must be arrangedin descending powers. Thus12 + X2 must be written asX2 + 12.

2) If there are any missing exponentsin your dividend , you make space for them by adding a zeroterm.

Division by a Polynomial with 2 or more terms.

Quotient

X+3 X2 + 5X + 12

Divide the first term X2 by thefirst term in the divisor, X. Writethe result above 5X.

Multiply X by the divisor X + 3 and write the answer below the dividendmatching like terms as you go.

Subtract the bottom line by changing the signs of the bottomline you just wrote. When finishedbring down the next term, which is12.

We are not finished yet so continue onto the next slide!

Example 1: Divide (X2 + 5X + 12) by ( X + 3)

Set up the long divisionproblem.

X2 + 5X + 12X2 + 3X

X+3X

X2 + 5X + 12-X2 - 3X 2X + 12

X+3X

2X + 6

Divide the first term 2X by the firstterm X. The answer is 2. Write 2 above the 12. (click mouse)

Multiply 2(X + 3) = 2X + 6Write answer below 2X + 12.(click mouse)Subtract by changing signs.(Click mouse twice)

X2 + 5X + 12-X2 - 3X 2X + 12 -2X - 6 6

X + 3X + 2 + 6

X+3

Write the final answer with the remainder in the form below.

X2 + 5X + 12-X2 - 3X 2X + 12

X+3X + 2

- -

6

The remainder is 6.

X- 2 X2 + 0X - 5

Divide the first term X2 by thefirst term in the divisor, X. Writethe result above 0X.

Multiply X by the divisor X - 2 and write the answer below the dividendmatching like terms as you go.

Subtract the bottom line by changing the signs of the bottomline you just wrote. When finishedbring down the next term, which is-5

We are not finished yet so continue onto the next page!

Example 2: Divide (X2 - 5) by ( X - 2)Set up the long divisionproblem.

X2 + 0X - 5X2 - 2x

X- 2X

X2 + 0X - 5-X2 + 2X +2X - 5

X- 2X

Divide the first term 2X by the firstterm X. The answer is 2. Write +2 above the -5.

Multiply 2(X - 2) = 2X - 4Write answer below 2X - 5.

Subtract by changing signs.The remainder is -1.

Write the final answer in theform on left.

X2 + 0X - 5-X2 + 2X > 2X - 5 2X - 4

X- 2X + 2

X2 + 0X - 5-X2 - 2X 2X - 5 -2X + 4 -1

X - 2X + 2 + -1

x - 2

- +-1

8X3 + 0X2 + 0X - 12X + 1 4X2

We are not finished yet so continue onto the next page!

Example 3: Divide 8X3 - 1 2X + 1Set up the long division problem before you click the mouse.

Step 1: Divide 8X3 by 2XWrite answer before you click.

Step2: Multiply 4X2 by divisor. Write answer before you click.

8X3 + 0X2 + 0X - 18X3 + 4X2

2X+1 4X2

8X3 + 0X2 + 0X - 12X + 1

2X + 1 8X3 + 0X2 + 0X - 18X3 + 4X2

Repeat Steps 1 - 3 againStep 1: Divide - 4X2 by 2X

Step2: Multiply -2X by divisor 2x + 1.

8X3 + 0X2 + 0X - 1-8X3 - 4X2

- 4X2 + 0X - 1

2X+1 4X2 - 2X

Step3: Subtract

4X2

Step 3: Subtract by changingsigns and bring down leftover terms.

- -

-4X2 - 2X+ +

- 4X2 + 0X - 1

Repeat Steps 1 - 3 againStep 1: Divide 2X by 2X

Step2: Multiply 1 by divisor.

Step 3: Subtract by changingsigns and bring down leftover terms.

Go To Practice Problems

+ +

8X3 + 0X2 + 0X - 1-8X3 - 4X2

- 4X2 + 0X - 1

2X+1 4X2 - 2X

-4x2 - 2x

+ +

8X3 + 0X2 + 0X - 1-8X3 - 4X2

- 4X2 + 0X - 1

2X+1 4X2 - 2X

4x2 2x

2x -1

+ 1

2x + 1- -

-2

+ -2

2x +1

2x - 1

Step 3: Subtract