Adding and subtracting Polynomials

Lesson 8-1

TOPIC IX: Quadratic Equations and Functions

POLYNOMIALS

What does each prefix mean?mono

onebi

twotri

three

MONOMIALMonomial is a real number, a variable, or a product of a real number and one or more variables with whole-number exponent.Here are some examples of monomials

18 𝑧−4 𝑥22.5 𝑥 𝑦2𝑎3

What about poly?

one or moreA polynomial is a monomial or a sum/difference of monomials.

POLYNOMIAL

Important Note!!An expression is not a polynomial if there

is a variable in the denominator.

You can name a polynomial based on its degree or the number of monomials it contains

State whether each expression is a polynomial. If it is, identify it.

1) 7y - 3x + 4trinomial

2) 10x3yz2

monomial3)

not a polynomial2

57

2y

y

Which polynomial is represented by

X2

11

X

XX

1. x2 + x + 12. x2 + x + 23. x2 + 2x + 24. x2 + 3x + 25. I’ve got no idea!

The degree of a monomial is the sum of the exponents of the variables.

Find the degree of each monomial.1) 5x2 22) 4a4b3c 83) -3 0

DEGREE OF A POLYNOMIAL

To find the degree of a polynomial, find the largest degree of the terms.

1) 8x2 - 2x + 7Degrees: 2 1 0Which is biggest?

2) y7 + 6y4 + 3x4m4

Degrees: 7 4 8

2 is the degree!

8 is the degree!

Find the degree of x5 – x3y2 + 4

1. 02. 23. 34. 55. 10

A polynomial is normally put in ascending or descending order.

What is ascending order?Going from small to big exponents.

What is descending order?Going from big to small exponents.

Means that the degrees of its monomial term decrease from left to right

STANDARD FORM OF A POLYNOMIAL

Put in descending order:

1) 8x - 3x2 + x4 - 4 x4 - 3x2 + 8x - 4

2) Put in descending order in terms of x:12x2y3 - 6x3y2 + 3y - 2x

-6x3y2 + 12x2y3 - 2x + 3y

3) Put in ascending order in terms of y: 12x2y3 - 6x3y2 + 3y - 2x

-2x + 3y - 6x3y2 + 12x2y3

4) Put in ascending order:5a3 - 3 + 2a - a2

-3 + 2a - a2 + 5a3

Write in ascending order in terms of y:x4 – x3y2 + 4xy – 2x2y3

1. x4 + 4xy – x3y2– 2x2y3

2. – 2x2y3 – x3y2 + 4xy + x4 3. x4 – x3y2– 2x2y3 + 4xy

4. 4xy – 2x2y3 – x3y2 + x4

Adding and Subtracting Polynomials

You can add and subtract monomial by adding and subtracting like terms. Examples:

• =

• =

A polynomial is a monomial or a sum of monomial. The following polynomial is the sum of the monomial ,

3 𝑥4+5 𝑥2−7 𝑥+1

4210Degree of

each monomial

You can add polynomials by adding like terms

What is the simpler form of12)

12 8

Line up like terms then add the coefficients

Method 1 – Add vertically

ADDING POLYNOMIAL

Method 2 – Add horizontally

12)= 812

Group like terms then add the coefficients

Recall that subtraction means to add the opposite. So when you subtract a polynomial, change each of the term to its opposite. Then add the coefficients

What is the simpler form of12)

12

Line up like terms

Method 1 – Subtract vertically

12

Then add the opposite of each term in the polynomial being

subtracted

SUBTRACTING POLYNOMIAL

What is the simpler form of12)

( )

Method 2 – Subtract horizontally

Write the opposite of each term in the polynomial being

subtracted

SUBTRACTING POLYNOMIAL

=

= (

=

Group like term

Simplify

1. Add the following polynomials:(9y - 7x + 15a) + (-3y + 8x - 8a)

Group your like terms.(9y - 3y) + (- 7x + 8x) + (15a - 8a)

= 6y + x + 7a

Examples:

Combine your like terms.(3a2) + (3ab + 4ab) + (6b2 - b2)

3a2 + 7ab + 5b2

2. Add the following polynomials:(3a2 + 3ab - b2) + (4ab + 6b2)

Add the polynomials.+

X2

11XX

XYYY

YY

1 11

XYY

Y 111

1. x2 + 3x + 7y + xy + 82. x2 + 4y + 2x + 33. 3x + 7y + 84. x2 + 11xy + 8

Line up your like terms. 4x2 - 2xy + 3y2

+ -3x2 - xy + 2y2

_________________________

x2 - 3xy + 5y2

3. Add the following polynomials using column form (vertically):

(4x2 - 2xy + 3y2) + (-3x2 - xy + 2y2)

Rewrite subtraction as adding the opposite.

9y - 7x + 15a + 3y - 8x + 8aGroup the like terms.

9y + 3y -7x - 8x + 8a +15a12y - 15x + 23a

4. Subtract the following polynomials:(9y - 7x + 15a) - (-3y + 8x - 8a)

Rewrite subtraction as adding the opposite.

(7a - 10b) + (- 3a - 4b)Group the like terms.

7a - 3a - 10b - 4b4a - 14b

5. Subtract the following polynomials:(7a - 10b) - (3a + 4b)

Line up your like terms and add the opposite

4x2 - 2xy + 3y2

+ (+ 3x2 + xy - 2y2) 7x2 - xy + y2

6. Subtract the following polynomials using column form:

(4x2 - 2xy + 3y2) - (-3x2 - xy + 2y2)

Find the sum or difference.(5a – 3b) + (2a + 6b)

1. 3a – 9b2. 3a + 3b3. 7a + 3b4. 7a – 3b

Find the sum or difference.(5a – 3b) – (2a + 6b)

1. 3a – 9b2. 3a + 3b3. 7a + 3b4. 7a – 9b