Algebra Honors

Adding and Subtracting Polynomials 8-1

VOCABULARY

Monomial: a constant, a variable or product of variables and WHOLE number exponents – one term

Binomial: The SUM of TWO monomials

Trinomial: The SUM of THREE monomials

Polynomial: a monomial or the sum of monomials

Degree of monomial: The sum of the exponents of all the variables in the term

Degree of polynomial: The highest degree of any term in a polynomial

Standard form  (Descending form): Place variables in alphabetical order with the highest power  first

Leading coefficient: The coefficient of the term that has the highest degree.

Degree	Name
0	constant
1	linear
2	quadratic
3	cubic
4	quartic
5	quintic
6	6th degree

ADDING AND SUBTRACTING POLYNOMIALS
This is nothing more than combining LIKE TERMS so this is DÉJÀ VU!

LIKE TERMS = same variable AND same power

You can do this using 3 different strategies:
1. Simply do it in your head, but keep track by crossing out the terms as you use them.
2. Rewrite, putting the like terms together (commutative and associative property)
3. Rewrite in COLUMN form, putting like terms on top of each other like you do when adding a column of numbers.

EXAMPLE OF COLUMN FORM:
[5x4 - 3x2 - (-4x) + 3] + [-10x4 + 3x3- 3x2 - x + 3]

Rewrite in column form, lining up like terms:
LEAVE A SPACE IF ONE OF THE POWERS ARE MISSING AND MAKE SURE BOTH OF THEM ARE IN DESCENDING ORDER!

                                                                                                                                          

                                5x4           - 3x2 - (-4x)  + 3                                                                        

                         + -10x4  + 3x3 - 3x2 -     x + 3                                                                                 

                           -----------------------------------                               

                              -5x4  + 3x3 - 6x2  + 3 x  + 6

SUBTRACTING POLYNOMIALS
You can use the ADDITIVE INVERSE PROPERTY (our BFF) with polynomials!
Subtracting is simply adding the opposite so.............
DISTRIBUTE THE NEGATIVE SIGN TO EACH TERM!!
(Change all the signs of the second polynomial!)
After you change all the signs,

use one of your ADDING POLYNOMIAL strategies!

EXAMPLE OF COLUMN FORM:
[5x4 - 3x2 - (-4x) + 3] - [-10x4 + 3x3- 3x2 - x + 3]
Rewrite in column form, lining up like terms:

                                                                                                                          

      5x4          - 3x2 - (-4x) + 3

- ( -10x4 + 3x3 - 3x2 -      x + 3)

        -----------------------------------
                                                                                                 

DISTRIBUTE THE NEGATIVE, THEN ADD:
                                                                                                                         

     5x4          - 3x2 -  (-4x) + 3   

+10x4 - 3x3 + 3x2 +      x - 3

                                ----------------------------------- 15x4 - 3x3    + 5 x

Algebra Honors ( Period 4)

Adding & Subtracting Polynomials 8-1

VOCABULARY

Monomial: a constant, a variable or product of variables and WHOLE number exponents – one term

Binomial: The SUM of TWO monomials

Trinomial: The SUM of THREE monomials

Polynomial: a monomial or the sum of monomials

Determine whether each expression is a polynomial. If it is a polynomial, find the degree and determine whether it is a monomial, binomial or trinomial.

Example 1:  4y – 5xz 
  Yes 4y and -5xz is really the sum of 4y and -5xz. Degree: 2  binomial

Example 2:  -6.5
Yes, -6.5 is a real number. Degree: 0 monomial

Example 3: 7a^-3 + 9b   No it is not a polynomial because 7a^-3 the exponent is NOT a whole number exponent.

Example 4: 6x³ + 4x + x + 3 Yes  it actually simplifies to 6x³ + 5x  + 3 the Degree is 3 and it is a trinomial

Degree of monomial: The sum of the exponents of all the variables in the term

Degree of polynomial: The highest degree of any term in a polynomial

Although the terms of a polynomial can be written in any order, polynomials in one variable are usually written in standard form.

Standard form  (Descending form): Place variables in alphabetical order with the highest power  first

Leading coefficient: The coefficient of the term that has the highest degree.

Write each polynomial in standard form Identify the leading coefficient

Example 1: 3x² + 4x⁵ -7x
Find the degree of each term
The greatest degree is 5.  so    4x⁵ + 3x² -7x is its standard form , with a leading coefficient of 4

Example 2: 5y -9-2y⁴ -6y³ 
Find the degree of each term
The greatest degree is 4 so,  -2y⁴ -6y³ +5y -9   is its standard form, with a leading coefficient of -2. Notice that sign of the coefficient is kept!

Degree	Name
0	constant
1	linear
2	quadratic
3	cubic
4	quartic
5	quintic
6	6th degree

Add and Subtract Polynomials

Adding polynomials involves adding like terms. You can group by using a horizontal or vertical format.

Find each sum
Example 1; ( 2x² + 5x -7) + (3 – 4x² + 6x)
Horizontal method:
Group and combine like terms

[2x² + (– 4x² )] + [5x + 6x] + {-7 + 3]

– 2x²  + 11x -4

Example 2: (3y + y³ -5) + (4y²- 4y+ 2y³ +8)

Align like terms in columns and combine. Insert a placeholder to help align your terms, if necessary

 y³ + 0y² + 3y -5
+
2y³ + 4y² – 4y + 8

3y³ + 4y²  – y  + 3

You can subtract a polynomial by ADDING its additive inverse. To find the additive invers of a polynomial, write the opposite of each term. That is, distribute the negative to each of the terms.

-(3x² + 2x - 6) = -3x² - 2x + 6

Find each difference

Example 1 (3 – 2x + 2x²) – (4x -5 + 3x²)
Horizontal Method
Subtract 4x -5 + 3x² by ADDING its additive inverse so
( 3 – 2x + 2x²) – (4x -5 + 3x²) = ( 3 – 2x + 2x²) + (-4x +5  -3x²)

Group like terms
[2x²  + (– 3x²)] + [( -2x ) + (-4x)] + [3 + 5]
x² – 6x + 8

Example 2 (7p + 4p³ -8) – (3p² + 2 -9p)
Vertical Method

Align like terms in columns and subtract by adding the additive inverse

4p³ + 0p² + 7p – 8        ADD THE OPPOSITE   4p³ + 0p² + 7p – 8       
(-)     3p² - 9p + 2                                               (+)    - 3p² + 9p - 2
                                                                              4p³ -3p²+ 16p -10

Adding or subtracting integers results in an integers so the set of integers is closed under addition and subtraction. Similarly adding and subtracting polynomials results in a polynomial so the set of polynomials is closed under addition and subtraction.  

Algebra ( Period 5)

Adding and  & Subtracting Polynomials 8-1

VOCABULARY

Monomial: a constant, a variable or product of variables and WHOLE number exponents – one term

Binomial: The SUM of TWO monomials

Trinomial: The SUM of THREE monomials

Polynomial: a monomial or the sum of monomials

Degree of monomial: The sum of the exponents of all the variables in the term

Degree of polynomial: The highest degree of any term in a polynomial

Standard form  (Descending form): Place variables in alphabetical order with the highest power  first

Leading coefficient: The coefficient of the term that has the highest degree.

Degree	Name
0	constant
1	linear
2	quadratic
3	cubic
4	quartic
5	quintic
6	6th degree