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.