2-1 Adding and Subtracting Polynomials

� How to identify parts of a polynomial.

� How to add and subtract polynomials.

� That polynomials are “closed” under addition and subtraction.

442415672 xxxx −+−−+

How many terms are there?There are 6 terms – a term is a constant or a variable or a

product of a constant and a variable separated by + and –

signs.

Give an example of like terms.-6, 1 and are like terms – terms with the

same variable to the same power.

Give an example of a coefficient.2, 7, -5, and -1 are coefficients – when the term

contains a number and a variable, the number part is the

coefficient.

444,5,2 xxx −−

442415672 xxxx −+−−+

Give an example of a constant.-6 and 1 are constants – a term that does not have a

variable.

Simplify the expression.

57424

−+− xx

32325375 xxxxx −+−−+

How many terms are there?

6

What are the like terms?

5��, ���� � 5�, ��

List the coefficients.

5, 7, -5, 1, and -1

What are the constants?

-3

Simplify the expression:

�6� � 6�� � 7� � 3

An expression formed by adding a finite number of “same base” unlike terms.

Example:

Exponents must be positive integers (no fractions), there can be no square roots, and no variables in the denominator (no negative exponents).

- Not a polynomial, why?

16423

+− xx

5213

2

+−−

xx

� The exponent of a term is the degree of the term.

Example has a degree of 5

The value of the largest exponent is the degreeof a polynomial.

Example has a degree of 3

The leading coefficient is the coefficient of the first term when the polynomial is written in standard form (largest degree first).

59x

16423

+− xx

4264 yy −+1)

��� � 4�� � 6, Deg: 4 LC: -1 2) 9 + 3x

3x + 9, Deg: 1 LC: 3

3) Not a polynomial

73423

−+−

zz

)15()23(4545

+−+− xxxx

Drop parentheses and add like terms.

Make sure answer is in standard form

)15()23(4545

+−+− xxxx

Line up terms by degree

15

23

45

45

+−

−

xx

xx

)65()65(232

xxxx −++++

� � 6�� � 6� � 11

11�� � 3� � 4�

Degree = 5

Leading Coefficient = 11

3�� � 2� � �8�� � 5� � 4��

3� � 2� � 8

Degree = 3

Leading Coefficient = 3

2� � 2 � �� � 2� � 6�

)753()283(22

−+−−+− xxxx

Change the signs of the second polynomial (you distribute the -1) and then add.

)753()283(22

−+−−+− xxxx

2832

+− xx

)753(2

−+−− xx

)8()145(22

xxx −−+−

6�� � 4� � 7

�� � 7� � 8��

Degree = 5

Leading Coefficient = 1

6� � 5�� � �8�� � 4�� � ��

3�� � 9� � 6

Degree = 4

Leading Coefficient = 3

� � 8� � �� � �� � 6 � 2�� � ��

� If you add or subtract polynomials your answer is also a polynomial.

� This means polynomials are “closed” under addition and subtraction.