Section 5.4

Adding and Subtracting Polynomials

5.4 Lecture Guide: Adding and Subtracting Polynomials Objective 1: Use the terminology associated with polynomials.

Monomials

Polynomials

Verbally

A monomial is a real number, a variable, or a product of real numbers and variables with _______________ number exponents.

A polynomial is a monomial or a sum of a finite number of _______________.

Algebraic Examples

– 5, , x, A, 5x, 7xy

and are monomials.

and are polynomials.

is a polynomial.

2r

2 27 9 3x xy y

5 7xy

Binomial

Trinomial

Verbally

A binomial is a polynomial containing _______________ terms.

A trinomial is a polynomial containing _______________ terms.

Algebraic Examples

3 4x

22 5 7x x

Degree of a Monomial

The degree of a monomial is the sum of the _______________ for all the variables in this term. A nonzero constant is understood to have degree _______________ with exponent , but no degree is assigned to the monomial 0.

0(4 4x 0)

1. Determine whether each expression is a monomial. If the expression is a monomial, give its coefficient and the degree of each monomial.

43x

Monomial? Yes / NoCoefficient: ______Degree: ______

Monomial? Yes / NoCoefficient: ______Degree: ______

Monomial? Yes / NoCoefficient: ______Degree: ______

Monomial? Yes / NoCoefficient: ______Degree: ______

(a) (b) (c) (d)43x xyz

2. Determine whether each expression is a polynomial. Classify each polynomial according to the number of terms it contains.

(a) (b) (c) (d)4 7x 92x 2

9x312 4x x

Polynomial? Yes / No

Polynomial? Yes / No

Polynomial? Yes / No

Polynomial? Yes / No

Classification:____________

Classification:____________

Classification:____________

Classification:____________

Standard Form of a PolynomialA polynomial is in standard form if (1) the variables in each term are written in ____________ order and (2) the terms are arranged in ____________ powers of the first variable.

Degree of a PolynomialThe degree of a polynomial is the same as the degree of the term with the ____________ degree. To find this highest degree, examine each term individually---do not sum the degrees of the terms.

3. Write each polynomial in standard form, and give the degree of each polynomial.

(a) 29 3 8x x

Standard form:

Degree:

3. Write each polynomial in standard form, and give the degree of each polynomial.

(b) 2 59y zx

Standard form:

Degree:

3. Write each polynomial in standard form, and give the degree of each polynomial.

(c) 5 2 34 9 3 1v v v v

Standard form:

Degree:

Objective 2: Add and subtract polynomials.

When adding or subtracting polynomials, remember to use the __________________ property to remove any grouping symbols and then add like terms.

Remember that like terms have exactly the same ________ factors.

Identify each pair of terms as like or unlike.

4. 23x y and 23xy 5. and 32x y 311x y

6. 8 5 2 9x x

Determine the sum or difference and write the result in standard form.

7. 2 24 2 2 6x x x x

Determine the sum or difference and write the result in standard form.

8. 2 23 3 5 2 3 10x x x x

Determine the sum or difference and write the result in standard form.

9.

Determine the sum or difference and write the result in standard form.

2 25 3 2 2 7x x x x

10.

Determine the sum or difference and write the result in standard form.

2 23 3 5 2 3 10x x x x

11.

Determine the sum or difference and write the result in standard form.

2 22 3 4 5 3 4 5 1x x x x

12.

Determine the sum or difference and write the result in standard form.

2 2 23 7 3 4 2 2 1x x x x x

13.

Determine the sum or difference and write the result in standard form.

2 2 2 22 6 1 5 4 7 2x xy y x xy y

14. The profit in dollars made by selling x units is given by the polynomial

2( ) 25 46.P x x x

(a) Evaluate 0P

Evaluate and interpret each expression.

Interpret 0P

(b) Evaluate

Interpret

2P

2P

14. The profit in dollars made by selling x units is given by the polynomial

2( ) 25 46.P x x x Evaluate and interpret each expression.

(c) Evaluate 13P

13PInterpret

14. The profit in dollars made by selling x units is given by the polynomial

2( ) 25 46.P x x x Evaluate and interpret each expression.

15. Write a polynomial for the perimeter of the polygon.

3x

2 4 5x x

3 4x

5

22 1x