1. Polynomials and Polynomial Functions
5.3
Polynomials and Polynomial Functions

2. Polynomial Vocabulary
Term – a number or a product of a number and variables raised to powers
Coefficient – numerical factor of a term
Constant – term which is only a number
Polynomial – a sum of terms involving variables raised to a whole number exponent, with no variables appearing in any denominator.

3. Polynomial Vocabulary
In the polynomial 7x5 + x2y2 – 4xy + 7
There are 4 terms: 7x5, x2y2, -4xy and 7.
The coefficient of term 7x5 is 7,
of term x2y2 is 1,
of term –4xy is –4 and
of term 7 is 7.
7 is a constant term.

4. Types of Polynomials Monomial is a polynomial with one term.
Binomial is a polynomial with two terms.
Trinomial is a polynomial with three terms.

5. Degrees Degree of a term
The degree of a term is the sum of the exponents on the variables contained in the term.
Degree of a constant is 0.
Degree of the term 5a4b3c is 8 (remember that c can be written as c1).

6. Degrees Degree of a polynomial
The degree of a polynomial is the greatest degree of all its terms.
Degree of 9x3 – 4x2 + 7 is 3.

7. Evaluating Polynomials
Evaluating a polynomial for a particular value involves replacing the value for the variable(s) involved.
Example:
Find the value of 2x3 – 3x + 4 when x = 2.
2x3 – 3x + 4
= 2(2)3 – 3(2) + 4
= 2(8)
= 6

8. Combining Like Terms Example:
Like terms are terms that contain exactly the same variables raised to exactly the same powers.
Warning!
Only like terms can be combined through addition and subtraction.
Example:
Combine like terms to simplify.
x2y + xy – y + 10x2y – 2y + xy
= x2y + 10x2y + xy + xy – y – 2y
(Like terms are grouped together)
= (1 + 10)x2y + (1 + 1)xy + (– 1 – 2)y =
11x2y + 2xy – 3y

9. Adding Polynomials To Add Polynomials
To add polynomials, combine all like terms.

10. Example Add: (3x – 8) + (4x2 – 3x + 3). (3x – 8) + (4x2 – 3x + 3)

11. Example
Add: using a vertical format.

12. Subtracting Polynomials
To Subtract Polynomials
To subtract a polynomial, add its opposite.

13. Example Subtract 4 – (– y – 4). 4 – (– y – 4) = 4 + y + 4 = y + 4 + 4

14. Example Subtract (– a2 + 1) – (a2 – 3) + (5a2 – 6a + 7).

15. Example
Subtract:

16. Example
Subtract:

17. Types of Polynomials a > 0 a < 0
Using the degree of a polynomial, we can determine what the general shape of the function will be, before we ever graph the function.
A polynomial function of degree 1 is a linear function. We have examined the graphs of linear functions in great detail previously in this course and prior courses.
A polynomial function of degree 2 is a quadratic function.
In general, for the quadratic equation of the form y = ax2 + bx + c, the graph is a parabola opening up when a > 0, and opening down when a < 0.
a > 0 x
a < 0 x

18. Types of Polynomials
Polynomial functions of degree 3 are cubic functions. Cubic functions have four different forms, depending on the coefficient of the x3 term.
x3 coefficient is negative
x3 coefficient is positive