1. Polynomials

2. Polynomials
A polynomial is a term or the sum or difference of two or more terms.
A polynomial has no variables in the denominator.
The “degree of a term” is the exponent of the variable (4x3 is a 3rd degree term).
The “degree of the polynomial” is the same as the degree of the term with the highest degree.
(x5 + 4x3 – 3x + 2 is a fifth degree polynomial)
Polynomials in standard form are in order of degree from highest to lowest with the constant at the end.

3. Polynomials By Terms A polynomial with one term is called a monomial.
A polynomial with two terms is called a binomial.
A polynomial with three terms is called a trinomial.

4. Polynomials by Degrees
A first degree polynomial is linear.
A second degree polynomial is quadratic.
A third degree polynomial is cubic.
A polynomial with no variable is called a constant.

5. Examples 1 and 2
Name the degree of each term and each polynomial. Put them in standard form.
Degree of each term 5, 3, 1, and 0
Degree of the polynomial 5
It’s in standard form.
New Problem:
Degree of each term 1, 2, 0, and 3
Degree of the polynomial 3rd
In standard form, it is:

6. Model Polynomial Addition and Subtraction

7. Algebra Tiles 1 unit or -1 unit x units or -x units x2 units or

8. Model the following with Algebra Tiles
(2x2 – x) + (x2 + 3x – 1)
3x2 + 2x - 1

9. (2x2 + 6) – 4x2
-2x2 + 6

10. What is the expression modeled below?
(2x2 – 2x – 4) + (-x2 + 3x + 2)

11. Adding Polynomials Collect like terms.
In order to have like terms, the variable parts must be exactly the same.
Combine the coefficients (the numbers in front of the variable).

12. Subtracting Polynomials
Drop the first set of parentheses.
Distribute a –1 in the second set of parentheses.
Combine like terms.

13. To subtract polynomials, remember that subtracting is the same as adding the opposite. To find the opposite of a polynomial, you must write the opposite of each term in the polynomial:
–(2x3 – 3x + 7)= –2x3 + 3x – 7

14. Subtract (3x2 + 2x – 1) – (x2 + 4x – 2) Distribute the -1
Combine like terms
2x2 – 2x + 1

15. Example 3 An egg is thrown off the top of a building.
Its height in meters above the ground can be
approximated by the polynomial t – 4.9t2,
where t is the time since it was thrown in seconds.
How high is the egg above the ground after 5 seconds?
187.5
How high is the egg above the ground after 6 seconds?
135.6

17. Example 4
A firework is launched from a platform 6 feet above the ground at a speed of 200 feet per second. The firework has a 5-second fuse. The height of the firework in feet is given by the polynomial -16t t + 6, where t is the time in seconds. How high will the firework be when it explodes?
606

18. Try these…
Find the degree of each polynomial.
1. 7a3b2 – 2a4 + 4b – 15
2. 25x2 – 3x4
Write each polynomial in standard form. Then give the leading coefficient.
3. 24g g5 – g2
4. 14 – x4 + 3x2 5 4
7g5 + 24g3 – g2 + 10; 7
–x4 + 3x2 + 14; –1

19. Try these…
Classify each polynomial according to its degree and number of terms.
5. 18x2 – 12x + 5
quadratic trinomial
6. 2x4 – 1
quartic binomial
7. The polynomial 3.675v v2 is used to estimate the stopping distance in feet for a car whose speed is v miles per hour on flat dry pavement. What is the stopping distance for a car traveling at 70 miles per hour? ft

20. Try these… (5x2 – 2x + 3) + (6x2 + 5x + 6)