1. 4.1 Introduction to Polynomials Objectives:
• Classify polynomials by the number of terms.
• Identify the Leading coefficient of a polynomial.
• Identify the degree of a polynomial.
• Write polynomials in Standard Form.
• Add and Subtract polynomials.

2. Concept: Classifying Polynomials
a number
x, y
Term: An expression that is a constant, a variable, or a product of a constant and variable(s) that are raised to whole number powers.
Multiply a number and a variable
The exponent can’t be negative!
The exponent can’t have fractions!

3. Concept: Classifying Polynomials cont…
Polynomials are classified by the
number of their terms:
Monomial:
1 term
2 terms
Binomial:
3 terms
Trinomial:
Polynomial:
4 or more terms

4. Concept: Coefficient of a term
The coefficient of a term is the numerical factor of the term.
Ex: Given
The coefficient is 4
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5. Concept: Degree of a term
The degree of a term is the sum of the exponents of all variables in a term.
Ex: Given
The exponents of this term are 2 and 3 so the degree of this term is 5
2 + 3 = 5
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6. Got it? Let’s see! Identify the coefficient and degree of term: C: 8

7. Concept: Degree of a polynomial
The degree of a polynomial is the same as the greatest degree of any of the terms in the polynomial.
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8. Got it? Let’s see! Identify the type of polynomial and the degree:
Monomial
D: 3
Binomial
D: 2
When an equation in one variable is solved the answer is a point on a line.
Trinomial D: 3
Polynomial D: 3
Polynomial D: 4
Not a polynomial

9. Concept: Evaluate Polynomials
To evaluate a polynomial for given values of the variables, you will substitute the values into the polynomial and then simplify.
Ex: Evaluate 2x2 + 3xy3 for x = 2 and y = 3
2(2)2 + 3(2)(3)3
2(4) + 3(2)(9) =
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10. Got it? Let’s see! Evaluate each of the following: -2(-1)2(4) = -8
(-4)2 – (-4) – 3 = – 3 = 17
- (-1)2 (2) = - (1)(2) = -2

11. Concept: Polynomials in Standard Form
To write polynomials in Standard Form you just arrange the terms from largest to smallest degree.
Ex: m3 + 5m2 – 16m – 14m
–16m4
+ 22m3
+ 5m2
– 14m
+ 21

12. Got it? Let’s see!
Write the polynomial in descending order.

13. Concept: Leading Coefficient
The leading coefficient of the polynomial is the coefficient of the highest degree term.
It is called the leading coefficient because when written in Standard Form, the highest degree term will LEAD the polynomial
Ex: m3– 14m – 16m m2
–16m m3 + 5m2 – 14m + 21
Leading Coefficient = –16

14. Concept: Adding Polynomials
Addition is just combining like terms: (4x2 - 2xy + 3y2) + (-3x2 - xy + 2y2)
Line up your like terms. 4x2 - 2xy + 3y2 + -3x2 - xy + 2y2 _________________________ x2 - 3xy + 5y2

15. Rewrite subtraction as adding the opposite.
Concept: Subtracting Polynomials
Subtraction just adds one extra step: Subtract: (9y –7x + 15a) – (-3y + 8x – 8a)
Rewrite subtraction as adding the opposite.
(9y – 7x + 15a) + (+ 3y – 8x + 8a)
Group the like terms.
9y + 3y - 7x - 8x + 15a + 8a
12y - 15x + 23a

16. Concept: Evaluating Made Easy
Step #1a:
Write the terms of the polygon in Standard Form.
Ex: Find f(3):
Step #1b:
Write the terms of the polygon in Standard Form.
Ex: Find f(3):

17. Concept: Setting It Up cont. . .
Step #2:
Write the value you are using in the box at the left and write down all of the coefficients.
Ex: find 5x4 + 0x3 –4x2 +x +6

18. Step #3: Bring down the first coefficient, (5).
Concept: Doing it
Step #3: Bring down the first coefficient, (5).
Ex:

19. Step #4: Multiply the first coefficient by r (3*5).
Concept: Doing it cont…
Step #4: Multiply the first coefficient by r (3*5).
Ex:

20. Concept: Doing it cont…
Step #5:
Add the column straight down
Add the column
Ex:

21. Concept: Doing it cont…
Ste p #6:
Multiply the sum (15) by r. (15 • 3 = 45) and place this number under the next coefficient. Then add the column again.
Ex:

22. Concept: Doing it cont…
Step #7: Repeat step #6 until you run out of … coefficients..

23. The bottom row of numbers in the division problem was: 5 15 41 124 378
Concept: Writing the answer cont…
The bottom row of numbers in the division problem was:
The value is the last number in the bottom row.
f (3) = 378

24. Find f(-3): f (x)= 2x 3 + x 2 – 8x + 5
Concept: Got it? Let’s see!
Find f(-3): f (x)= 2x 3 + x 2 – 8x + 5
SOLUTION:
– –
– –21
2 – –16
f (–3) = –16
ANSWER:

25. Homework:
PM 4.1

26. Classify the expression
a) Monomial
b) Binomial
c) Trinomial
d) None of these
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5.2

27. Classify the expression
a) Monomial
b) Binomial
c) Trinomial
d) None of these
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5.2

28. Evaluate when x = –3. a) –118 b) –10 c) 10 d) 134 5.2
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5.2

29. Evaluate when x = –3. a) –118 b) –10 c) 10 d) 134 5.2
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5.2

30. Identify the degree of the polynomial.
b) 5
c) 6
d) 7
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5.2

31. Identify the degree of the polynomial.
b) 5
c) 6
d) 7
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5.2

32. Identify the leading coefficient.
b) 5
c) 6
d) 7
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5.2

33. Identify the degree of the polynomial.
b) 5
c) 6
d) 7
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5.2