1. Polynomials
Topic 6.1.1

2. Polynomials 6.1.1 1.1.1 California Standard: What it means for you:
Lesson
1.1.1
Topic
6.1.1
Polynomials
California Standard:
10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques.
What it means for you:
You’ll learn what polynomials are, and you’ll simplify them.
Key words:
monomial
polynomial
like terms
degree

3. Topic
6.1.1
Lesson
1.1.1
Polynomials
Polynomial — another math word that sounds a lot harder than it actually is.
Read on and you’ll see that actually polynomials are not as complicated as you might think.

4. Polynomials 6.1.1 1.1.1 A Monomial is a Single Term
Lesson
1.1.1
Topic
6.1.1
Polynomials
A Monomial is a Single Term
A monomial is a single-term expression.
It can be either a number or a product of a number and one or more variables.
For example, 13, 2x2, and –x3yn4 are all monomials.

5. A binomial is a two-term polynomial, such as x2 + 1.
Topic
6.1.1
Lesson
1.1.1
Polynomials
A Polynomial Can Have More Than One Term
A polynomial is an algebraic expression that has one or more terms (each of which is a monomial).
For example, x + 1 and –3x2 + 2x + 1 are polynomials.
There are a couple of special types of polynomial:
A binomial is a two-term polynomial, such as x2 + 1.
A trinomial is a polynomial with three terms, such as –3x2 + 2x + 1.

6. 6.1.1 1.1.1 Polynomials Guided Practice
Topic
6.1.1
Lesson
1.1.1
Polynomials
Guided Practice
For each of the polynomials below, state whether it is a monomial, a binomial, or a trinomial.
1. 5x x2 + 4
3. 2y x2y
5. 2y x – 2
7. 7x x2 + y
monomial
binomial
monomial
monomial
binomial
binomial
monomial
binomial
Solution follows…

7. Polynomials 6.1.1 1.1.1 Guided Practice
Lesson
1.1.1
Topic
6.1.1
Polynomials
Guided Practice
For each of the polynomials below, state whether it is a monomial, a binomial, or a trinomial.
9. x2 + 2x x2 + 4x – 8
x – 8.9x x x2y
x2y + xy2 + 4xy
15. a2 + b a – 14.2c
trinomial
trinomial
trinomial
monomial
trinomial
monomial
binomial
binomial
Solution follows…

8. Polynomials 6.1.1 1.1.1 Use Like Terms to Simplify Polynomials
Lesson
1.1.1
Topic
6.1.1
Polynomials
Use Like Terms to Simplify Polynomials
Like terms are terms that have exactly the same variables — for example, –2x2 and 5x2 are like terms.
Like terms always have the same variables, but may have different coefficients.
A polynomial can often be simplified by combining all like terms.

9. Polynomials 6.1.1 Simplify the expression 2x2 + 4y + 3x2. Solution
Topic
6.1.1
Polynomials
Example 1
Simplify the expression 2x2 + 4y + 3x2.
Solution
Notice that there are two like terms, 2x2 and 3x2:
You can combine the like terms: 2x2 + 3x2 = 5x2
So 2x2 + 4y + 3x2 = 5x2 + 4y
Solution follows…

10. Polynomials 6.1.1 1.1.1 Guided Practice
Lesson
1.1.1
Topic
6.1.1
Polynomials
Guided Practice
Simplify each of the following polynomials.
17. x x y + 2y x2 + 4x + 7x x2 + x + x2
3x + 1 5y
9x2 + 11x
2x2 + x
Solution follows…

11. 6.1.1 1.1.1 Polynomials Guided Practice
Topic
6.1.1
Lesson
1.1.1
Polynomials
Guided Practice
Simplify each of the following polynomials, then state whether your answer is a monomial, binomial, or trinomial.
21. 3x2 + 4 – 8 + x x3 + x4 – 6x x2y – 2x2y – 2y + 3 – x3 + 7 – x3 + 4 – 3x3 – 11
26. 5x2 + 9x y 27. 3xy + 4xy + 5x2y – 4xy2
28. 9x5 + 2x2 + 4x4 + 5x5 – 3x4 – x2
4x2 – 4, binomial
x4 + 2x3 + 4, trinomial
x2y + 8, binomial
–2y, monomial
0, monomial
14x2 + 2y + 4, trinomial
7xy + 5x2y – 4xy2, trinomial
14x5 + x4 + x2, trinomial
Solution follows…

12. 6.1.1 1.1.1 Polynomials Finding the Degree of a Polynomial
Topic
6.1.1
Lesson
1.1.1
Polynomials
Finding the Degree of a Polynomial
The degree of a polynomial in x is the size of the highest power of x in the expression.
If you see the phrase “a fourth-degree polynomial in x,” you know that it will contain at least one term with x4, but it won’t contain any higher powers of x than 4.
For example:
2x + 1 has degree 1 — it’s a first-degree polynomial in x
y2 + y – 3 has degree 2 — it’s a second-degree polynomial in y
x4 – x2 has degree 4 — it’s a fourth-degree polynomial in x

13. Polynomials 6.1.1 1.1.1 Guided Practice
Lesson
1.1.1
Topic
6.1.1
Polynomials
Guided Practice
State the degree of each of the following polynomials.
29. 3x x x2 + 2x x x2
1st degree
4th degree
2nd degree
2nd degree
Solution follows…

14. 6.1.1 1.1.1 Polynomials Guided Practice
Topic
6.1.1
Lesson
1.1.1
Polynomials
Guided Practice
Simplify and state the degree of each of the following polynomials.
33. 2x + x2 + x – a3 + 4a – 2a3 + 4a2
35. 4x3 + 4x8 – 3x8 + 2x y + 2y – 5y2 + 6y
37. b13 + 2b13 – – 3b z3 + z3 – z6 + z7 + 3z7
39. c4 + c3 + c3 – c4 + c – 2c3 40. x – 2x9 – 8x4 + 13x2
x2 + 3x – 3, 2nd degree
a3 + 4a2 + 4a, 3rd degree
x8 + 6x3, 8th degree
–5y2 + 11y, 2nd degree
–4, degree 0
4z7 – z6 + 2z3, 7th degree
c, 1st degree
–2x9 – 8x4 + 13x2 + x, 9th degree
Solution follows…

15. Polynomials 6.1.1 Independent Practice
Topic
6.1.1
Polynomials
Independent Practice
For the polynomials below state whether they are a monomial, a binomial, or a trinomial.
1. 19a c – 4a + 6
3. 42xy a2b + 4ab2
binomial
trinomial
monomial
binomial
Simplify each of the following polynomials.
5. 0.7x – x2
6. 17x2 – 14x9 + 7x9 – 7x2 + 7x9
7. 0.8x x – x2 – 9x x2
–0.3x
10x2
–8.2x x
Solution follows…

16. Polynomials 6.1.1 Independent Practice
Topic
6.1.1
Polynomials
Independent Practice
State the degree of the following polynomials.
8. x – 9x x8 + 16x10 + 4x8
10. 2x2 – 4x4 + 7x x2 – 4x + 8
6th degree
10th degree
5th degree
2nd degree
Simplify each polynomial, state the degree of the polynomial, and determine whether it is a monomial, a binomial, or a trinomial.
12. 93a – 4a – 81a2 + 7
x2 – 13x4 – 1.5x x2
14. 5x9 – 6x x9 – 3 – 1
12a2 – 4a + 176, 2nd degree, trinomial
–14.5x x2, 4th degree, binomial
0, degree 0, monomial
Solution follows…

17. x9 – 2x3 + , 9th degree, trinomial
Topic
6.1.1
Polynomials
Independent Practice
Simplify each polynomial, state the degree of the polynomial, and determine whether it is a monomial, a binomial, or a trinomial.
x9 – x3 – x x9 +
x10 – x6 – x x10 – 7 9 1 2 3 4 5
x9 – 2x , 9th degree, trinomial 7 9 5 6 2 9
x10 – x6 – , 10th degree, 3 4 27 20
trinomial
17. When a third degree monomial is added to a second degree binomial, what is the result?
A third degree trinomial
18. When a 4th degree monomial is added to a 6th degree binomial, what are the possible results?
6th degree polynomial that could be either trinomial, binomial, or monomial
Solution follows…

18. Topic
6.1.1
Polynomials
Round Up
This Topic gets you started on manipulating polynomials, by simplifying them.
In the next couple of Topics you’ll see how to add and subtract polynomials.