# Adding and Subtracting Polynomials

I. Types of Polynomials

A. Basic Definitions

1. In the term bx ^{m}, b is called the **coefficient**, x is
called the **variable**, and m is

called the ** exponent on the variable **.

2. Whenever we have just one term, that term is called a **monomial.**

3. Examples

a. 6x The coefficient is 6, the variable is x , the
exponent on the variable

is 1.

b. –17 The coefficient is –17, the variable is x (or anything else you

would like ), and the exponent on the variable is 0.

B. A **polynomial** is a sum of monomials.

Examples

1. 15x^{3} – 8x^{2} + 5x – 7

2. 28y^{12} + 16y^{6} – 25y^{3} + 4

C. The **degree** of a term is the sum of the exponents on the
variables of the term.

D. The **degree of a polynomial** is the highest degree of any of the terms of the

polynomial.

E. A polynomial is written in **standard form** when the term with the highest
degree is

written first and the degrees of the terms go in descending order .

F. When a polynomial is written in standard form, the first coefficient is
called the

**leading coefficient**. Then if the last term has no variable, it is called a
**constant term.**

II. Names of Polynomials

A. A polynomial of one term is called a **monomial.**

B. A polynomial of two terms is called a ** binomial .**

C. A polynomial of three terms is called a ** trinomial .**

D. A polynomial of more than three terms is just called a polynomial.

A. We simplify polynomials by using the distributive, associative, and
commutative

properties to combine the like terms .

B. Examples – simplify each polynomial.

1. 3y – 2y – y

**Answer: 0**

2. 3x^{3} – x^{2} + x^{3} – 2x^{2}

3x^{3} + x^{3} – x^{2} – 2x^{2}

**Answer: 4x ^{3} – 3x^{2}**

3. Now you try one: -2y – 1 – 3y – 4

**Answer: −5y – 5**

4.

First we distribute to get :

**Answer :
**

5. Now you try one: −3.2(2.1 – 1.2w) + 2.1w

**Answer: 5.94w – 6.72**

6. (8t^{5} + 3t^{3} + 5t) − (19t^{5} − 6t^{3} + t)

On the first parentheses , we distribute +1, which effectively does what?

On the second parentheses, we distribute −1, which effectively does what?

So we get:

8t^{5} + 3t^{3} + 5t − 19t^{5} + 6t^{3} − t

OR

**Answer: −11t ^{5} + 9t^{3} + 4t**

7. Now you try one: (8m^{2} − 7m) − (3m^{2} + 7m − 6)

**Answer: 5m ^{2} −14m + 6**

8. Note that this: Subtract

Means the same thing as this: (13y^{5} − y^{3}) − (−7y^{5} + 5y^{3})

9. As you complete more years of education, you can count on a greater

income. The bar graph (bottom of the right-hand column, page 330) shows

the median, or middle-most, annual income for Americans, by level of

education, in 2004.

Here are polynomial models that describe the median annual income for men,

M, and for women, W, who have completed x years of education:

M = 177x^{2} + 288x + 7075

W = 255x^{2} – 2956x + 24,336

M = −18x^{3} + 923x^{2} – 9603 x + 48,446

W = 17x^{3} – 450x^{2} + 6392x – 14,764

Use the equations defined by polynomials of degree 3 to find a mathematical

model for M – W (page 331, #104a)

A polynomial of degree 3 means that the highest exponent on the polynomial

is 3. This is the case with the second set of formulas . So we have:

M – W = (−18x^{3} + 923x^{2} – 9603 x + 48,446) – (17x^{3} – 450x^{2} + 6392x – 14,764)

M – W = −18x^{3} + 923x^{2} – 9603 x + 48,446 – 17x^{3} + 450x^{2} – 6392x + 14,764

**Answer: M – W = −35x ^{3} + 1373x^{2} – 15,995x + 63,210**

b. According to the model in part (a), what is the difference in median annual

income between men and women with 16 years of education?

Remember that x represents the number of years of education. So we need

to substitute 16 in for x in the answer that we got in part (a).

M – W = −35(16)^{3} + 1373(16)^{2} – 15,995(16) + 63,210

M – W = −35(4096) + 1373(256) – 15,995(16) + 63,210

M – W = −143,360 + 351,488 – 255,920 + 63,210

M – W = 15,418

**Answer: The difference in median annual income between men and women
with 16 years of education is $15,418.**

c. According to the data displayed by the graph on page 330, what is the actual

difference in the median annual income between men and women with 16

years of education? Did the model in part (b) underestimate or overestimate

this difference? By how much?

Looking at the bar graph on page 330, we see that the median annual income

for men with 16 years of education is $57,220; for women it is $41,681. The

difference is

57,220 – 41,681 = 15,539

So our number in part (b) is an underestimate of $121 (15,539 – 15,418).

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