### Key Concepts

## 6.1 Add and Subtract Polynomials

**Monomials**- A monomial is a term of the form $a{x}^{m}$, where $a$ is a constant and $m$ is a whole number

**Polynomials****polynomial**—A monomial, or two or more monomials combined by addition or subtraction is a polynomial.**monomial**—A polynomial with exactly one term is called a monomial.**binomial**—A polynomial with exactly two terms is called a binomial.**trinomial**—A polynomial with exactly three terms is called a trinomial.

**Degree of a Polynomial**- The
**degree of a term**is the sum of the exponents of its variables. - The
**degree of a constant**is 0. - The
**degree of a polynomial**is the highest degree of all its terms.

- The

## 6.2 Use Multiplication Properties of Exponents

**Exponential Notation**

**Properties of Exponents**- If $a,b$ are real numbers and $m,n$ are whole numbers, then

$\begin{array}{cccccc}\mathbf{\text{Product Property}}\hfill & & & \hfill {a}^{m}\xb7{a}^{n}& =\hfill & {a}^{m+n}\hfill \\ \mathbf{\text{Power Property}}\hfill & & & \hfill {\left({a}^{m}\right)}^{n}& =\hfill & {a}^{m\xb7n}\hfill \\ \mathbf{\text{Product to a Power}}\hfill & & & \hfill {\left(ab\right)}^{m}& =\hfill & {a}^{m}{b}^{m}\hfill \end{array}$

- If $a,b$ are real numbers and $m,n$ are whole numbers, then

## 6.3 Multiply Polynomials

**FOIL Method for Multiplying Two Binomials**—To multiply two binomials:- Step 1.
Multiply the
**First**terms. - Step 2.
Multiply the
**Outer**terms. - Step 3.
Multiply the
**Inner**terms. - Step 4.
Multiply the
**Last**terms.

- Step 1.
Multiply the
**Multiplying Two Binomials**—To multiply binomials, use the:- Distributive Property (Example 6.34)
- FOIL Method (Example 6.39)
- Vertical Method (Example 6.44)

**Multiplying a Trinomial by a Binomial**—To multiply a trinomial by a binomial, use the:- Distributive Property (Example 6.45)
- Vertical Method (Example 6.46)

## 6.4 Special Products

**Binomial Squares Pattern**- If $a,b$ are real numbers,

- ${\left(a+b\right)}^{2}={a}^{2}+2ab+{b}^{2}$
- ${\left(a-b\right)}^{2}={a}^{2}-2ab+{b}^{2}$
- To square a binomial: square the first term, square the last term, double their product.

- If $a,b$ are real numbers,
**Product of Conjugates Pattern**- If $a,b$ are real numbers,

- $\left(a-b\right)\left(a+b\right)={a}^{2}-{b}^{2}$
- The product is called a difference of squares.

- If $a,b$ are real numbers,
**To multiply conjugates:****square the first term square the last term**write it as a difference of squares

## 6.5 Divide Monomials

**Quotient Property for Exponents:**- If $a$ is a real number, $a\ne 0$, and $m,n$ are whole numbers, then:

$\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},m>n\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{m-n}},n>m$

- If $a$ is a real number, $a\ne 0$, and $m,n$ are whole numbers, then:
**Zero Exponent**- If $a$ is a non-zero number, then ${a}^{0}=1$.

- If $a$ is a non-zero number, then ${a}^{0}=1$.
**Quotient to a Power Property for Exponents**:- If $a$ and $b$ are real numbers, $b\ne 0,$ and $m$ is a counting number, then:

${\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}}$ - To raise a fraction to a power, raise the numerator and denominator to that power.

- If $a$ and $b$ are real numbers, $b\ne 0,$ and $m$ is a counting number, then:
**Summary of Exponent Properties**- If $a,b$ are real numbers and $m,n$ are whole numbers, then

$\begin{array}{ccccc}\mathbf{\text{Product Property}}\hfill & & \hfill {a}^{m}\xb7{a}^{n}& =\hfill & {a}^{m+n}\hfill \\ \mathbf{\text{Power Property}}\hfill & & \hfill {\left({a}^{m}\right)}^{n}& =\hfill & {a}^{m\xb7n}\hfill \\ \mathbf{\text{Product to a Power}}\hfill & & \hfill {\left(ab\right)}^{m}& =\hfill & {a}^{m}{b}^{m}\hfill \\ \mathbf{\text{Quotient Property}}\hfill & & \hfill \frac{{a}^{m}}{{a}^{n}}& =\hfill & {a}^{m-n},a\ne 0,m>n\hfill \\ & & \hfill \frac{{a}^{m}}{{a}^{n}}& =\hfill & \frac{1}{{a}^{n-m}},a\ne 0,n>m\hfill \\ \mathbf{\text{Zero Exponent Definition}}\hfill & & \hfill {a}^{o}& =\hfill & 1,a\ne 0\hfill \\ \mathbf{\text{Quotient to a Power Property}}\hfill & & \hfill {\left(\frac{a}{b}\right)}^{m}& =\hfill & \frac{{a}^{m}}{{b}^{m}},b\ne 0\hfill \end{array}$

- If $a,b$ are real numbers and $m,n$ are whole numbers, then

## 6.6 Divide Polynomials

**Fraction Addition**- If $a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c$ are numbers where $c\ne 0$, then

$\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}$

- If $a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c$ are numbers where $c\ne 0$, then
**Division of a Polynomial by a Monomial**- To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.

## 6.7 Integer Exponents and Scientific Notation

**Property of Negative Exponents**- If $n$ is a positive integer and $a\ne 0$, then $\frac{1}{{a}^{\text{\u2212}n}}={a}^{n}$

**Quotient to a Negative Exponent**- If $a,b$ are real numbers, $b\ne 0$ and $n$ is an integer , then ${\left(\frac{a}{b}\right)}^{\text{\u2212}n}={\left(\frac{b}{a}\right)}^{n}$

**To convert a decimal to scientific notation:**- Step 1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
- Step 2. Count the number of decimal places, $n$, that the decimal point was moved.
- Step 3.
Write the number as a product with a power of 10. If the original number is:
- greater than 1, the power of 10 will be ${10}^{n}$
- between 0 and 1, the power of 10 will be ${10}^{\text{\u2212}n}$

- Step 4.
Check.

**To convert scientific notation to decimal form:**- Step 1. Determine the exponent, $n$, on the factor 10.
- Step 2.
Move the decimal $n$places, adding zeros if needed.
- If the exponent is positive, move the decimal point $n$ places to the right.
- If the exponent is negative, move the decimal point $\left|n\right|$ places to the left.

- Step 3. Check.