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.

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
  $\begin{matrix} Product Property & a^{m} \cdot a^{n} & = & a^{m + n} \\ Power Property & {(a^{m})}^{n} & = & a^{m \cdot n} \\ Product to a Power & {(a b)}^{m} & = & a^{m} b^{m} \end{matrix}$

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

Binomial Squares Pattern
- If $a, b$ are real numbers,
- ${(a + b)}^{2} = a^{2} + 2 a b + b^{2}$
- ${(a - b)}^{2} = a^{2} - 2 a b + b^{2}$
- To square a binomial: square the first term, square the last term, double their product.
Product of Conjugates Pattern
- If $a, b$ are real numbers,
- $(a - b) (a + b) = a^{2} - b^{2}$
- The product is called a difference of squares.
To multiply conjugates:
- square the first term square the last term write it as a difference of squares

Quotient Property for Exponents:
- If $a$ is a real number, $a \neq 0$ , and $m, n$ are whole numbers, then:
  $\frac{a^{m}}{a^{n}} = a^{m - n}, m > n and \frac{a^{m}}{a^{n}} = \frac{1}{a^{m - n}}, n > m$
Zero Exponent
- 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, $b \neq 0,$ and $m$ is a counting number, then:
  ${(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}}$
- To raise a fraction to a power, raise the numerator and denominator to that power.
Summary of Exponent Properties
- If $a, b$ are real numbers and $m, n$ are whole numbers, then
  $\begin{matrix} Product Property & a^{m} \cdot a^{n} & = & a^{m + n} \\ Power Property & {(a^{m})}^{n} & = & a^{m \cdot n} \\ Product to a Power & {(a b)}^{m} & = & a^{m} b^{m} \\ Quotient Property & \frac{a^{m}}{a^{n}} & = & a^{m - n}, a \neq 0, m > n \\ \frac{a^{m}}{a^{n}} & = & \frac{1}{a^{n - m}}, a \neq 0, n > m \\ Zero Exponent Definition & a^{o} & = & 1, a \neq 0 \\ Quotient to a Power Property & {(\frac{a}{b})}^{m} & = & \frac{a^{m}}{b^{m}}, b \neq 0 \end{matrix}$

Fraction Addition
- If $a, b, and c$ are numbers where $c \neq 0$ , then
  $\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c} and \frac{a + b}{c} = \frac{a}{c} + \frac{b}{c}$
Division of a Polynomial by a Monomial
- To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.

Property of Negative Exponents
- If $n$ is a positive integer and $a \neq 0$ , then $\frac{1}{a^{− n}} = a^{n}$
Quotient to a Negative Exponent
- If $a, b$ are real numbers, $b \neq 0$ and $n$ is an integer , then ${(\frac{a}{b})}^{− n} = {(\frac{b}{a})}^{n}$
To convert a decimal to scientific notation:
1. Step 1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
2. Step 2. Count the number of decimal places, $n$ , that the decimal point was moved.
3. 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 $10^{− n}$
4. Step 4. Check.
To convert scientific notation to decimal form:
1. Step 1. Determine the exponent, $n$ , on the factor 10.
2. Step 2.
  Move the decimal nnplaces, 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 $| n |$ places to the left.
3. Step 3. Check.