Chapter+8

= = [|exponents.docx]

= **8-1** =
 * ==== Multiply monomials. ====
 * ==== Simplify expressions involving powers of monomials. ====

Monomial- one term with only multiplication (no division, subtraction, or addition) Constant- a monomial without a variable


 * = ** ...................................................... Identifying Monomials ** = ||
 * ** Expression ** || ** Monomial? ** || ** Reason ** ||
 * -5 || Yes || -5 is a real number and an example of a constant. ||
 * p + q || No || The expression involves the addition, not the product, of two variables. ||
 * x || Yes || Single variables are monomials. ||
 * ....... 5/9 || No || The expression is the quotient, not the product, of two variables. ||

__You Always Need To...__
Multiply Coefficents (constants) (#'s) Base stays the same Add exponents

x^2 * x^5 1.Multiply numbers 2. Add the exponents 3. Base stays the same

c*c*c*c*d*d*x*x*x = c^4 * d^2 * x^3

**Numbers first and variables in alphabetical order** **ex. 8m3 p6 **

= **8.2** =
 * Simplify expressions involving the quotient of monomials.
 * Simplify expressions containing negative exponents.

*Simplify the exponents outside of the parenthesis first. *Simplify the constants. *Simplify the exponents inside the parenthesis. *All answers should have only positive exponents.

**X to the zero power is always 1**

**dividing exponents: x13/x7=x6** ............................... just subtract the exponents

// Negative exponents drop to the denominator and turn to positive **(put them in their happy place)** //

= **8.3** =

. Standard form- how we see a number (ex. 53,000,000,000,000,000 not 5.3 * 10^16) Scientific notation- an abriviated number (ex. a * 10^n)
 * Express numbers in scientific notation and standard notation.
 * Find products and quotients of numbers expressed in scientific notation.

__Steps__
 * 1) Move the decimal behind the first number that is not a zero
 * 2) multiply that by ten to a power (the power is the amount of zeros in the number)

n = integer (how many zeros) a = more than 1 less than ten

Examples:
 * 6.5 * 10^4 = 6.5 * 10,000
 * (1.23 * 10^5) (2 * 10^6) = 2.46 * 10^11

Rules for Division in Scientific Notation: 1) Divide the coefficients 2) Subtract the exponents (base 10 remains) . Ex: __5.4 * 10^5__ ...... 2 * 10^ -4 = 2.7 * 10^9 = **8.4** =
 * Find the degrees of a polynomial.
 * Arrange the terms of a polynomial in ascending or descending order.

Monomial- one term (xy) Binomial- two terms (xy + 3) Trinomial- three terms (xy + 3 + 2) Polynomial- any number of terms

Degree- the sum of the value of the exponents ** ............ 3 = 0 = any number has a value of 0** ** ............ x = 1 = any variable by itself** ** ......... x^2 = 2 = the same number as the exponent**

__Degree of a Polynomial__
 * 1) find the degree of each term
 * 2) the value of the largest degree given term equals the degree of a polynomial

Example: x + 2x + 4xy + zxy .............. 1 .. + . 1 .. + .. 2 ... + .. 3  .............. Polynomial degree = 3

//Write Polynomials in//... Ascending- smallest to largest .................................. Descending- largest to smallest

= **8.5** =
 * Add polynomials.
 * Subtract polynomials.

//Write final answer in this order//
 * 1) Find like terms
 * 2) Combine like terms
 * 3) DONE!!!
 * Sort || The || Terms ||
 * x^2 || . x ||  ... # ||

*When **subtracting**, if there's not a matching term in the polynomials __before__, then you change the sign, if there's not a matching term in the polynomials __after__, then its the same.

= 8.6 =
 * Find the product of a monomial and polynomial.
 * Solve equations involving polynomials.

__Steps__
 * 1) Distribute
 * 2) Multiply
 * 3) Simplify

Example: 2x^2 (3x^2 + 4x - 6) ............... 6x^4 + 8x^3 - 12x^2 = 8.7 =
 * Multiply two binomials by using the FOIL method.
 * Multiply two polynomials by using the Distributive Property.

//Key- make sure **each** term in one polynomial gets multiplied by **all** the terms in the other polynomial// //.// F- first O- outer I- inner L- last

// ... (a + b) (c + d) // // F ...... O ...... I ....... L // // ac + ad + bc + bd //

R A I N B O W method :)

= 8.8 =
 * Find squares of sums and differences.
 * Find the product of a sum and a difference.

__Square of a Sum__ (x + 1)^2 = (x + 1)(x + 1)

__Square of a Difference__ (x - 1)2 = (x - 1)(x - 1)

// **Always** write it out first, then FOIL //