Elementary Algebra

Elementary Algebra Exam 4 Material Exponential Expressions & Polynomials

Exponential Expression • An exponential expression is: where is called the base and is called the exponent • An exponent applies only to what it is immediately adjacent to (what it touches) • Example:

Meaning of Exponent • The meaning of an exponent depends on the type of number it is • An exponent that is a natural number (1, 2, 3,…) tells how many times to multiply the base by itself • Examples:

Rules of Exponents • Product Rule: When two exponential expressions with the same base are multiplied, the result is an exponential expression with the same base having an exponent equal to the sum of the two exponents • Examples:

Rules of Exponents • Power of a Power Rule: When an exponential expression is raised to a power, the result is an exponential expression with the same base having an exponent equal to the product of the two exponents • Examples:

Rules of Exponents • Power of a Product Rule: When a product is raised to a power, the result is the product of each factor raised to the power • Examples:

Rules of Exponents • Power of a Quotient Rule: When a quotient is raised to a power, the result is the quotient of the numerator to the power and the denominator to the power • Example:

Rules of Exponents • Don’t Make Up Your Own Rules • Many people try to make these rules: • Proof:

Using Combinations of Rules to Simplify Expression with Exponents • Examples:

Homework Problems • Section: 4.1 • Page: 261 • Problems: Odd: 5 – 11, 25 – 79 • MyMathLab Section 4.1 for practice • MyMathLab Homework Quiz 4.1 is due for a grade on the date of our next class meeting

Integer Exponents • Thus far we have discussed the meaning of an exponent when it is a natural (counting) number: 1, 2, 3, … • An exponent of this type tells us how many times to multiply the base by itself • Next we will learn the meaning of zero and negative integer exponents • Examples:

Integer Exponents • Before giving the definition of zero and negative integer exponents, consider the pattern:

Definition of Integer Exponents • The patterns on the previous slide suggest the following definitions: • These definitions work for any base, , that is not zero:

Quotient Rule for Exponential Expressions • When exponential expressions with the samebase are divided, the result is an exponential expression with the same base and an exponent equal to the numerator exponent minus the denominator exponent Examples: .

“Slide Rule” for Exponential Expressions • Whenboth the numeratoranddenominator of a fraction are factored then any factor may slide from the top to bottom, or vice versa, by changing the sign on the exponent Example: Use rule to slide all factors to other part of the fraction: • This rule applies to all types of exponents • Often used to make all exponents positive

Simplify the Expression:(Show answer with positive exponents)

Scientific Notation • A number is written in scientific notation when it is in the form: Examples: • Note: When in scientific notation, a single non-zero digit precedes the decimal point

Converting from Normal Decimal Notation to Scientific Notation • Given a decimal number: • Move the decimal to the right of the first non-zero digit to get the “a” • Count the number of places the decimal was moved • If it was moved to the right “n” places, use “-n” as the exponent on 10 • If it was moved to the left “n” places, use “n” as the exponent on 10 • Examples: .

Converting from Scientific Notation to Decimal Notation • Given a number in scientific notation: • Move the decimal in “a” to the right “n” places, if “n” is positive • Move the decimal in “a” to the left “n” places, if “n” is negative • Examples: .

Applications of Scientific Notation • Scientific notation is often used in situations where the numbers involved are extremely large or extremely small • In doing calculations involving multiplication and/or division of numbers in scientific notation it is best to use commutative and associative properties to rearrange and regroup the factors so as to group the “a” factors and powers of 10 separately and to use rules of exponents to end up with an answer in scientific notation • It is also common to round the answer to the least number of decimals seen in any individual number

Example of Calculations Involving Scientific Notation • Perform the following calculations, round the answer to the appropriate number of places and in scientific notation

Homework Problems • Section: 4.3 • Page: 278 • Problems: Odd: 1 – 9, 13 – 49, 63 – 75 • MyMathLab Section 4.3 for practice • MyMathLab Homework Quiz 4.3 is due for a grade on the date of our next class meeting

Review of Terminology of Algebra • Constant – A specific number Examples of constants: • Variable – A letter or other symbol used to represent a number whose value varies or is unknown Examples of variables:

Review of Terminology of Algebra • Expression – constants and/or variables combined with one or more math operation symbols for addition, subtraction, multiplication, division, exponents and roots in a meaningful way Examples of expressions: • Only the first of these expressions can be simplified, because we don’t know the numbers represented by the variables

Review of Terminology of Algebra • Term – an expression that involves only a single constant, a single variable, or a product (multiplication) of a constant and variables Examples of terms: • Note: When constants and variables are multiplied, or when two variables are multiplied, it is common to omit the multiplication symbol Previous example is commonly written:

Review of Terminology of Algebra • Every term has a “coefficient” • Coefficient – the constant factor of a term • (If no constant is seen, it is assumed to be 1) • What is the coefficient of each of the following terms?

Terminology of Algebra • Every term has a “degree” • Degree – the sum of the exponents on the variables in the term • (constant terms always have degree 0) • What is the degree of each of the following terms?

Review of Like Terms • Recall that a term is a constant, a variable, or a product of a constant and variables • Like Terms: terms are called “like terms” if they have exactly the same variables with exactly the same exponents, but may have different coefficients • Example of Like Terms:

Review of Like Terms • Given the term: • Which of the following are like terms to this one?

Adding and Subtracting Like Terms • When “like terms” are added or subtracted, the result is a like term and its coefficient is the sum or difference of the coefficients of the other terms • Examples:

Polynomial • Polynomial – a finite sum of terms • Examples:

Number of Terms One term: Two terms: Three terms: Special Name Special Names for Certain Polynomials

Evaluating Polynomials • To “evaluate” a polynomial is to replace variables with parentheses containing specific numbers and simplify • Evaluate the polynomial for :

Adding and Subtracting Polynomials • To add or subtract polynomials horizontally: • Distribute to get rid of parentheses • Combine like terms • Example:

Adding and Subtracting Polynomials • To add or subtract polynomials vertically: • Line up like terms in vertical columns • Add or subtract terms in each column • Example:

Homework Problems • Section: 4.4 • Page: 289 • Problems: Odd: 1 – 55, 59 – 69, 73 – 77 • MyMathLab Section 4.4 for practice • MyMathLab Homework Quiz 4.4 is due for a grade on the date of our next class meeting

Multiplying Polynomials • To multiply polynomials: • Get rid of parentheses by multiplying every term of the first by every term of the second using the rules of exponents • Combine like terms • Examples:

Multiplying Binomials by FOIL • As seen by the last example, we already know how to multiply binomials by the general rule (every term of first by every term of the second) • With binomials, this is sometimes called the FOIL method: • First times First • Outside times Outside • Inside times Inside • Last times Last

Squaring a Binomial • To square a binomial means to multiply it by itself • Although a binomial can be squared by foiling it by itself, it is best to memorize a shortcut for squaring a binomial:

Finding Higher Powers of Binomials • To find powers of binomials higher than the second we use the definition of exponents and the rules already learned • Example:

Conjugate Binomials • Two binomials are called “conjugates” if they are exactly the same except for the sign in the middle • Examples: What is the conjugate of the given binomial?

Multiplying Conjugate Binomials • Conjugate binomials can be multiplied by foil: • However, it is best to memorize a formula for multiplying conjugate binomials:

Dividing a Polynomial by a Monomial • Write problem so that each term of the polynomial is individually placed over the monomial in “fraction form” • Simplify each fraction by dividing out common factors

Dividing a Polynomial by a Polynomial • First write each polynomial in descending powers • If a term of some power is missing, write that term with a zero coefficient • Complete the problem exactly like a long division problem in basic math

Example

Elementary Algebra