PLEASE HELP US OUT WITH THIS When you go to the open lab next door in 203 please make sure you sign in on the log sheet and enter your instructor s name and your section number We need to collect this information to document lab usage and ensure future funding for tutors Any questions on the Section 5 1 homework Now please CLOSE YOUR LAPTOPS and turn off and put away your cell phones Sample Problems Page Link Dr Bruce Johnston Section 5 2 More Exponents Scientific Notation Power Rule am n amn Note that you MULTIPLY the exponents in this case Example Simplify each of the following expressions 23 3 23 3 29 512 x4 2 x4 2 x8 CAUTION Notice the importance of considering the effect of the parentheses in the preceding example Compare the result of 23 3 to the result of 23 23 23 23 23 3 26 64 Compare the result of x4 2 to the result of x4x2 x4 x2 x4 2 x6 Power of a Product Rule ab n an bn Example Simplify 5x2y 3 53 x2 3 y3 125x6 y3 Example from today s homework Example from today s homework Power of a Quotient Rule n n a a n b b b 0 Example Simplify the following expression 2 p 3 3r 4 p 3r 2 4 3 4 p 3 r 2 4 4 3 4 p8 12 81r Power of product Power rule rule in this step in this step Example from today s homework Summary of exponent rules If m and n are integers and a and b are real numbers then Product Rule for exponents am an am n Power Rule for exponents am n amn Power of a Product ab n an bn n an a Power of a Quotient n b 0 b b am m n a a 0 Quotient Rule for exponents n a Zero exponent a0 1 a 0 1 n Negative exponent a n a 0 a Example Simplify the following expression using only positive exponents in your answer Note Problems like this are much easier to solve if you start by simplifying the part inside the parentheses by combining the exponents of identical bases using the quotient rule and then apply the power rule using the exponent outside of the parentheses 2 3 ab 4 7 3 3 a b 3 2 3 2 4 a3 7 b1 3 2 32 a 4 b 4 2 32 2 a 4 2 b4 2 3 4 a8 b 8 a8 34b8 a8 81b8 Multiplying and dividing with numbers written in scientific notation involves using properties of exponents Example Perform the following operations 1 7 3 10 2 8 1 105 7 3 8 1 10 2 105 59 13 103 5 913 x 104 sci notation 59 130 standard form 4 1 2 10 1 2 10 0 3 10 5 3 10 6 sci not 9 2 9 4 10 4 10 0 000003 std form 4 Example Note A number is not in scientific notation if it has more than one digit in front of the decimal point Example problem Calculate 4 2 x 104 6 3 x 107 Solution 4 2 6 3 x 104 107 24 46 x 104 7 24 46 x 1011 Not in scientific notation 2 446 x 1012 Example A number is not in scientific notation if it has no nonzero digit in front of the decimal point Example problem Calculate 4 2 x 10 4 8 4 x 107 Solution 4 2 8 4 x 10 4 107 0 5 x 10 4 7 0 5 x 10 11 Not in scientific notation 5 x 10 12 Example from today s homework Example from today s homework Section 5 3 Polynomials and Polynomial Functions Polynomial vocabulary Term a number or a product of a number and variables raised to powers the terms in a polynomial are separated by or signs Coefficient numerical factor of a term Constant term which is only a number A polynomial is a sum of terms involving coefficients numbers times variables raised to a whole number 0 1 2 exponent with no variables appearing in any denominator Consider the polynomial 7x5 x2y2 4xy 7 How many TERMS does it have There are 4 terms 7x5 x2y2 4xy and 7 What are the coefficients of those terms The coefficient of term 7x5 is 7 The coefficient of term x2y2 is 1 The coefficient of term 4xy is 4 The coefficient of term 7 is 7 7 is a constant term no variable part like x or y A Monomial is a polynomial with 1 term A Binomial is a polynomial with 2 terms A Trinomial is a polynomial with 3 terms Degree of a term To find the degree take the sum of the exponents on the variables contained in the term Degree of the term 7x4 is 4 Degree of a constant like 9 is 0 because you could write it as 9x0 since x0 1 Degree of the term 5a4b3c is 8 add all of the exponents on all variables remembering that c can be written as c1 Degree of a polynomial To find the degree take the largest degree of any term of the polynomial Example The degree of 9x3 4x2 7 is 3 More examples 1 Consider the polynomial 7x5 x3y3 4xy Is it a monomial binomial or trinomial What is the degree of the polynomial 2 Which of the following expressions are NOT polynomials 5x4 5x 5x 3y7 2xy 10 1 x 5 3x 5 5x3y7 2xy 10 3 y2 6y 8 Problem from today s homework We can use function notation to represent polynomials Example P x 2x3 3x 4 is a polynomial function Evaluating a polynomial for a particular value involves replacing the value for the variable s involved Example Find the value P 2 2x3 3x 4 P 2 2 2 3 3 2 4 2 8 6 4 This means that the ordered pair 6 2 6 would be one point on the graph of this function Don t forget how to work with fractions Example For the polynomial function f x 7x2 x 2 Calculate f Answer Calculate f Answer 14 9 Like terms Terms that contain exactly the same variables raised to exactly the same powers Warning Only like terms can be combined by combining their coefficients Example Combine like terms to simplify x2y xy y 10x2y 2y xy x2y 10x2y xy xy y 2y like terms are grouped together 1 10 x2y 1 1 xy 1 2 y 11x2y 2xy 3y Adding polynomials Combine all the like terms Subtracting polynomials Change the signs of the terms of the polynomial being subtracted and then combine all the like terms Example Add or subtract each of the following as indicated 1 3x 8 4x2 3x 3 3x 8 4x2 3x 3 4x2 3x 3x 8 3 4x2 5 2 4 y 4 4 y 4 y 4 4 y 8 3 a2 1 a2 3 5a2 6a 7 a2 1 a2 3 5a2 6a 7 a2 a2 5a2 …