Eby, MATH 0310 Spring 2017 Page 53. Parentheses are IMPORTANT!! Exponents only change what they! So if a is not inside parentheses, then it

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1 Eby, MATH 010 Spring 017 Page Eponents Parentheses are IMPORTANT!! Eponents only change what they! So if a is not inside parentheses, then it get raised to the power! Eample 1 4 b) 4 c) 4 ( ) d) 4 e) 4y when y= 5 f) Rules for Eponents There are a total of rules for eponents, we will look at five of them today.

2 Eby, MATH 010 Spring 017 Page 54 Product Rule If we wanted to multiply 4 we could use. Then it would look like:, which would give us a total of s. How else could we get this number of s using & 4? This is how we get the product rule for eponents. The says: If m and n are positive integers and a is a real number, then. When multiplying terms with more than one variable then put to multiply. Eample Use the product rule to simplify each epression. Write the results using eponents a a b) ( 4z )( 9z 5 ) c) ( y )( 5 4 y 6 ) d) ( 9abc 4 )( 11ab )( bc 5 )

3 Eby, MATH 010 Spring 017 Page 55 Power Rule If we wanted to do ( ) 4 we could use. Then it would look like:, which would give us a total of s. How else could we get this number of s using & 4? This is how we get the power rule for eponents. The says: If m and n are positive integers and a is a real number, then. Power of a Product Rule If we wanted to do ( ) 4 ab we could use. Then it would look like:, which would give us a total of a s AND b s! This is how we get the power of a product rule for eponents. The says: If n is a positive integer and a and b are real numbers, then. Remember: Parentheses are Important! Eponents only change what they! Eample 9 ( b ) 11 b) ( y ) c) ( ) abc

4 Eby, MATH 010 Spring 017 Page 56 The power of a product rule works the same way for a quotient! So that means that the parentheses gets! Eample 4 7 ab c b) y z 4 4 Quotient Rule If we wanted to do 1 7 we could use. Then it would look like:, which would give us a total of s. How else could we get this number of s using 1 & 7? This is how we get the quotient rule for eponents. The says: If m and n are positive integers and a is a real number, then. Eample 5 ( 6) ( 6) 11 9 b) 8 8abc 5 18abc

5 Eby, MATH 010 Spring 017 Page 57 Zero Eponent Rule So now that we know the quotient rule what happens if we do 5 5? But we also know that anything divided by itself. That means: 5 5 = =! This is how we get the zero eponent rule. The says: a 0 =, as long as a is not 0. ( 0 0 is undefined) Remember: Parentheses are Important! Eponents only change what they! Eample b) ( 5 y ) 0 c) d) 7( a bc ) 0 e) The Rules for Eponents (so far) Product Rule Power Rule Power of a Product Rule Quotient Rule Zero Eponent Rule

6 Eby, MATH 010 Spring 017 Page Polynomial Functions and Adding and Subtracting Polynomials A is a sum of terms involving variables raised to a whole number eponent, with no variables appearing in any denominator. A is a number or a product of a number and variables raised to powers. The is the numerical factor of a term. The is the of all of the eponents of all of the variables in a single term. A is a term with no variable. All polynomials must be written in based on degree. If the degrees are all the, put the terms in order. Once the polynomial is in descending order you can find the, which is the first term of the polynomial. From the leading term you can find two things: the, and the. There are three different named polynomials: monomials, binomials, and trinomials. A monomial has. Binomials have. And trinomials have.

7 Eby, MATH 010 Spring 017 Page 59 Eample 1 Find the indicated information for each polynomial. 4 yz b) + Leading coefficient: Degree of the polynomial: Monomial/binomial/trinomial Leading term: Degree of the polynomial: Leading coefficient: Coefficient of : Constant: Monomial/binomial/trinomial c) 4a a 6 + d) 7 1 y+ y y Leading term: Degree of the polynomial: Leading coefficient: Coefficient of a : Monomial/binomial/trinomial Leading term: Degree of the polynomial: Leading coefficient: Monomial/binomial/trinomial e) y 1y y + y 6+ y + 5y 4 4 Leading term: Degree of the polynomial: Leading coefficient: Coefficient of Constant: y : Monomial/binomial/trinomial

8 Eby, MATH 010 Spring 017 Page 60 Evaluating Polynomials If you are asked to evaluate a polynomial function then you need to all of the variables in the function with what is. Eample Evaluate each polynomial for P(0) and P(-). P( ) = 4 7 b) P y y y ( ) = Adding and Subtracting To add polynomials:. To subtract two polynomials: to the terms of the polynomial being subtracted ( ) and then add. Eample ( 5w 4 w 8w + w 14) + ( w 4 + w + 1w+ 5) b) ( 16y 4 + 1y + 4y 8) ( 5y 5 y 4 + y + 9y+ 1)

9 Eby, MATH 010 Spring 017 Page Multiplying Polynomials In 5.1 we multiplied monomials, we just didn t know that was what they were called yet. Now we will look at multiplying all different types of polynomials. The nice thing is the rules are eactly the same! Eample 1 ( t 4 )( 5t ) b) a a 7 8 We also have already kind of seen multiplying a monomial times a bi or trinomial when we used in Chapter. Eample Multiply the monomial by the polynomial. 5(a 4) b) (+ 7) b) abc( 4abc abc+ 5)

10 Eby, MATH 010 Spring 017 Page 6 Multiplying other polynomials There are several different ways to multiply two larger polynomials together. You can like we did with the monomial polynomial, if you have two binomials you can use the method, or you can use the bo method which I will show you. Binomial Binomial You are more than welcome to use any method you like to multiply polynomials in this class. For bi bi I will show you two ways: FOIL and bo. Eample ( )( 4) + b) ( 4)( + 5) b) ( + )

11 Eby, MATH 010 Spring 017 Page 6 Binomial Trinomial Again, you can use any method you like, but I will focus only on the bo method. Eample 4 ( a b)( 4a 6ab+ 8b ) Trinomial Trinomial ( y + 6y+ 9)( y 4y+ 8)

12 Eby, MATH 010 Spring 017 Page Special Products In the process of using the FOIL method on products of certain types of binomials, we see specific patterns that lead to. Squaring a Binomial (Know 1 to 16!!!!!) A binomial squared is equal to the square of the first term plus or minus twice the product of both terms plus the square of the second term, the resulting polynomial is called a. ( a+ b) = ( a b) = Eample 1 ( + ) b) ( 5y) c) ( ) 7a 4

13 Eby, MATH 010 Spring 017 Page 65 Multiplying the sum and difference of two terms When we have two binomials which are the sum and difference of the same two terms they have a special name:. The resulting polynomial is called. Eample Find the conjugate of the given binomial and multiply them. ( 4) b) ( 5+ 7y) So we can say that when we multiply, conjugates, the product of the sum and difference of two terms is the square of the first term minus the square of the second term. Eample ( a+ b)( a b) = a a+ 5 5 b) ( 6+ 11)( 6 11) b) ( 5 + 7y)( 5 7y)

14 Eby, MATH 010 Spring 017 Page Negative Eponents and Scientific Notation Remember from 5.1 we had a total of rules for eponents, but only talked about five of them. The Rules for Eponents Product Rule Power Rule Power of a Product Rule Quotient Rule Zero Eponent Rule Negative Eponents We will now add the last rule for negative eponents. If we wanted to do 7 1 we could use. Then it would look like:, which would give us a total of s. From the quotient rule we know that when we divide eponents we them. So that means 7 1 = =. This is how we get the quotient rule for eponents. The says: If a is a real number other than 0, and n is an integer, then. Remember: Eponents only change what they! Eample 1 Simplify each epression. Write each result using positive eponents only. b) 8a c) y 4

15 Eby, MATH 010 Spring 017 Page 67 Putting all the rules together Eample Simplify each epression. Write each result using positive eponents only. 4 ( ) 15 8 b) ( y ) b) 6 y y d) 5 y y 4 7 d) a b 6a b 5 f) ( r s t) ( 5t 4 )

16 Eby, MATH 010 Spring 017 Page 68 Scientific Notation In many fields of science we encounter very large or very small numbers. is a convenient shorthand for epressing these types of numbers. For a number to be written in scientific notation it must follow certain rules. 1. There can be only digit from 1-9 to the left of the decimal point.. The number must be multiplied by. Powers of = 1 10 = 10 = 10 = etc Positive Eponents make 1 10 = 10 = 10 = 4 10 = etc Negative Eponents make Converting from Standard Form to Scientific Notation Steps: 1. Move the decimal point until there is only digit from 1-9 to its left.. Re-write the number getting rid of all zeros at the beginning or the end and multiply by 10 to the power of.. If the number was your eponent will be, if the number was your eponent will be. Eample Convert the following numbers from standard form to scientific notation. 40,95,000,000 b)

17 Eby, MATH 010 Spring 017 Page 69 Converting from Scientific Notation to Standard Form Steps: 1. Look at the power of 10. This tells you to move the decimal.. If the power is positive you will have a! Move the decimal to the.. If the power is negative you will have a! Move the decimal to the. Eample 4 Convert the following numbers in scientific notation to standard form b)

18 Eby, MATH 010 Spring 017 Page Dividing Polynomials We ve already talked about how to divide monomial by monomial so we will now talk about how to divide larger polynomials. Dividing a Polynomial by a Monomial Divide each term of the polynomial separately by the monomial. Eample b) Dividing Polynomials by Polynomials To divide a polynomial by a polynomial other than a monomial, we use a process known as. Polynomial long division is similar to number long division. 1 4

19 Eby, MATH 010 Spring 017 Page 71 Eample Find each quotient using long division. Don t forget to write the polynomials in descending order and fill in any missing terms b) c) ( + 81) ( ) END OF EXAM MATERIAL