Ch. 7.1 Polynomial Degree & Finite Differences

Transcription

1 Ch. 7.1 Polynomial Degree & Finite Differences Learning Intentions: Define terminology associated with polynomials: term, monomial, binomial & trinomial. Use the finite differences method to determine the degree of a polynomial. Find a polynomial function that models a set of data.

2 Vocabulary

3 Vocabulary: Given: u n :

4 m n = u n u n 1 = difference r = m n m n 1 = common ratio of 1 st differences

5 Polynomials are sums of these "variables and eponents" epressions. Each piece of the polynomial, each part that is being added, is called a "term". Polynomial terms have variables which are raised to whole-number eponents (or else the terms are just plain numbers); there are no square roots of variables, no fractional powers, and no variables in the denominator of any fractions. Here are some eamples: 6 This is NOT a polynomial term......because the variable has a negative eponent. Eample of a typical polynomial: 1 / sqrt() 4 This is NOT a polynomial term... This is NOT a polynomial term... This IS a polynomial term......because the variable is in the denominator....because the variable is not to an integer power ^(1/)....because it obeys all the rules. Notice the eponents on the terms. The first term has an eponent of ; the second term has an "understood" eponent of 1; and the last term doesn't have any variable at all. Polynomials are usually written this way, with the terms written in "decreasing" order; that is, with the largest eponent first, the net highest net, and so forth, until you get down to the plain old number. Any term that doesn't have a variable in it is called a "constant" term because, no matter what value you may put in for the variable, that constant term will never change. In the picture above, no matter what might be, 7 will always be just 7. The first term in the polynomial, when it is written in decreasing order, is also the term with the biggest eponent, and is called the "leading term". The eponent on a term tells you the "degree" of the term. For instance, the leading term in the above polynomial is a "second-degree term" or "a term of degree two". The second term is a "first degree" term. The degree of the leading term tells you the degree of the whole polynomial; the polynomial above is a "second-degree polynomial". Here are a couple more eamples:

6 Review: Adding Like-Terms Simplify each epression or system by finding the given sum or difference ) (4 4 ) (5 5 7 a a a a p p p p

7 SOLUTION: Review: Adding Like-Terms Simplify each epression or system by finding the given sum or difference ) (4 4 ) (5 5 7 a a a a p p p p

8 Review: FOIL (Distribution) Use a rectangle diagram to prove the product of each pair of factors. ( )( 4) ( 1)( 5) ( y)( y)

9 SOLUTIONS: Review: FOIL (Distribution) Use a rectangle diagram to prove the product of each pair of factors. ( )( 4) + ( 1)( 5) 4 ( y)( y) y y y y y y y y y

10 Identify the degree of each polynomial.

11 Solutions: Identify the degree of each polynomial.

12 Determine which of these epressions are polynomials. For each polynomial, state the degree and write it in general form. If it is not a polynomial, eplain why not.

13 Solutions: Determine which of these epressions are polynomials. For each polynomial, state the degree and write it in general form. If it is not a polynomial, eplain why not.

14 y 4 y 5 7 y y y y What relationship is there between the degree of the polynomial and the number of differences up through the first column of constants?

15 Solution: Find the differences between the terms up through the first constant-valued difference. y 4 y 5 7 y y y y The number of differences, including the first column of constants, equals the degree of each polynomial.

16 Eample p. 79 Find a polynomial function that models the relationship between the number of sides and the number of diagonals of a conve polygon. Use the function to find the number of diagonals of a dodecagon (a 1 sided polygon). Number of sides () Number of diagonals (y)

17 Number of sides () Number of diagonals (y)

18 Solution con d.

19 E. p.8 #5 n = number of rows (height) s = number of pennies 1 n s a.) Calculate the finite differences to find the degree of the polynomial. b.) Describe how the degree of this polynomial function is related to the finite differences. c.) What is the minimum number of data points required to determine the degree of this particular polynomial function? d.) Find the polynomial function that models these data and use it to find s when n is 1.

20 Solutions: p. 8 #5 n s (See net page for work)

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