Bell Quiz 2-3. Determine the end behavior of the graph using limit notation. Find a function with the given zeros , 2. 5 pts possible.

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Bell Quiz 2-3 2 pts Determine the end behavior of the graph using limit notation. 5 2 1.

1 Bell Quiz pts Determine the end behavior of the graph using limit notation g( ) = pts Find a function with the given zeros , 2 5 pts possible Ch 2A Big Ideas 1

2 Questions on 2-3 Section 2-4 2

3 2.4(a) Real Zeros of Polynomial Functions Section 2-4 3

4 Quick Review Rewrite the epression as a polynomial in standard form Factor the polynomial into linear factors Section 2-4 4

5 Quick Review Solutions Rewrite the epression as a polynomial in standard form Factor the polynomial into linear factors ( + 4)( 4) 4 ( + 1)( + 2)( 2) ( )( ) Section 2-4 5

6 What you ll learn about Long Division and the Division Algorithm Remainder and Factor Theorems Synthetic Division and why These topics help identify and locate the real zeros of polynomial functions. Section 2-4 6

7 Starting Easy 211 ( ) ( ) 32 ( ) Divide ( ) ( ) = + ( ) ( ) 3587 ( ) 32 = = = + ( ) 3587= Section 2-4 7

8 Division Algorithm for Polynomials Let f ( ) and d( ) be polynomials with the degree of f greater than or equal to the degree of d, and d( ) 0. Then there are unique polynomials q( ) and r( ), called the quotient and remainder, such that f ( ) = d( ) q( ) + r( ) where either r( ) = 0 or the degree of r is less than the degree of d. The function f ( ) in the division algorithm is the dividend, and d( ) is the divisor. If r( ) = 0, we say d( ) di vides evenly into f ( ). POLYNOMIAL FORM = + ( ) FRACTIONAL FORM ( ) ( ) = + ( ) ( ) Section 2-4 8

9 Eample Using Polynomial Long Division Use long division t o find the quotient and remainder when is divided by + both polynomial and fractional form Write your answer in Section 2-4 9

10 Eample Using Polynomial Long Division Use long division t o find the quotient and remainder when 3 f ( ) = 1and d ( ) = both polynomial and fractional form Write your answer in Section

11 Remainder Theorem If polynomial f ( ) is divided by k, then the remainder is r = f ( k). When we divided (previous eample) we got the remainder to be - 2. Let's use the remainder theorem to see if the same answer. 3 f ( ) = 1and d( ) = + 1, we get Section

12 Eample Using the Remainder Theorem Find the remainder when f ( ) 2 = is divided by + 3. Section

13 Factor Theorem A polynomial function f ( ) has a factor k if and only if f ( k) = 0. Applying the Factor Theorem To find out if + 4 is a factor, we can first apply the factor theorem Since the remainder is 0, then we can use long division to find the other factor. 2 So, f ( ) = = ( + 4)(3 5) For polynomials of degree 3 and higher, these methods can be very helpful in solving equations. Section

14 Fundamental Connections for Polynomial Functions For a polynomial function f and a real number k, the following statements are equivalent: 1. = k is a solution (or root) of the equation f() = k is a zero of the function f. 3. k is an -intercept of the graph of y = f(). 4. k is a factor of f(). Section

15 Synthetic Division When the divisor is k ( can only be raised to the first power) we can use a short cut called Synthetic Division. Set k = 0 and solve for. Divide by 1 using synthetic division. Section

16 Eample Using Synthetic Division Divide using synthetic division. Section

17 Real Numbers Section

18 Rational Zeros Theorem f ( ) = p q a n n + a n 1 n 1 Potential Rational Zeros : = integers factors of = integer factors of +... a a n 0 + a 0 = p q Find all the potential rational zeros of f ( ) 3 2 = Section

19 Eample Finding the Real Zeros of a Polynomial Function Find all of the real zeros of Potential Rational Zeros (P.R.Z.) Look at graph. 4 3 f ( ) = Compare the -intercepts of the graph to the list of P.R.Z. 2 Section

20 Upper and Lower Bound Tests for Real Zeros Let f be a polynomial function of degree n 1 with a positive leading coefficient. Suppose f ( ) is divided by k using synthetic division. If k then If k 0 and every number in the last line is nonnegative (positve or zero), k is an upper bound 0 and the numbers in the last line are alternately nonnegative and nonpositive, then k is a lower for the real zeros of f. bound for the real zeros of f. Section

21 Upper and Lower Bound Tests for Real Zeros Prove that all of the real zeros of 3 2 f ( ) = must lie in the interval[-4, 2]. Then find all the real zeros. Section

22 Upper and Lower Bound Tests for Real Zeros (cont.) Section

23 HOMEWORK Section 2-4(a) (page 223) 1, 2, 5-10, 13-16, 19-28, 33, 34, 37, 38, 41, 42, 57-60, 61(a, b,c), 71(a, b, c) Section

24 HOMEWORK Section 2-4(b) (page 224) 1, 2, 5-10, 13-16, 19-28, 33, 34, 37, 38, 41, 42, 57-60, 61(a, b,c), 71(a, b, c) Section

More Polynomial Equations Section 6.4

MATH 11009: More Polynomial Equations Section 6.4 Dividend: The number or expression you are dividing into. Divisor: The number or expression you are dividing by. Synthetic division: Synthetic division