Math3. Lesson 31 Finding Zeroes of NOT nice 3rd Degree Polynomials


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1 Math Lesson  Finding Zeroes of NOT nice rd Degree Polynomials
2 f ( ) Is this one of the nice rd degree polynomials? a) Sum or difference of two cubes: y 8 5 y 7 b) rd degree with no constant term. c) nice pattern that can be factored by grouping: y y 4 We are left with guessing the first degree polynomial that divides it evenly!
3 Try the bo method ( )( a b c)
4 YES, we can find the quadratic factor! ( )( a b c) ( )( 8)
5 Try: = linear factor would be ( ) y
6 Try: = linear factor would be ( ) y There s got to be an easier way!!!
7 Long Division Look at the left numbers divides 54 4 times = 48 Subtract 48 from 54 Bring the down. 5 times = 60; subtract 60 from Remainder Divisor
8 Your turn: Look at the left numbers 7 divides 44 times 7 = Subtract from 4 Bring the down. 4 times 7 = 8; subtract 8 from Remainder Divisor
9 Steps ) Look at leftmost numbers ) What # times left = left? ) Multiply 4) Subtract 5) Bring down. 6) Repeat steps Remainder Divisor
10 Vocabulary Polynomial Long division: One method used to divide polynomials similar to long division for numbers ( ) a b c Divide Evenly: A divisor divides evenly if there is a zero for the remainder.
11 Polynomial Long Division ) Look at leftmost numbers ) What # times left = left? ) Multiply? ( ) 4) Subtract ( )
12 Polynomial Long Division ( ) 4) Subtract Careful with the negatives! 5 8 5) Bring down.
13 Polynomial Long Division ( ) 5 8? ) Multiply ( ) 6) Repeat steps 5. ) Look at leftmost numbers ) What # times left = left? 4) Subtract ( )
14 Polynomial Long Division ( ) 5 8 ( ) 4) Subtract Careful of the negatives 8 5) Bring down.
15 Polynomial Long Division ( ) 5) Bring down. 5 8 ( ) 8 8
16 Polynomial Long Division ( ) 5 8 ( 8 ) 8 8 6) Repeat steps 5. ) Look at leftmost numbers ) What # times left = left? 8? ) Multiply 8 8( ) 8 8 4) Subtract ( 8 8)
17 Polynomial Long Division ( ) 5 8 ( ) 8 8 ( 8 8) 0 4) Subtract ( 8 8) Remainder = 0 ( ) divides evenly. ( ) is a factor ( )( 8)
18 Is there an easier way to do this? Yes! st step: Write the polynomial with only its coefficients. nd step: Write the zero of the linear factor. rd step: Bring down the lead coefficient
19 Is there an easier way to do this? Yes! th step: Multiply the zero by the lead coefficient. 5th step: Write the product under the net term to the right. 6 th step: add the second column downward
20 Is there an easier way to do this? Yes! th step: Multiply the zero by the second number 8th step: Write the product under the net term to the right. 9 th step: add the net column downward
21 Is there an easier way to do this? Yes! th step: Multiply the zero by the rd number th step: Write the product under the net term to the right. th step: add the net column downward
22 Is there an easier way to do this? Yes! This last number is the remainder when you divide: by
23 Because the remainder = 0, then ( ) is a factor AND = is a zero of the original polynomial!
24 The Remainder Theorem: When dividing a polynomial epression by a lower degree polynomial epression, If the remainder is zero, then the divisor is a factor of the original polynomial.
25 We call this synthetic division Look at the numbers at the bottom. Are these familiar? ( )( 8) They are the coefficients of the quadratic factor! (wow)
26 ( )( 8) ( )( 6)( )
27 Synthetic Division ) )( ( th degree poly = ( st degree poly)( rd degree poly) 0 Is the rd degree polynomial a nice one?
28 f ( )? ( ) Synthetic Substitution
29 f ( )? ( ) Synthetic Substitution
30 f ( )? ( ) Synthetic Substitution
31 f ( )? ( ) Synthetic Substitution
32 f ( )? ( ) Synthetic Substitution
33 ( )? f ( ) Synthetic Substitution
34 f ( )? ( ) Synthetic Substitution
35 f ( )? ( ) Synthetic Substitution
36 f ( )? ( ) Synthetic Substitution (Since (+) has a remainder of 5, then ( + ) is not a factor).
37 Synthetic Division & Substitution 4 5 ) ( 4 f
38 Good to here.
39 Your turn: Use Synthetic division. 8 8
40 Your turn: Use Synthetic Substitution 4. f ( ) 8 8 f ( 5)?
41 Possible Rational Zeroes f ( ) 4 5 8,,, 6, 9, 8 Factors of the last term divided by the factors of the first term = is a zero of f() ( ) is a factor of f(). Coefficients of quadratic factor Remainder ( 4 5 8) ( )( 8)
42 Polynomial Long Division ( ) 5 8 ( ) 8 8 ( 8 8) 0
43 Polynomial Long Division ( ) 5 8 ( ) 8 8 ( 8 8)
44 Can you follow this? What are the zeroes? 0 ( 5)( )( 7)  Multiply the three binomials (convert to std. form) ,, 7 What do you notice about the first and last terms and the zeroes?
45 0 ( 5)( )( 7) 57,, The denominators of the solutions are factors of the lead coefficient. The numerators of the solutions are factors of the constant term.
46 root = zero of a polynomial = inputs that cause outputs to = 0 The Rational Roots Theorem: the possible rational roots of a polynomial are factors of the constant term divided by factors of the lead coefficient. 0 ( 5)( )( 0 ( 5)( 7) ) ,, 5, 7,0, 5, 70,
47 The Rational Roots Theorem: the possible rational roots of a polynomial are factors of the constant divided by factors of the lead coefficient. 0 4,, 5, 7,0, 5, 70, , , 7,, 5, 7,0, 5,70,,, 5, 7,..., 5, 70 We always check the easy ones st.
48 Your turn: What are the possible rational roots of the following polynomial? y 4 5 8,,, 6, 9,8 If = is a zero, what factor did = come from? ( ) If ( ) is a factor, then the rd degree factors as: y ( )( a b c)
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