Math- Lesson - Finding Zeroes of NOT nice rd Degree Polynomials
f ( ) 4 5 8 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!
Try the bo method 4 5 8 8 ( )( a b c) 8 8 8 8 8 8 8 8
YES, we can find the quadratic factor! 4 5 8 8 8 ( )( a b c) 8 8 8 8 8 8 8 8 4 5 8 ( )( 8)
Try: = linear factor would be ( ) y 4 6 6 6 6 6 6 6 6 6 6
Try: = linear factor would be ( ) y 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 There s got to be an easier way!!!
54 54 54 45 54 48 6 6 60 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 6 54 45 Remainder Divisor
Your turn: 4 7 7 7 4 4 4 7 4 8 Look at the left numbers 7 divides 44 times 7 = Subtract from 4 Bring the down. 4 times 7 = 8; subtract 8 from 4 4 5 5 7 7 Remainder Divisor
4 7 7 7 7 4 4 4 8 5 Steps ) Look at left-most numbers ) What # times left = left? ) Multiply 4) Subtract 5) Bring down. 6) Repeat steps -5. 4 5 4 7 7 Remainder Divisor
Vocabulary Polynomial Long division: One method used to divide polynomials similar to long division for numbers. 4 5 8 ( ) a b c Divide Evenly: A divisor divides evenly if there is a zero for the remainder.
Polynomial Long Division 4 5 8 ) Look at left-most numbers ) What # times left = left? ) Multiply? ( ) 4) Subtract ( )
Polynomial Long Division 4 5 8 ( ) 4) Subtract Careful with the negatives! 5 8 5) Bring down.
Polynomial Long Division 4 5 8 ( ) 5 8? ) Multiply ( ) 6) Repeat steps -5. ) Look at leftmost numbers ) What # times left = left? 4) Subtract ( )
Polynomial Long Division 4 5 8 ( ) 5 8 ( ) 4) Subtract Careful of the negatives 8 5) Bring down.
Polynomial Long Division 4 5 8 ( ) 5) Bring down. 5 8 ( ) 8 8
Polynomial Long Division 4 5 8 ( ) 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)
Polynomial Long Division 8 4 5 8 ( ) 5 8 ( ) 8 8 ( 8 8) 0 4) Subtract ( 8 8) Remainder = 0 ( ) divides evenly. ( ) is a factor 4 5 8 ( )( 8)
Is there an easier way to do this? Yes! 4 5 8-4 -5 8 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
Is there an easier way to do this? Yes! 4 5 8-4 -5 8-4 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
Is there an easier way to do this? Yes! 4 5 8-4 -5 8 - - -8 7 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
Is there an easier way to do this? Yes! 4 5 8-4 -5 8 - -8 - -8 0 0 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
Is there an easier way to do this? Yes! 4 5 8 8-4 -5 8 - -8 - -8 0 This last number is the remainder when you divide: 4 5 8 by
4 5 8-4 -5 8 - -8 - -8 0 Because the remainder = 0, then ( ) is a factor AND = is a zero of the original polynomial!
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.
We call this synthetic division. 4 5 8-4 -5 8 - -8 - -8 0 Look at the numbers at the bottom. Are these familiar? 4 5 8 ( )( 8) They are the coefficients of the quadratic factor! (wow)
4 5 8 8-4 -5 8 - -8 - -8 0 4 5 8 ( )( 8) 4 5 8 ( )( 6)( )
Synthetic Division 6 9 5 4-5 9 6 6 - - - -6 6) )( ( 6 9 5 4 4 th degree poly = ( st degree poly)( rd degree poly) 0 Is the rd degree polynomial a nice one?
f ( )? ( ) Synthetic Substitution 4 5 4 - - -5-4 -
f ( )? ( ) Synthetic Substitution 4 5 4 - - -5-4 -
f ( )? ( ) Synthetic Substitution 4 5 4 - - -5-4 - -
f ( )? ( ) Synthetic Substitution 4 5 4 - - -5-4 6 - -
f ( )? ( ) Synthetic Substitution 4 5 4 - - -5-4 6 - - 4
( )? f ( ) Synthetic Substitution 4 5 4 - - -5-4 6 - - - 4
f ( )? ( ) Synthetic Substitution 4 5 4 - - -5-4 6 - - - 4-8
f ( )? ( ) Synthetic Substitution 4 5 4 - - -5-4 6-4 - - 4-8
f ( )? ( ) Synthetic Substitution 4 5 4 - - -5-4 6-4 - - 4-8 5 (Since (+) has a remainder of 5, then ( + ) is not a factor).
Synthetic Division & Substitution 4 5 ) ( 4 f - - -5-4 - - 6 4 - -8 4 5 5 8 4 4 5 4
Good to here.
Your turn: Use Synthetic division. 8 8
Your turn: Use Synthetic Substitution 4. f ( ) 8 8 f ( 5)?
Possible Rational Zeroes f ( ) 4 5 8,,, 6, 9, 8 Factors of the last term divided by the factors of the first term -4-5 8 - -8 = is a zero of f(). - -8 0 ( ) is a factor of f(). Coefficients of quadratic factor Remainder ( 4 5 8) ( )( 8)
Polynomial Long Division 8 4 5 8 ( ) 5 8 ( ) 8 8 ( 8 8) 0
Polynomial Long Division 8 4 5 8 ( ) 5 8 ( ) 8 8 ( 8 8) 0 4 5 8 8
Can you follow this? What are the zeroes? 0 ( 5)( )( 7) - Multiply the three binomials (convert to std. form) 0... 70 5,, 7 What do you notice about the first and last terms and the zeroes?
0 ( 5)( )( 7) 5-7,, 0... 70 The denominators of the solutions are factors of the lead coefficient. The numerators of the solutions are factors of the constant term.
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) ) 0 4 8 70,, 5, 7,0, 5, 70,
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, 8 70 5, -, 7,, 5, 7,0, 5,70,,, 5, 7,..., 5, 70 We always check the easy ones st.
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)