2.32.3 real zeros of polynomial functions. quick review
TRANSCRIPT
2.32.32.32.3
Real Zeros of Polynomial Real Zeros of Polynomial FunctionsFunctions
Quick Review
3 2
5 3 2
2
3
3 2
2
Rewrite the expression as a polynomial in standard form.
2 31.
2 82.
2Factor the polynomial into linear factors.
3. 16
4. 4 4
5. 6 24
x x x
xx x x
x
x x
x x x
x
Quick Review Solutions
3 2
5 3 2
2
3
3 2
2
3
Rewrite the expression as a polynomial in standard form.
2 31.
2 82.
2Factor the polynomial into linear factors.
3. 16
4. 4
2 3 1
14
2
4 4
4 2 1 2
x x
x x
x
x x x
xx x x
x
x x
x x
x x
x x xx
25. 6 24 2 2 6 x xx
What you’ll learn about
• Long Division and the Division Algorithm• Remainder and Factor Theorems• Synthetic Division• Rational Zeros Theorem• Upper and Lower Bounds
… and whyThese topics help identify and locate the real
zeros of polynomial functions.
Example Using Polynomial Long
Division4 3
2
Use long division to find the quotient and remainder when 2 3
is divided by 1.
x x
x x
Example Using Polynomial Long
Division
4 3
2
Use long division to find the quotient and remainder when 2 3
is divided by 1.
x x
x x
2
2 4 3 2
4 3 2
3 2
3 2
2
2
4 3 2 2
2
2 11 2 0 0 3
2 2 2
2 0 3
+ 3
1
2 2
2 22 3 1 2 1
1
x xx x x x x x
x x x
x x x
x x x
x x
x x
x
xx x x x x x
x x
Remainder Theorem If polynomial ( ) is divided by , then the remainder is ( ).f x x k r f k
Example Using the Remainder Theorem
2Find the remainder when ( ) 2 12 is divided by 3.f x x x x
Example Using the Remainder Theorem
2Find the remainder when ( ) 2 12 is divided by 3.f x x x x
2
( 3) 2 3 3 12 =33r f
Example Using Synthetic Division
3 2Divide 3 2 5 by 1 using synthetic division.x x x x
Example Using Synthetic Division
3 2Divide 3 2 5 by 1 using synthetic division.x x x x
1 3 2 1 5
3
1 3 2 1 5
3 1 2
3 1 2 3
3 2
23 2 5 33 2
1 1
x x xx x
x x
Example Finding the Real Zeros of a Polynomial
Function4 3 2Find all of the real zeros of ( ) 2 7 8 14 8.f x x x x x
Example Finding the Real Zeros of a Polynomial
Function
4 3 2Find all of the real zeros of ( ) 2 7 8 14 8.f x x x x x
:
Factors of 8 1, 2, 4, 8 1: 1, 2, 4, 8,
Factors of 2 1, 2 2
Compare the -intercepts of the graph and the list of possibilities,
and decide that 4 and -1/2 are potential rational
Potential Rational Zeros
x
zeros.
Example Finding the Real Zeros of a Polynomial
Function4 3 2Find all of the real zeros of ( ) 2 7 8 14 8.f x x x x x
Example Finding the Real Zeros of a Polynomial
Function
4 3 2 3 2
4 2 7 8 14 8
8 4 16 8
2 1 4 2 0
This tells us that 2 7 8 14 8 ( 4)(2 4 2).
1/ 2
x x x x x x x x
4 3 2 2
2 1 4 2
1 0 2
2 0 4 0
1This tells us that 2 7 8 14 8 2( 4) 2 .
2
1The real zeros are 4,
x x x x x x x
, 2.2
4 3 2Find all of the real zeros of ( ) 2 7 8 14 8.f x x x x x