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Warm-up
• Determine if the following are polynomial functions
in one variable. If yes, find the LC and degree
1
3
4
2
f ( x)  16  3x  4 x  2 x 
2x
5
2
2
g ( x)  12  4 x  2 x  3x
Given the following polynomial function, find the following
information to help graph it! Degree? LC? End Behavior? Y-int, and
Factors with Multiplicity? Graph it!
f ( x)  x  x  6 x
4
3
2
1
Warm-up
1
f ( x)  16  3x  4 x  2 x 
2x
3
4
2
g ( x)  12  4 x  2 x  3x
5
2
f ( x)  x  x  6 x
4
3
Left: Up
Y-Int: (0, 0)
Right: Up
Yes, LC: -4;
Degree: 5
2
End Behavior
Degree: 4;
LC 1
2
NO
Factors: x2(x-3)(x+2)
x = 0 (m=2, even)
x = 3 (m = 1; odd)
x = -2 (m = 1; odd)
2
Section 2-3
Using Synthetic
Division to find Zeros
Objectives
• I can use synthetic division to
find factors of a polynomial
• I can use synthetic division to
find zeros of a polynomial
4
Dividing Numbers
Quotient
Divisor
4
4 16
When you divide a number
by another number and there
is NO REMAINDER:
Then the DIVISOR is a
factor!!
Also the QUOTIENT
becomes another factor!!!
Dividend
5
Find: (6x3- 19x2 + x + 6)  (x-3)
That means (x – 3) is a
factor and (6x2 – x – 2) is
also a factor
•
6x3 – 19x2 + 1x + 6
6
That means (3, 0) is a zero.
3
6
-19
1
6
18
-3
-6
-1
-2
0
6x2 – 1x – 2 (No remainder)
6
Find: (4x4- 5x2 + 2x + 4)  (x+1)
That means (x + 1) is
NOT a factor
• 4x4 + 0x3 – 5x2 + 2x + 4
4
-1
4
0
-5
2
4
-4
4
1
-3
-4
-1
3
1
1
4 x  4 x  1x  3 
x 1
3
2
7
Finding Additional Factors or
Zeros
• Sometimes you will know one factor or
zero, but need to find the remaining factors
or zeros
• Then using synthetic division we would
divide by the known factor or zero and the
quotient will be a new factor.
8
Given:
3
x
-
2
x
- 5x - 3; (x + 1)
1x3 – 1x2 – 5x – 3
1
-1
1
-1
-5
-3
-1
2
3
-2
-3
0
1x2 – 2x - 3
1x2 – 2x - 3
(
)(
)
(x – 3)(x + 1)
(3, 0) (-1, 0)
9
Given:
3
x
+
2
5x
- 12x - 36; (3, 0)
1x3 + 5x2 – 12x – 36
1
3
1
1x2
5
-12
-36
3
24
36
8
12
0
+ 8x - 12
1x2 + 8x - 12
(
)(
)
(x + 6)(x + 2)
(-6, 0) (-2, 0)
10
Factor Theorem: A polynomial f(x) has a factor (x – k) if and
only if f(k) = 0.
Example: Show that (x + 2) and (x – 1) are factors of
f(x) = 2x 3 + x2 – 5x + 2.
–2
2
2
1
–5
2
–4
6
–2
–3
1
0
1
2
2
–3
1
2
–1
–1
0
The remainders of 0 indicate that (x + 2) and (x – 1) are factors.
The complete factorization of f is (x + 2)(x – 1)(2x – 1).
11
Read the Question
•
•
•
•
Find the remaining factors
Find all the factors
Find the remaining zeros
Find all the zeros
12
Homework
• WS 4-2
13