polynomial p(x) binomial factors polynomial p(x) binomial factors solutions of p(x)=0 zeros of p(x)
TRANSCRIPT
2 2 0x x
1 2 0x x
1 2x or x
1 2 0x x
1 2x or x
2 2 ( )x x P x Polynomial P(x)
2 2 0x x
2 2 0x x
1 2 0x x
1 2x or x
Polynomial P(x)
When 0P x
2 2 ( )x x P x
1 2 0x x
1 2x or x
Polynomial P(x)
When 0P x
Binomial Factors
2 2 ( )x x P x 2 2 0x x
1 2 0x x
1 2x or x
Polynomial P(x)
When 0P x
Binomial Factors
Solutions of P(x)=0
Zeros of P(x)
2 2 ( )x x P x 2 2 0x x
1 2 0x x
1 2x or x
Polynomial P(x)
When 0P x
Binomial Factors
Solutions of P(x)=0
Zeros of P(x)
2 2 ( )x x P x 2 2 0x x
If in polynomial P(x), P(c) = 0
is a solution to the equation ( ) 0c Px x
is a zero of ( )x c P x
is a factor of ( )x c P x
1 2x or x
Zero Zero
P(x)=0 P(x)=0
1 2P x x x
Zero Zero
P(x)=0 P(x)=0
Binomial Factors
1 2x or x
2 4 4 0x x
2 2 0x x
2 2x or x
2( ) 4 4P x x x Polynomial P(x)
When 0P x
Binomial Factors
Solutions of P(x)=0
Zeros of P(x)
2 2x or x Solutions of P(x)=0
Zeros of P(x)
2 2x or x Solutions of P(x)=0
Zeros of P(x)
2, . .d r
multiplicity 22,
2 2x or x Solutions of P(x)=0
Zeros of P(x)
mult. 22,
Zeros
P(x)=0
mult. 22,
The graph is tangent (touches) the x-axis but doesn’t cross the x-axis at that point
Zeros
P(x)=0
mult. 22,
The graph is tangent (touches) the x-axis but doesn’t cross the x-axis at that point
( ) 2 2P x x x Binomial Factors
2( ) 2P x x x 2( ) 4 4P x x x
( ) 1 2P x x x ( ) 2 2P x x x
( ) 1 2P x x x 2( ) 2P x x
( ) 1 2P x x x 2( ) 2P x x
( ) 1 2P x x x 2( ) 2P x x
2( ) 1 2P x x x
2
( ) 1 2P x x x
-1
P(x)=0
mult. 11,
2
( ) 1 2P x x x
P(x)=0
mult. 22,-1
P(x)=0
mult. 11,
2( ) 1 2P x x x
2( ) 1 2P x x x
( ) 1 2 2P x x x x
2( ) 1 4 4P x x x x
2( ) 1 2P x x x
( ) 1 2 2P x x x x
2( ) 1 4 4P x x x x
3 2 2( ) 4 4 4 4P x x x x x x
3 2( ) 3 4P x x x
3 2( ) 3 4P x x x
P(x)=0
mult. 22,-1
P(x)=0
mult. 11,
P(x)=0
mult. 22,-1
P(x)=0
2( ) 1 2P x x x
mult. 11,
2
( ) 2 1P x x x
mult. 22,
2( ) 2 1P x x x
2
( ) 2 1P x x x
mult. 22,
2( ) 2 1P x x x
2
( ) 2 1P x x x
mult. 22,
2( ) 2 1P x x x
mult. 11,
2
( ) 2 1P x x x
mult. 22, mult. 11,
2
( ) 2 1P x x x
mult. 22, 1mult. 11,
3 2( ) 3 4P x x x
mult. 22, 1mult. 11,
2
( ) 2 1P x x x
mult. 22, 1mult. 11,
2 2
( ) 2 1P x x x
mult. 22,
2 2
( ) 2 1P x x x
mult. 22,
2 2
( ) 2 1P x x x
mult. 22, mult. 21,
2 2
( ) 2 1P x x x
mult. 22, mult. 21,
2 2
( ) 2 1P x x x
mult. 22, mult. 21,
4 3 2( ) 2 3 4 4P x x x x x
mult. 22, mult. 21,
2 2
( ) 2 1P x x x
mult. 22, mult. 21,
– 3, multiplicity 2
2, multiplicity 1
– 3, multiplicity 2
2, multiplicity 1
2( ) 3 2P x x x
– 3, multiplicity 2
2, multiplicity 1
2( ) 3 2P x x x
2( ) 3 2P x x x
2( 6) 29xP x x x
2( ) 3 2P x x x
2( 6) 29xP x x x
2( ) 3 2P x x x
2( 6) 29xP x x x
x2 6x 9
x
–2
3x 26x 9x22x 12x 18
x2 6x 9
x
–2
3x 26x 9x22x 12x 18
x2 6x 9
x
–2
3x 26x 9x22x 12x 18
3 2( ) 4 3 18P x x x x
3
( ) 2P x x
Mother Cubic Function
Shifted 2 to the left
3( )f x x 3
( ) 2P x x
3
( ) 2P x x
mult. 32,
The graph intersects the x-axis and
flattens out
as it passes through
that point.
3( ) 2P x x ( ) 2 2P x x x
3( ) 2P x x ( 2 2)P x x x
1
3( ) 2P x x ( 2 2)P x x x
1
If the degree of the binomial is odd and greater than 1, the graph will intersect the x-axis and flatten out
as it passes through that point.
If the degree of the binomial is odd and greater than 1, the graph will intersect the x-axis and flatten out
as it passes through that point.
–1 1 2 3
–1 1 2 3
3 2( ) 1 2 3P x x x x
mult. 31,
–1 1 2 3
3 2( ) 1 2 3P x x x x
mult. 22,
3 2
( ) 1 2 3P x x x x
–1 1 2 3
mult. 13,
1
4
( ) 2P x x
4
( ) 2P x x
4
( ) 2P x x
4
( ) 2P x x
The graph is tangent (touches) the x-axis but doesn’t cross the x-axis at that point
4
( ) 2P x x
mult. 42,
4
( ) 2P x x
mult. 42,
The graph is tangent (touches) the x-axis but doesn’t cross the x-axis at that point
4( ) 2P x x 2
( ) 2P x x
2( ) 3P x x
4( ) 3P x x
Example 1
4 3 26 9f x x x x
– 3
– 3 2nd TABLE
2 2 223 0 3f x x x x x
2 2 223 0 3f x x x x x
2nd TABLE
2 2 223 0 3f x x x x x
2 2 223 0 3f x x x x x
Example 2
4 23 2f x x x x
ZOOM 1: ZBox
22 0 1f x x x x
22 0 1f x x x x
22 0 1f x x x x
22 0 1f x x x x
22 0 1f x x x x
22 0 1f x x x x
2
2 1f x x x x
4 3 23) 5 9 7 2f x x x x x
5 4 3 24) 5 5 25 4 20f x x x x x x
4 3 23) 5 9 7 2f x x x x x
5 4 3 24) 5 5 25 4 20f x x x x x x
32 1f x x x
4 3 23) 5 9 7 2f x x x x x
5 4 3 24) 5 5 25 4 20f x x x x x x
32 1f x x x
5 2 1 1 2f x x x x x x