Multiplicity of Zero of Polynomial Function and Turning Around
Let f ( x)  2( x  3)( x  4).
Find the zeros for f ( x)  2( x  3)( x  4).
Solution :
f ( x)  2( x  3)( x  4).
f (3)  2(3  3)(3  4)  2(0)(1)  0
f (4)  2(4  3)(4  4)  2(1)(0)  0
3 and 4 are called the zeros of the function f ( x)  2( x  3)( x  4).
Note that graph crosses x-axis at 3 and 4; thus x-intercepts are (3, 0) and (4, 0).
Note: f ( x)  2( x  3)( x  4)  2( x  3)1 ( x  4)1.
Since 3 is a zero with multiplicity 1 (odd), graph touches the x-axis at 3
and then crosses the x-axis.
Since 4 is a zero with multiplicity 1 (odd), graph touches the x-axis at 4
and then crosses the x-axis.
Let f ( x)  2( x  1) 2 ( x  5)3 .
Find the zeros for f ( x)  2( x  1) 2 ( x  5)3
Solution :
f ( x)  2( x  1) 2 ( x  5)3 .
f (1)  2(1  1) 2 (1  5)3  2(0) 2 (6)3  0
f (5)  2(5  1) 2 (5  5)3  2( 6) 2 (0) 3  0
1 and -5 are called the zeros of the function f ( x)  2( x  1) 2 ( x  5)3 .
1 is a zero with multiplicity 2.
5 is a zero with multiplicity 3.
Note that x-intercepts are (-5, 0)
and (1, 0)
Let f ( x)  2( x  1)2 ( x  5)3.
Since 1 is a zero with multiplicity 2 (even), graph touches the x-axis at 1
and then turn around.
Since  5 is a zero with multiplicity 3 (odd), graph touches the x-axis at -5
and then crosses the x-axis.
Let f ( x)  4( x  2)( x  14 ) 4 .
Find the zeros of f ( x)  4( x  2)( x  14 ) 4 .
Solution :
f ( x)  4( x  2)( x  14 ) 4 .
f (2)  4(2  2)(2  14 ) 4  4(0)(2.25) 4  0.
f (1/ 4)  4(1/ 4  2)( 1/ 4  14 ) 4  4( 2.25)(0) 4  0.
2 and -1/4 are called the zeros of the function f ( x)  4( x  2)( x  14 ) 4 .
2 is a zero with multiplicity 1.
1/ 4 is a zero with multiplicity 4.
Note that x-intercepts are
(-1/4, 0) and (2, 0)
Let f ( x)  4( x  2)( x  14 )4 .
2 is a zero with multiplicity 1 (odd), graph touches x-axis at 2 and
then crosses the x-axis.
Since  1/ 4 is a zero with multiplicity 4, graph touches x-axis at -1/4 and
then turns around.
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