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Answer» Hi,
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Factor Theorem :
If p( x ) is a polynomial of degree n ≥ 1 and
' a ' is any real number , then
i )( x - a ) is a factor of p( x ) , if p( a ) = 0,
ii ) and it's converse " if ( x - a ) is a factor of
a polynomial p( x ) then p ( a ) = 0,
***********"***********************
Now ,
Let p( x ) = 12x² - 7x + 1 ,
= 12x² - 4x - 3x + 1
= 4x ( 3x - 1 ) - 1( 3x - 1 )
= ( 3x - 1 )( 4x - 1 )
TAKE 3x - 1 = 0 or 4x - 1 = 0
x = 1/3 or x = 1/4
p ( 1/3 ) = 12 × ( 1/3 )² - 7 ( 1/3 ) + 1
= 12/9 - 7/3 + 1
= 4/3 - 7/3 + 1
= ( 4 - 7 + 3 )/3
= 0
Therefore ,
x - 1/3 is a factor of p ( x ) .
2 ) p( 1/4 ) = 12 ( 1/4 )² - 7( 1/4 ) + 1
= 12/16 - 7/4 + 1
= 3/4 - 7/4 + 1
= ( 3 - 7 + 4 ) / 4
= 0
Therefore ,
x - 1/4 is a factor of p( x ).
Now , we CONCLUDE that ,
x - 1/3 , x - 1/4 are two FACTORS of p( x ).
I hope this helps you.
: )
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