In the polynomial 9x³ - kx + 4, where k is an integer, if 3x - 2 is a factor of the polynomial, what is the value of k?
Answer (2)
To find the value of k in the polynomial 9 x 3 − k x + 4 given that 3 x − 2 is a factor, we can use the Factor Theorem.
The Factor Theorem states that a polynomial f ( x ) has a factor ( x − r ) if and only if f ( r ) = 0 . Since 3 x − 2 is a factor, let's find r such that 3 r − 2 = 0 . Solving for r :
3 r − 2 = 0 3 r = 2 r = 3 2
Now, substitute x = 3 2 into the polynomial:
f ( 3 2 ) = 9 ( 3 2 ) 3 − k ( 3 2 ) + 4
Calculate 9 ( 3 2 ) 3 :
9 ( 3 2 ) 3 = 9 × 27 8 = 27 72 = 3 8
Now, substitute back:
f ( 3 2 ) = 3 8 − k ( 3 2 ) + 4 = 0
Multiply everything by 3 to clear the denominators:
8 − 2 k + 12 = 0
Simplify:
20 − 2 k = 0
Solve for k :
2 k = 20 k = 10
Therefore, the value of k is 10 .
The value of k in the polynomial 9x³ - kx + 4, where 3x - 2 is a factor, is 10. This is determined using the Factor Theorem, which involves substituting the value of r and solving for k. After following the necessary steps, we find that k equals 10.
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