Find \( k \) if \((x-3)\) is a factor of [tex] k^{2} x^{2} - kx - 2[/tex].

w ( x ) = k 2 x 2 − k x − 2 ( x − 3 ) i s a f a c t or w ( x ) t h e n w ( 3 ) = 0. S u b s t i t u t e : k 2 ⋅ 3 2 − k ⋅ 3 − 2 = 0 9 k 2 − 3 k − 2 = 0 9 k 2 − 6 k + 3 k − 2 = 0 3 k ( 3 k − 2 ) + 1 ( 3 k − 2 ) = 0 ( 3 k − 2 ) ( 3 k + 1 ) = 0 ⟺ 3 k − 2 = 0 ∨ 3 k + 1 = 0 3 k = 2 ∨ 3 k = − 1 k = 3 2 ​ ∨ k = − 3 1 ​ ​ ​

Answered by Anonymous | 2024-06-10

k can be (-1/3) or (2/3). since (x-3) is a factor. if you put x=3 in the equation, it must be equal to zero. So the equation becomes (9k2 - 3k -2 = 0). if you solve this you will get k=(-1/3) and k=(2/3). You can substitute this k in the original equation and check, (x-3) is a factor.

Answered by songaraspcm | 2024-06-10

The values of k such that ( x − 3 ) is a factor of the polynomial k 2 x 2 − k x − 2 are 3 2 ​ and − 3 1 ​ . This is determined by applying the Factor Theorem and solving the resulting quadratic equation. By substituting x = 3 , we can find the necessary conditions for k .
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Answered by Anonymous | 2024-12-24