Find the value of k if x - 3 is a factor of k2x3 - kx2 + 3kx - k.

1.	Find the value of k if x - 3 is a factor of k2x3 - kx2 + 3kx - k.
Answer» Let, f (x) = k²x³ - kx² + 3kx - k By factor theorem, If (x – 3) is a factor of f (x) then f (3) = 0 k² (3)³ – k (3)² + 3 k (3) – k = 0 27k² – 9k + 9k – k = 0 k (27k – 1) = 0 k = 0 or (27k – 1) = 0 k = 0 or k = \(\frac{1}{27}\) Hence, (x – 3) is a factor of f (x) when k = 0 or k = \(\frac{1}{27}.\)

Answer»

Let, f (x) = k²x³ - kx² + 3kx - k

By factor theorem,

If (x – 3) is a factor of f (x) then f (3) = 0

k² (3)³ – k (3)² + 3 k (3) – k = 0

27k² – 9k + 9k – k = 0

k (27k – 1) = 0

k = 0 or (27k – 1) = 0

k = 0 or k = \(\frac{1}{27}\)

Hence,

(x – 3) is a factor of f (x) when k = 0 or k = \(\frac{1}{27}.\)

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