Question

If a polynomial x^4-3x^3-6x^2+kx-16is exactly divisible by x^2-3x+2 then find value of K.

Answer 1

Hello Mate!

What we have?

Polynomial p(x) = x⁴ - 3x³ - 6x² + kx - 16 and divisior x² - 3x + 2.

What can we find?

Factors of x² - 3x + 2.

What we have to find?

Value of k, just find factors and then value of x. Keep values accordingly in p(x) and we get value of k.

Answer : Value of k is 24.

For solution please refer to attachment.

Have great future ahead!

Answer 2

Find the factor :

x² - 3x + 2

⇒ x² - ( 1 + 2 ) x + 2

⇒ x² - x - 2x + 2

⇒ x ( x - 1 ) - 2 ( x - 1 )

⇒ ( x - 1 ) ( x - 2 )

∴ x = 2 and x = 1 are the zeros of polynomial x⁴ - 3 x³ - 6 x² + kx - 16.

Putting the value of x = 1 in polynomial :

x⁴ - 3 x³ - 6 x² + kx - 16 = 0

⇒ 1⁴ - 3 ( 1 )³ - 6 ( 1 )² + k × 1 - 16 = 0

⇒ 1 - 3 - 6 + k - 16 = 0

⇒ k - 24 = 0

⇒ k = 24.          ........... ( i )

Putting the value of x = 2 in polynomial :

x⁴ - 3 x³ - 6 x² + kx - 16 = 0

⇒ 2⁴ - 3 ( 2 )³ - 6 ( 2 )² + k × 2 - 16 = 0

⇒ 16 - 24 - 24 + k - 16 = 0

⇒ 2k - 48 = 0

⇒ 2k = 48

⇒ k = 24       ............. ( ii )

From ( i ) and ( ii ), we get ;

∴  k = 24

Answer : The value of k is 24.

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Thanks for the question !

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