Question

find the value of a for which the polynomial 2x^4-ax^3+4x^2+2x+1 is divisible by 1-2x

Answer 1

Hey!!!!!

We have

=> 2x⁴ - ax³ + 4x² + 2x + 1 = p(x)

Thus for the divisiblity of 1 - 2x

=> 1 - 2x = 0

=> x = 1/2

Thus P(1/2) = 0

=> 2(1/2)⁴ - a(1/2)³ + 4(1/2)² + 2(1/2) + 1 = 0

=> 2(1/16) - a(1/8) + 4(1/4) + 1 + 1 = 0

=> 1/8 - a/8 + 1 + 1 + 1 = 0

=> 1/8 - a/8 + 3 = 0

=> a/8 = 25/8

=> a = 25

Hope this helps ✌️

Answer 2

Value of a = 25  if Polynomial 2x⁴-ax³+4x²+2x+1 is divisible by 1-2x

Given:

Polynomial 2x⁴-ax³+4x²+2x+1 is divisible by 1-2x

To Find:

Value of a

Solution:

Factor Theorem. x – a is a factor of the polynomial p(x), if p(a) = 0.  

Also, if x – a is a factor of p(x), then p(a) = 0,  

where a is any real number.

Step 1:

Equate 1 -2x with 0 and solve for x

1 - 2x = 0

=> 2x = 1

=> x = 1/2

Step 2:

Substitute x = 1/2 in 2x⁴-ax³+4x²+2x+1 , equate with 0  and solve for a

2(1/2)⁴ - a(1/2)³ + 4(1/2)² + 2(1/2) + 1 = 0

=> 2/16 - a/8  + 4/4 + 2/2 + 1 = 0

=> 1/8 - a/8 + 1 + 1 + 1 = 0

=> 1/8  + 3  = a/8

=> 1  + 24 = a

=> 25 = a

Value of a = 25