Question

What must be subtracted from f(x) = 3x4 + 30x3 – 62x2 + 17x – 10 so that resulting polynomial is divisible by g(x) = x2 – x + 1?

Answer 1

Answer:

-48x + 22

Step-by-step explanation:

f(x) = 3x⁴ + 30x³  - 62x²  + 17x - 10

g(x) = x² - x + 1

q(x) = ax² + bx + c

r(x) = dx + e

f(x) - r(x) = g(x) q(x)

g(x) q(x)

= (x² - x + 1)(ax² + bx + c)

=  ax⁴  + x³(b-a) + x²(c + a - b) + x(b -c) + c

comparing  with

3x⁴ + 30x³  - 62x²  + 17x - 10 - (dx + e)

= 3x⁴ + 30x³  - 62x²  + (17-d)x - (10 + e)

a = 3

b-a = 30 => b = 33

c + a - b = -62 => c -30 = -62 => c = -32

b-c = 17-d  => 33 -(-32) = 17 - d => d = -48

c = -(10 + e) => -32 = -(10 + e) => e = 22

Remainder = -48x + 22

f(x) - r(x)

3x⁴ + 30x³  - 62x²  + 17x - 10 -(-48x + 22 )

= 3x⁴ + 30x³  - 62x²  + 65x - 32

= x²(3x² + 33x  -32)   -x(3x² + 33x -32)   + 1(3x² + 33x -32)

= (x² - x + 1)(3x² + 33x -32)

= g(x) q(x)