Question

Find the GCD of the following pairs of polynomials using division algorithm.

Answer 1

GCD OF PAIR OF POLYNOMIALS BY DIVISION ALGORITHM :

FOR DIVISION PROCESS SEE PIC ATTACHED :

STEPS : A
1) We have,
${x}^{3} - 9 {x}^{2} + 23x - 15 = \\ 4( {x}^{2} - 4x + 3)(x - 5) + 0$
Here,
Second Low degree polynomial completely divides the Higher degree polynomial on dividing.
So,
Since, coefficient of higher degree is 1 :
Low Degree polynomial :
$( {x}^{2} - 4x + 3)$
is GCD of pair of polynomials.
See pic 1:
-------------
STEP :B
1 ) We have,
$(3 {x}^{3} + 18 {x}^{2} + 33x + 18) \\ = (3 {x}^{2} + 13x + 10)(x + \frac{5}{3} ) + \frac{4}{3} (x + 1) \\ = > (3 {x}^{2} + 13x + 10) = (3x + 10)(x + 1) + 0$
2)

Since, (x+1) is the last remainder.
So, GCD is (x+1)
See pic : 2
------------

STEP :C
1 ) We have,
Clearly,
$(6 {x}^{3} + 12 {x}^{2} + 6x + 12) = \\ 3(2 {x}^{3} + 2 {x}^{2} + 2x + 2) + 6( {x}^{2} +1 ) \\ = > (2 {x}^{3} + 2 {x}^{2} + 2x + 2) = 2(x( {x}^{2} + 1) )+ 2 {x}^{2} + 2$
Since,
2(x^2 + 1) is the last non -zero remainder.
So, GCD = 2(x^2 +1 )

STEP : D
1)
${x}^{4} + {x}^{3} + 4 {x}^{2} + 4x = \\ \: \:( {x}^{3} - 3 {x}^{2} + 4x - 12 )(x + 4) + 12( {x}^{2} + 4) \\ = > ( {x}^{3} - 3 {x}^{2} + 4x - 12) = ( {x}^{2} + 4)(x - 3) + 0$
Since,
(x^2 + 4) is the last non-zero remainder.
So,
GCD = (x^2 + 4)