Question

Find the GCD of the polynomials 3x^4+6x^3-12x^2-24x, 4x^4 +14x^3+8x^2-8x​

Answer 1

Greatest common divisor (GCD) is X × (X+2)²

Step-by-step explanation:

• Greatest common divisor is nothing but highest common factor.

• Here is two polynomial is given

• First polynomial

        $\mathbf{P(X)= 3X^{4}+6X^{3}-12X^{2}-24X}$

        Taking common 3 and X in above polynomial

        P(X) = 3 X {X³ + 2X²-4X-8}

        P(X)= 3 X { X²(X+2) -4(X+2)}

        P(X) = 3 X { (X+2)(X²-4)}

        P(X) = 3× X × (X+2)× (X-2)× (X+2)

• Means it can be written as

        P(X) = 3× X × (X+2)²× (X-2)       ...1)

• Second polynomial

       $\mathbf{Q(X)= 4X^{4}+14X^{3}+8X^{2}-8X}$

      Taking common 2 and X in above polynomial

       Q(X) = 2 X {2 X³ + 7 X²+4 X-4}    

       Here 7 X² and 4 X can be split in two term as given below

       Q(X) = 2 X {2 X³ + 4 X²+3 X²+6 X-2 X-4}

       Q(X) = 2 X {2 X²(X + 2) +3 X (X+2) -2 (X +2)}

       Q(X) = 2 X (X+2) ( 2 X² + 3X -2)

        Again 3 X can be split in two term as given below

        Q(X) = 2 X (X+2) ( 2 X² + 4 X -X  -2)

        On solving, we will get

        Q(X) = 2× X ×(X+2) ×( 2 X-1) ×(X+2)

• Means it can be written as

        Q(X) = 2× X ×(X+2)² ×( 2 X-1)       ...2)

• Again writing P(X)

        P(X) = 3× X × (X+2)²× (X-2)            ...1)

• Here we see that common factor in both polynomial is  X × (X+2)²

• So GCD or HCF =  X × (X+2)²