Question

Find the GCD of 6x3-30x2+60x – 48 and 3x3-12x2+21x-18. ​

Answer 1

Step-by-step explanation:

Given:$6 {x}^{3} - 30 {x}^{2} + 60x - 48$ and $3 {x}^{3} - 12 {x}^{2} + 21x - 18$

To find: GCD of both polynomial.

Solution:

Step 1: Assume one as f(x) and other as g(x)

Let

$f(x) = 6 {x}^{3} - 30 {x}^{2} + 60x - 48$

and

$g(x) = 3{x}^{3} - 12{x}^{2} + 21x - 18$

Step 2: Divide f(x) by g(x)

$3{x}^{3} - 12 {x}^{2} + 21x - 18 )6 {x}^{3} - 30 {x}^{2} + 60x - 48(2 \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 6 {x}^{3} - 24 {x}^{2} + 42x - 36 \\ ( - ) \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: - - - - - - - - - - - \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: - 6 {x}^{2} + 18x - 12 \\ or \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: - 6( {x}^{2} - 3x + 2)$

Step 3: Divide $3 {x}^{3} - 12 {x}^{2} + 21x - 18$ by x²-3x+2

${x}^{2} - 3x + 2) \: 3 {x}^{3} - 12 {x}^{2} + 21x - 18(3x - 3 \\ 3 {x}^{3} - 9 {x}^{2} + 6x \\ ( - ) \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ - - - - - - - - \\ - 3 {x}^{2} + 15x - 18 \\ - 3{x}^{2} + 9x - 6 \\ ( - ) \\ - - - - - - - - - \\ 6x - 12 \\ or \\ 6(x - 2)$

Step 4: Divide x²-3x+2 by (x-2)

$x - 2) {x}^{2} - 3x + 2( x - 1 \\ {x}^{2} - 2x \\ ( - ) \\ - - - - - - \\ -x + 2 \\ - x + 2 \\ ( - ) \\ - - - - - - - \\ 0 \\ - - - - - - - -$

Step 5: Remainder is 0 now,Thus (x-2) is a common factor of f(x) and g(x).

On analysis it is clear that 3 is also common in both.

Final answer:

GCD of $6 {x}^{3} - 30{x}^{2} + 60x - 48$ and $3 {x}^{3} - 12 {x}^{2} + 21x - 18$ is 3(x-2).

Hope it helps you.

Note*: It is better to take any common constant term out before start division.

After every division leave the constant term.

To learn more on brainly:

find the GCD of the polynomials

45(x4-x³-x²) and 75(8x5+x²)

https://brainly.in/question/43783839