Question

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

Answer 1

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Answer 2

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: Take 3 common from f(x) by g(x)

$f(x) = 3(2 {x}^{3} - 10 {x}^{2} + 20x - 16)$

$g(x) = 3({x}^{3} - 4{x}^{2} + 7x - 6)$

Now,Divide f(x) and g(x)

$({x}^{3} - 4 {x}^{2} + 7x - 6 )2 {x}^{3} - 10 {x}^{2} + 20x - 16(2 \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 2 {x}^{3} - 8 {x}^{2} + 14x - 12\\ ( - ) \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: - - - - - - - - - - - \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: - 2 {x}^{2} + 6x - 4 \\ or \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: - 2( {x}^{2} - 3x + 2)$

Step 3: Divide ${x}^{3} - 4 {x}^{2} + 7x - 6$ by x²-3x+2

${x}^{2} - 3x + 2) \: {x}^{3} - 4 {x}^{2} + 7x - 6(x - 1\\ {x}^{3} - 3 {x}^{2} + 2x \\ ( - ) \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ - - - - - - - - \\ -{x}^{2} + 5x - 6 \\ - {x}^{2} +3x - 2 \\ ( - ) \\ - - - - - - - - - \\ 2x - 4 \\ or \\ 2(x - 2)$

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).

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*: After every division leave the constant term.

To learn more on brainly:

find the GCD of the polynomials45(x4-x³-x²) and 75(8x5+x²) https://brainly.in/question/43783839