Question

INSE -Maths Test 2] Find the Gcd of the following Polynomials: x3 + x²+x+1 and 2x²-3x2+2x-3​

Answer 1

$\large\underline{\sf{Solution-}}$

Given polynomials are

$\rm \: f(x) = {x}^{3} + {x}^{2} + x + 1$

and

$\rm \: g(x) = {2x}^{3} - 3{x}^{2} + 2x - 3$

Now, Consider

$\rm \: f(x) = {x}^{3} + {x}^{2} + x + 1$

can be rewritten as

$\rm \: f(x) = {x}^{2}(x +1) + 1(x + 1)$

$\rm \: f(x) = (x +1)( {x}^{2} + 1)$

Now, Consider

$\rm \: g(x) = {2x}^{3} - 3{x}^{2} + 2x - 3$

$\rm \: g(x) = {x}^{2} (2x - 3) +1( 2x - 3)$

$\rm \: g(x) = (2x - 3)( {x}^{2} + 1)$

So, f(x) and g(x) in factorized form is as

$\rm \: f(x) = (x + 1)( {x}^{2} + 1)$

and

$\rm \: g(x) = (2x - 3)( {x}^{2} + 1)$

So, GCD is the common factor of two polynomials f(x) and g(x).

$\rm\implies \:GCD \: = \: {x}^{2} + 1$

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ADDITIONAL INFORMATION

Algebraic Identities :

➢  (a + b)² = a² + 2ab + b²

➢  (a - b)² = a² - 2ab + b²

➢  a² - b² = (a + b)(a - b)

➢  (a + b)² = (a - b)² + 4ab

➢  (a - b)² = (a + b)² - 4ab

➢  (a + b)² + (a - b)² = 2(a² + b²)

➢  (a + b)³ = a³ + b³ + 3ab(a + b)

➢  (a - b)³ = a³ - b³ - 3ab(a - b)