Let `|[x^2+3x,x-1,x+3],[x+1,-2x,x-4],[x-3,x+4, 3x]|` =`a x^4+b x^3+c x^2+e` be an identity in `x ,` where `a , b , c , d , e` are independent of `xdot` Then the value of `e` is 4 (b) 0 (c) 1 (d) none of these

Explanation

Given,`|[x^2+3x,x-1,x+3],[x+1,-2x,x-4],[x-3,x+4, 3x]|`
=`a x^4+b x^3+c x^2+e`
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⇒` (x^2 + 3x)[-2x × 3x – (x – 4)(x + 4)] – (x – 1)[(x + 1) × 3x – (x – 4)(x – 3)] + (x + 3)[(x + 1)(x + 4) – (-2x)(x – 3)] = ax4^ + bx^3 + cx^2 + dx + e`

⇒ `(x^2 + 3x)[-6x – (x^2 – 16)] – (x – 1)[3x(x + 1) – (x^2 – 3x – 4x + 12)] + (x + 3)[x^2 + x + 4x + 4 + 2x(x – 3)] = ax^4 + bx^3 + cx^2 + dx + e `

⇒ `(x^2 + 3x)[-6x – x^2 + 16] – (x – 1)[3x^2 + 3x – x^2 + 7x – 12] + (x + 3)[x^2 + 5x + 4 + 2x^2 – 6x] = ax^4 + bx^3 + cx^2 + dx + e`

⇒ `-x^4 – 6x^3 + 16x^2 – 3x^3 – 18x^2 + 48x – (x – 1)[2x^2 + 10x – 12] + (x + 3)[3x^2 – x + 4] = ax^4 + bx^3 + cx^2 + dx + e `

⇒ `-x^4 – 9x^3 – 2x^2 + 48x – (2x^3 – 2x^2 + 10x^2 – 10x – 12x + 12) + 3x^3 + 9x^2 – x^2 – 3x + 4x + 12 = ax^4 + bx^3 + cx^2 + dx + e`

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