The minimum value of the polynomial x(x + 1)( | vedclass

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(C) Let $f(x) = x(x + 1)(x + 2)(x + 3)$.We can group the terms as follows:$f(x) = [x(x + 3)] \cdot [(x + 1)(x + 2)]$$f(x) = (x^2 + 3x)(x^2 + 3x + 2)$L

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$-1$

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(C) Let $f(x) = x(x + 1)(x + 2)(x + 3)$.We can group the terms as follows:$f(x) = [x(x + 3)] \cdot [(x + 1)(x + 2)]$$f(x) = (x^2 + 3x)(x^2 + 3x + 2)$Let $z = x^2 + 3x$. Then the expression becomes:$f(x) = z(z + 2) = z^2 + 2z$To find the minimum value,we complete the square:$f(x) = (z^2 + 2z + 1) - 1 = (z + 1)^2 - 1$Since $(z + 1)^2 \ge 0$,the minimum value of $f(x)$ is $-1$,which occurs when $z + 1 = 0$,i.e.,$z = -1$.Substituting back $z = x^2 + 3x$,we have $x^2 + 3x + 1 = 0$,which has real roots for $x$,so the minimum value $-1$ is attainable.

The minimum value of the polynomial $x(x + 1)(x + 2)(x + 3)$ is

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