If x^2 + x + 1 is a factor of ax^3 + bx^2 + c | vedclass

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(A) Let $P(x) = ax^3 + bx^2 + cx + d$. Since $x^2 + x + 1$ is a factor of $P(x)$,we can write $P(x) = (x^2 + x + 1)(ax + k)$ for some constant $k$. Ex

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$-\frac{d}{a}$

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(A) Let $P(x) = ax^3 + bx^2 + cx + d$. Since $x^2 + x + 1$ is a factor of $P(x)$,we can write $P(x) = (x^2 + x + 1)(ax + k)$ for some constant $k$. Expanding this,we get $P(x) = ax^3 + (a+k)x^2 + (a+k)x + k$. Comparing the constant term with the original polynomial,we have $k = d$. Thus,$P(x) = (x^2 + x + 1)(ax + d)$. Setting $P(x) = 0$,we get $(x^2 + x + 1)(ax + d) = 0$. The roots of $x^2 + x + 1 = 0$ are complex (discriminant $D = 1^2 - 4(1)(1) = -3 The real root comes from $ax + d = 0$,which gives $x = -\frac{d}{a}$.

If $x^2 + x + 1$ is a factor of $ax^3 + bx^2 + cx + d$,then what is the real root of $ax^3 + bx^2 + cx + d = 0$?

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