help

If x - a is a factor of x^3 - 3*a*x^2 + 2*a^2*x + b, then find the value of b.

Wenn x - a ein Faktor von x ^ 3 - 3 * a * x ^ 2 + 2 * a ^ 2 * x + b ist, dann finde den Wert von b.

Hello Guest!

\(\frac{x ^ 3 - 3 a x ^ 2 + 2 a ^ 2 x + b}{ x-a}\\ =\frac{x ^ 3 - 3 a x ^ 2 + 2 a ^ 2 x }{x-a}+\frac{b}{x-a}\\ -\frac{x ^ 3 - 3 a x ^ 2 + 2 a ^ 2 x }{x-a}=\frac{b}{x-a}\\ \)

\(b=-x ^ 3 +3 a x ^ 2 -2 a ^ 2 x\)

  !

asinus

If x - a is a factor of x^3 - 3*a*x^2 + 2*a^2*x + b, then find the value of b.

Hello Guest!

\(\frac{x ^ 3 - 3 a x ^ 2 + 2 a ^ 2 x + b}{ x-a}\\ =\frac{x ^ 3 - 3 a x ^ 2 + 2 a ^ 2 x }{x-a} +\frac{b}{x-a}\\ -\frac{x ^ 3 - 3 a x ^ 2 + 2 a ^ 2 x }{x-a}=\frac{b}{x-a}\\ \color{blue}b=-x^3+3ax^2-2a^2x \)

\(\frac{x ^ 3 - 3 a x ^ 2 + 2 a ^ 2 x }{x-a}=-2 a x + x^2\)

\((x-a)\ is\ a\ Divisor\ of\ (x ^ 3 - 3 a x ^ 2 + 2 a ^ 2 x).\\ (x-a)\ is\ not\ a\ Divisor\ of\ (x ^ 3 - 3 a x ^ 2 + 2 a ^ 2 x\color{blue}+b)\ (b\neq 0).\\ (x-a)\ is\ a\ Divisor\ of\ (x ^ 3 - 3 a x ^ 2 + 2 a ^ 2 x\color{blue}+0).\\ so \)

\( b=0\)

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