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# help

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

Jan 1, 2020

#1
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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

Jan 2, 2020
edited by asinus  Jan 2, 2020
#2
+9151
+2

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!

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

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

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

$$b=0$$

!

Jan 2, 2020