Solution of
ax
3
+
bx
2
+
cx
+
d
= 0 when d = 0
Eg.
x
3
− 2
x
2
+
x
= 0
In this case, $x$ is a common factor:
$ax^3+bx^2+cx = x(ax^2+bx+c)$ \t \gap[40]
Eg. $x^3+2x^2+x = x(x^2+2x+1)$
The equation is therefore $x(ax^2+bx+c)= 0$ and its solutions are $x = 0$ and the solutions of the quadratic $ax^2+bx+c = 0,$ if any.
Eg. $x(x^2+2x+1)=0$ \t $\quad \Rightarrow \quad x(x+1)^2 = 0$ \t $\quad \Rightarrow \quad x = 0, \ x = -1$
Example
The cubic equation $3x^3-5x^2+2x = 0$ has $a = 3, b = -5, c = 2, d = 0,$
$2x^3-5x^2+2x = 0 \quad$ \t $\Rightarrow \quad x(2x^2-5x+2) = 0$ \t $\Rightarrow \quad x(2x-1)(x-2) = 0$ \\ \t $\Rightarrow \quad x = 0,\ \ x = \dfrac{1}{2}, \ \ x = 2$
Some for you
Solve the following cubic equations. Enter each solution as a comma-separated list, for instance
-2, -1/2, 3
.
$%0;\ x ={}$ \t BOX* \t
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\\ \t $\ $ ACTIVEMATH
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$%1;\ x ={}$ \t BOX* \t
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\\ \t $\ $ ACTIVEMATH
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$%2;\ x ={}$ \t BOX* \t
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\\ \t $\ $ ACTIVEMATH
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RANDOMIZE