Identify the roots of the equation: `2x^3 + 12 x^2 - 72x - 432` State the multiplicity of each root. .

Monday, August 12, 2013

$\displaystyle{2}{{x}}^{{3}}+{12}{{x}}^{{2}}-{72}{x}-{432}={2}{{x}}^{{2}}{\left({x}+{6}\right)}-{72}{\left({x}+{6}\right)}$

$\displaystyle={\left({2}{{x}}^{{2}}-{72}\right)}{\left({x}+{6}\right)}$

The roots of an equation, f(x) is 'a' if, f(a) = 0.

Thus, the roots of the current equation can be calculated as:

$\displaystyle{2}{{x}}^{{2}}-{72}={0}$

$\displaystyle{2}{{x}}^{{2}}={72}$

$\displaystyle{{x}}^{{2}}=\frac{{72}}{{2}}={36}$

$\displaystyle{x}=\sqrt{{{36}}}={6}{\quad\text{and}\quad}-{6}$

And another root is:

x + 6 = 0

or, x = -6

Thus the roots of the equation are 6 and -6 (appears twice).

The equation can also be written in a simplified as: $\displaystyle{2}{{\left({x}+{6}\right)}}^{{2}}{\left({x}-{6}\right)}$

Since, the root x = -6 appears twice in the equation, the multiplicity of -6 is 2. Similarly, the multiplicity of x = 6 is 1.

Hope this helps.

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