How do you find all the real and complex roots of $f (x) = x^{3} - 4 x^{2} - 16 x + 64$ $f(x) = x^3 - 4x^2 - 16x + 64$ ?

Question

2961 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-03-01T02:55:52

Use Grouping ...

$f (x) = x^{3} - 4 x^{2} - 16 x + 64$ $f(x) = x^3 - 4x^2 - 16x + 64$

$f (x) = (x^{3} - 4 x^{2}) - (16 x - 64)$ $f(x) = (x^3 - 4x^2) - (16x - 64)$

Next, factor out common terms ...

$f (x) = x^{2} (x - 4) - 16 (x - 4)$ $f(x) =x^2 (x - 4) - 16(x - 4)$

Simplify and set equal to zero ...

$f (x) = (x^{2} - 16) (x - 4) = 0$ $f(x) =(x^2 - 16)(x - 4)=0$

Solve for x ...

$x^{2} - 16 = 0$ $x^2-16=0$
$x = \pm 4$ $x=+-4$

$x - 4 = 0$ $x-4=0$
$x = 4$ $x=4$

So, the solutions are $x = 4$ $x=4$ or $x = - 4$ $x=-4$ with $x = 4$ $x=4$ a double root .

Hope that helped

enter image source here

How do you find all the real and complex roots of f(x) = x^3 - 4x^2 - 16x + 64f(x)=x3−4x2−16x+64f(x) = x^3 - 4x^2 - 16x + 64?