How do you factor x^3 - 12x^2 + 35x?

Apr 22, 2018

The answer is $x \left(x - 5\right) \left(x - 7\right)$.

Explanation:

First we take out the common factor x:
${x}^{3} - 12 {x}^{2} + 35 x = x \left({x}^{2} - 12 x + 35\right)$

Then we use the quadratic formula:
$x = \setminus \frac{- b \setminus \pm \left(\setminus \sqrt{{b}^{2} - 4 a c \setminus}\right)}{2 a}$

$x = \setminus \frac{- \left(- 12\right) \setminus \pm \left(\setminus \sqrt{{\left(- 12\right)}^{2} - 4 \cdot 1 \cdot 35 \setminus}\right)}{2 \cdot 1}$

$x = \setminus \frac{12 \setminus \pm \left(\setminus \sqrt{144 - 140 \setminus}\right)}{2}$

$x = \setminus \frac{12 \setminus \pm \setminus \sqrt{4}}{2}$

$x = \setminus \frac{12 \setminus \pm \setminus 2}{2}$

${x}_{1} = 5$

${x}_{2} = 7$

$a {x}^{2} + b x + c = a \cdot \left(x - {x}_{1}\right) \left(x - {x}_{2}\right)$

Then we just write this instead of the quadratic equation we've just solved:

${x}^{3} - 12 {x}^{2} + 35 x = x \left(x - 5\right) \left(x - 7\right)$

Apr 22, 2018

$x \left(x - 7\right) \left(x - 5\right)$

Explanation:

Start by taking out the common factor of $x$

${x}^{3} - 12 {x}^{2} + 35 x$

$= x \left({x}^{2} - 12 x + 35\right)$

Find factors of $35$ which add to make $12$
$7 \times 5$ will do nicely: $7 \times 5 = 35 \mathmr{and} 7 + 5 = 12$
$x \left(x \text{ "7)(x" } 5\right)$
For the signs, they both have to be the same to get $+ 35$, but they need to add to $- 12$
$x \left(x - 7\right) \left(x - 5\right)$