Find the Roots (Zeros) p(x)=x^5-8x^4-3x^2+24x

$p (x) = x^{5} - 8 x^{4} - 3 x^{2} + 24 x$

Step 1

Set $x^{5} - 8 x^{4} - 3 x^{2} + 24 x$ equal to $0$ .

$x^{5} - 8 x^{4} - 3 x^{2} + 24 x = 0$

Step 2

Solve for $x$ .

Factor the left side of the equation.

Factor $x$ out of $x^{5} - 8 x^{4} - 3 x^{2} + 24 x$ .

Factor $x$ out of $x^{5}$ .

$x \cdot x^{4} - 8 x^{4} - 3 x^{2} + 24 x = 0$

Factor $x$ out of $- 8 x^{4}$ .

$x \cdot x^{4} + x (- 8 x^{3}) - 3 x^{2} + 24 x = 0$

Factor $x$ out of $- 3 x^{2}$ .

$x \cdot x^{4} + x (- 8 x^{3}) + x (- 3 x) + 24 x = 0$

Factor $x$ out of $24 x$ .

$x \cdot x^{4} + x (- 8 x^{3}) + x (- 3 x) + x \cdot 24 = 0$

Factor $x$ out of $x \cdot x^{4} + x (- 8 x^{3})$ .

$x (x^{4} - 8 x^{3}) + x (- 3 x) + x \cdot 24 = 0$

Factor $x$ out of $x (x^{4} - 8 x^{3}) + x (- 3 x)$ .

$x (x^{4} - 8 x^{3} - 3 x) + x \cdot 24 = 0$

Factor $x$ out of $x (x^{4} - 8 x^{3} - 3 x) + x \cdot 24$ .

$x (x^{4} - 8 x^{3} - 3 x + 24) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$x ((x^{4} - 8 x^{3}) - 3 x + 24) = 0$

Factor out the greatest common factor (GCF) from each group.

$x (x^{3} (x - 8) - 3 (x - 8)) = 0$

Factor.

Factor the polynomial by factoring out the greatest common factor, $x - 8$ .

$x ((x - 8) (x^{3} - 3)) = 0$

Remove unnecessary parentheses.

$x (x - 8) (x^{3} - 3) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 8 = 0$

$x^{3} - 3 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 8$ equal to $0$ and solve for $x$ .

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Add $8$ to both sides of the equation.

$x = 8$

Set $x^{3} - 3$ equal to $0$ and solve for $x$ .

Set $x^{3} - 3$ equal to $0$ .

$x^{3} - 3 = 0$

Solve $x^{3} - 3 = 0$ for $x$ .

Add $3$ to both sides of the equation.

$x^{3} = 3$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{3}$

The final solution is all the values that make $x (x - 8) (x^{3} - 3) = 0$ true.

$x = 0, 8, \sqrt[3]{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 0, 8, \sqrt[3]{3}$

Decimal Form:

$x = 0, 8, 1.44224957 \dots$

Step 4

Find the Roots (Zeros) p(x)=x^5-8x^4-3x^2+24x

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