Find the Roots (Zeros) x^4-x^3-11x^2+9x+18=0

$x^{4} - x^{3} - 11 x^{2} + 9 x + 18 = 0$

Step 1

Factor the left side of the equation.

Regroup terms.

$- x^{3} + 9 x + x^{4} - 11 x^{2} + 18 = 0$

Factor $- x$ out of $- x^{3} + 9 x$ .

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

$- x \cdot x^{2} + 9 x + x^{4} - 11 x^{2} + 18 = 0$

Factor $- x$ out of $9 x$ .

$- x \cdot x^{2} - x \cdot - 9 + x^{4} - 11 x^{2} + 18 = 0$

Factor $- x$ out of $- x (x^{2}) - x (- 9)$ .

$- x (x^{2} - 9) + x^{4} - 11 x^{2} + 18 = 0$

Rewrite $9$ as $3^{2}$ .

$- x (x^{2} - 3^{2}) + x^{4} - 11 x^{2} + 18 = 0$

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$- x ((x + 3) (x - 3)) + x^{4} - 11 x^{2} + 18 = 0$

Remove unnecessary parentheses.

$- x (x + 3) (x - 3) + x^{4} - 11 x^{2} + 18 = 0$

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$- x (x + 3) (x - 3) + {(x^{2})}^{2} - 11 x^{2} + 18 = 0$

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$- x (x + 3) (x - 3) + u^{2} - 11 u + 18 = 0$

Factor $u^{2} - 11 u + 18$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $18$ and whose sum is $- 11$ .

$- 9, - 2$

Write the factored form using these integers.

$- x (x + 3) (x - 3) + (u - 9) (u - 2) = 0$

Replace all occurrences of $u$ with $x^{2}$ .

$- x (x + 3) (x - 3) + (x^{2} - 9) (x^{2} - 2) = 0$

Rewrite $9$ as $3^{2}$ .

$- x (x + 3) (x - 3) + (x^{2} - 3^{2}) (x^{2} - 2) = 0$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$- x (x + 3) (x - 3) + (x + 3) (x - 3) (x^{2} - 2) = 0$

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

Factor $(x + 3) (x - 3)$ out of $- x (x + 3) (x - 3)$ .

$(x + 3) (x - 3) (- x) + (x + 3) (x - 3) (x^{2} - 2) = 0$

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

$(x + 3) (x - 3) (- x + x^{2} - 2) = 0$

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$(x + 3) (x - 3) (- u + u^{2} - 2) = 0$

Factor $- u + u^{2} - 2$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Write the factored form using these integers.

$(x + 3) (x - 3) ((u - 2) (u + 1)) = 0$

Factor.

Replace all occurrences of $u$ with $x$ .

$(x + 3) (x - 3) ((x - 2) (x + 1)) = 0$

Remove unnecessary parentheses.

$(x + 3) (x - 3) (x - 2) (x + 1) = 0$

Step 2

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

$x + 3 = 0$

$x - 3 = 0$

$x - 2 = 0$

$x + 1 = 0$

Step 3

Set $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 4

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

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Step 5

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Step 6

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 7

The final solution is all the values that make $(x + 3) (x - 3) (x - 2) (x + 1) = 0$ true.

$x = - 3, 3, 2, - 1$

Step 8

Find the Roots (Zeros) x^4-x^3-11x^2+9x+18=0

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