Find the Roots (Zeros) y=x^4+3x^3+x^2-12x-20

$y = x^{4} + 3 x^{3} + x^{2} - 12 x - 20$

Step 1

Set $x^{4} + 3 x^{3} + x^{2} - 12 x - 20$ equal to $0$ .

$x^{4} + 3 x^{3} + x^{2} - 12 x - 20 = 0$

Step 2

Solve for $x$ .

Factor the left side of the equation.

Regroup terms.

$3 x^{3} - 12 x + x^{4} + x^{2} - 20 = 0$

Factor $3 x$ out of $3 x^{3} - 12 x$ .

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

$3 x (x^{2}) - 12 x + x^{4} + x^{2} - 20 = 0$

Factor $3 x$ out of $- 12 x$ .

$3 x (x^{2}) + 3 x (- 4) + x^{4} + x^{2} - 20 = 0$

Factor $3 x$ out of $3 x (x^{2}) + 3 x (- 4)$ .

$3 x (x^{2} - 4) + x^{4} + x^{2} - 20 = 0$

Rewrite $4$ as $2^{2}$ .

$3 x (x^{2} - 2^{2}) + x^{4} + x^{2} - 20 = 0$

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 $b = 2$ .

$3 x ((x + 2) (x - 2)) + x^{4} + x^{2} - 20 = 0$

Remove unnecessary parentheses.

$3 x (x + 2) (x - 2) + x^{4} + x^{2} - 20 = 0$

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

$3 x (x + 2) (x - 2) + {(x^{2})}^{2} + x^{2} - 20 = 0$

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

$3 x (x + 2) (x - 2) + u^{2} + u - 20 = 0$

Factor $u^{2} + u - 20$ 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 $- 20$ and whose sum is $1$ .

$- 4, 5$

Write the factored form using these integers.

$3 x (x + 2) (x - 2) + (u - 4) (u + 5) = 0$

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

$3 x (x + 2) (x - 2) + (x^{2} - 4) (x^{2} + 5) = 0$

Rewrite $4$ as $2^{2}$ .

$3 x (x + 2) (x - 2) + (x^{2} - 2^{2}) (x^{2} + 5) = 0$

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 $b = 2$ .

$3 x (x + 2) (x - 2) + (x + 2) (x - 2) (x^{2} + 5) = 0$

Factor $(x + 2) (x - 2)$ out of $3 x (x + 2) (x - 2) + (x + 2) (x - 2) (x^{2} + 5)$ .

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

$(x + 2) (x - 2) (3 x) + (x + 2) (x - 2) (x^{2} + 5) = 0$

Factor $(x + 2) (x - 2)$ out of $(x + 2) (x - 2) (3 x) + (x + 2) (x - 2) (x^{2} + 5)$ .

$(x + 2) (x - 2) (3 x + x^{2} + 5) = 0$

Reorder terms.

$(x + 2) (x - 2) (x^{2} + 3 x + 5) = 0$

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

$x + 2 = 0$

$x - 2 = 0$

$x^{2} + 3 x + 5 = 0$

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

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

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

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.

$x = 2$

Set $x^{2} + 3 x + 5$ equal to $0$ and solve for $x$ .

Set $x^{2} + 3 x + 5$ equal to $0$ .

$x^{2} + 3 x + 5 = 0$

Solve $x^{2} + 3 x + 5 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = 3$ , and $c = 5$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot 5)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 5$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 5}}{2 \cdot 1}$

Multiply $- 4$ by $5$ .

$x = \frac{- 3 \pm \sqrt{9 - 20}}{2 \cdot 1}$

Subtract $20$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 11}}{2 \cdot 1}$

Rewrite $- 11$ as $- 1 (11)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 11}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (11)}$ as $\sqrt{- 1} \cdot \sqrt{11}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{11}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 3 \pm i \sqrt{11}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm i \sqrt{11}}{2}$

The final answer is the combination of both solutions.

$x = - \frac{3 - i \sqrt{11}}{2}, - \frac{3 + i \sqrt{11}}{2}$

The final solution is all the values that make $(x + 2) (x - 2) (x^{2} + 3 x + 5) = 0$ true.

$x = - 2, 2, - \frac{3 - i \sqrt{11}}{2}, - \frac{3 + i \sqrt{11}}{2}$

Step 3

Find the Roots (Zeros) y=x^4+3x^3+x^2-12x-20

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