Find the Roots (Zeros) 0=x^4-3x^3+10x^2-24x+16

$0 = x^{4} - 3 x^{3} + 10 x^{2} - 24 x + 16$

Step 1

Rewrite the equation as $x^{4} - 3 x^{3} + 10 x^{2} - 24 x + 16 = 0$ .

$x^{4} - 3 x^{3} + 10 x^{2} - 24 x + 16 = 0$

Step 2

Factor the left side of the equation.

Regroup terms.

$- 3 x^{3} - 24 x + x^{4} + 10 x^{2} + 16 = 0$

Factor $- 3 x$ out of $- 3 x^{3} - 24 x$ .

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

$- 3 x \cdot x^{2} - 24 x + x^{4} + 10 x^{2} + 16 = 0$

Factor $- 3 x$ out of $- 24 x$ .

$- 3 x \cdot x^{2} - 3 x \cdot 8 + x^{4} + 10 x^{2} + 16 = 0$

Factor $- 3 x$ out of $- 3 x (x^{2}) - 3 x (8)$ .

$- 3 x (x^{2} + 8) + x^{4} + 10 x^{2} + 16 = 0$

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

$- 3 x (x^{2} + 8) + {(x^{2})}^{2} + 10 x^{2} + 16 = 0$

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

$- 3 x (x^{2} + 8) + u^{2} + 10 u + 16 = 0$

Factor $u^{2} + 10 u + 16$ 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 $16$ and whose sum is $10$ .

$2, 8$

Write the factored form using these integers.

$- 3 x (x^{2} + 8) + (u + 2) (u + 8) = 0$

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

$- 3 x (x^{2} + 8) + (x^{2} + 2) (x^{2} + 8) = 0$

Factor $x^{2} + 8$ out of $- 3 x (x^{2} + 8) + (x^{2} + 2) (x^{2} + 8)$ .

Factor $x^{2} + 8$ out of $- 3 x (x^{2} + 8)$ .

$(x^{2} + 8) (- 3 x) + (x^{2} + 2) (x^{2} + 8) = 0$

Factor $x^{2} + 8$ out of $(x^{2} + 2) (x^{2} + 8)$ .

$(x^{2} + 8) (- 3 x) + (x^{2} + 8) (x^{2} + 2) = 0$

Factor $x^{2} + 8$ out of $(x^{2} + 8) (- 3 x) + (x^{2} + 8) (x^{2} + 2)$ .

$(x^{2} + 8) (- 3 x + x^{2} + 2) = 0$

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

$(x^{2} + 8) (- 3 u + u^{2} + 2) = 0$

Factor $- 3 u + u^{2} + 2$ 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 $- 3$ .

$- 2, - 1$

Write the factored form using these integers.

$(x^{2} + 8) ((u - 2) (u - 1)) = 0$

Factor.

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

$(x^{2} + 8) ((x - 2) (x - 1)) = 0$

Remove unnecessary parentheses.

$(x^{2} + 8) (x - 2) (x - 1) = 0$

Step 3

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

$x^{2} + 8 = 0$

$x - 2 = 0$

$x - 1 = 0$

Step 4

Set $x^{2} + 8$ equal to $0$ and solve for $x$ .

Set $x^{2} + 8$ equal to $0$ .

$x^{2} + 8 = 0$

Solve $x^{2} + 8 = 0$ for $x$ .

Subtract $8$ from both sides of the equation.

$x^{2} = - 8$

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

$x = \pm \sqrt{- 8}$

Simplify $\pm \sqrt{- 8}$ .

Rewrite $- 8$ as $- 1 (8)$ .

$x = \pm \sqrt{- 1 (8)}$

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{8}$

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

$x = \pm i \cdot \sqrt{8}$

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$x = \pm i \cdot \sqrt{4 (2)}$

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

$x = \pm i \cdot \sqrt{2^{2} \cdot 2}$

Pull terms out from under the radical.

$x = \pm i \cdot (2 \sqrt{2})$

Move $2$ to the left of $i$ .

$x = \pm 2 i \sqrt{2}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 2 i \sqrt{2}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2 i \sqrt{2}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2 i \sqrt{2}, - 2 i \sqrt{2}$

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 $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Step 7

The final solution is all the values that make $(x^{2} + 8) (x - 2) (x - 1) = 0$ true.

$x = 2 i \sqrt{2}, - 2 i \sqrt{2}, 2, 1$

Step 8

Find the Roots (Zeros) 0=x^4-3x^3+10x^2-24x+16

Related Questions

Support LCMGCF.com by disabling your adblocker.