Solve for x x^4+4x^3+3x^2+4x+4=0

$x^{4} + 4 x^{3} + 3 x^{2} + 4 x + 4 = 0$

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 2, \pm 4$

$q = \pm 1$

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 2, \pm 4$

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is $0$ , which means it is a root.

${(- 1)}^{4} + 4 {(- 1)}^{3} + 3 {(- 1)}^{2} + 4 (- 1) + 4$

Simplify the expression. In this case, the expression is equal to $0$ so $x = - 1$ is a root of the polynomial.

Simplify each term.

Raise $- 1$ to the power of $4$ .

$1 + 4 {(- 1)}^{3} + 3 {(- 1)}^{2} + 4 (- 1) + 4$

Raise $- 1$ to the power of $3$ .

$1 + 4 \cdot - 1 + 3 {(- 1)}^{2} + 4 (- 1) + 4$

Multiply $4$ by $- 1$ .

$1 - 4 + 3 {(- 1)}^{2} + 4 (- 1) + 4$

Raise $- 1$ to the power of $2$ .

$1 - 4 + 3 \cdot 1 + 4 (- 1) + 4$

Multiply $3$ by $1$ .

$1 - 4 + 3 + 4 (- 1) + 4$

Multiply $4$ by $- 1$ .

$1 - 4 + 3 - 4 + 4$

Simplify by adding and subtracting.

Subtract $4$ from $1$ .

$- 3 + 3 - 4 + 4$

Add $- 3$ and $3$ .

$0 - 4 + 4$

Subtract $4$ from $0$ .

$- 4 + 4$

Add $- 4$ and $4$ .

$0$

Since $- 1$ is a known root, divide the polynomial by $x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} + 4 x^{3} + 3 x^{2} + 4 x + 4}{x + 1}$

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$- 1$	$1$	$4$	$3$	$4$	$4$
	\xa0				\xa0

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$- 1$	$1$	$4$	$3$	$4$	$4$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 1)$ and place the result of $(- 1)$ under the next term in the dividend $(4)$ .

$- 1$	$1$	$4$	$3$	$4$	$4$
	\xa0	$- 1$			\xa0
	$1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 1$	$1$	$4$	$3$	$4$	$4$
	\xa0	$- 1$			\xa0
	$1$	$3$

Multiply the newest entry in the result $(3)$ by the divisor $(- 1)$ and place the result of $(- 3)$ under the next term in the dividend $(3)$ .

$- 1$	$1$	$4$	$3$	$4$	$4$
	\xa0	$- 1$	$- 3$		\xa0
	$1$	$3$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 1$	$1$	$4$	$3$	$4$	$4$
	\xa0	$- 1$	$- 3$		\xa0
	$1$	$3$	$0$

Multiply the newest entry in the result $(0)$ by the divisor $(- 1)$ and place the result of $(0)$ under the next term in the dividend $(4)$ .

$- 1$	$1$	$4$	$3$	$4$	$4$
	\xa0	$- 1$	$- 3$	$0$	\xa0
	$1$	$3$	$0$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 1$	$1$	$4$	$3$	$4$	$4$
	\xa0	$- 1$	$- 3$	$0$	\xa0
	$1$	$3$	$0$	$4$

Multiply the newest entry in the result $(4)$ by the divisor $(- 1)$ and place the result of $(- 4)$ under the next term in the dividend $(4)$ .

$- 1$	$1$	$4$	$3$	$4$	$4$
	\xa0	$- 1$	$- 3$	$0$	$- 4$
	$1$	$3$	$0$	$4$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 1$	$1$	$4$	$3$	$4$	$4$
	\xa0	$- 1$	$- 3$	$0$	$- 4$
	$1$	$3$	$0$	$4$	$0$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + 3 x^{2} + (0) x + 4$

Simplify the quotient polynomial.

$x^{3} + 3 x^{2} + 4$

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 3.35530139$

The polynomial can be written as a set of linear factors.

$(x + 1) (x + 3.35530139)$

These are the roots (zeros) of the polynomial $x^{4} + 4 x^{3} + 3 x^{2} + 4 x + 4$ .

$x = - 1, - 3.35530139$

Solve for x x^4+4x^3+3x^2+4x+4=0

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