Find the Roots (Zeros) 5x^4+x^3+3x^2-7

$5 x^{4} + x^{3} + 3 x^{2} - 7$

Step 1

Set $5 x^{4} + x^{3} + 3 x^{2} - 7$ equal to $0$ .

$5 x^{4} + x^{3} + 3 x^{2} - 7 = 0$

Step 2

Solve for $x$ .

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.

$p = \pm 1, \pm 7$

$q = \pm 1, \pm 5$

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

$\pm 1, \pm \frac{1}{5}, \pm 7, \pm \frac{7}{5}$

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.

$5 {(- 1)}^{4} + {(- 1)}^{3} + 3 {(- 1)}^{2} - 7$

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 $4$ .

$5 \cdot 1 + {(- 1)}^{3} + 3 {(- 1)}^{2} - 7$

Multiply $5$ by $1$ .

$5 + {(- 1)}^{3} + 3 {(- 1)}^{2} - 7$

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

$5 - 1 + 3 {(- 1)}^{2} - 7$

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

$5 - 1 + 3 \cdot 1 - 7$

Multiply $3$ by $1$ .

$5 - 1 + 3 - 7$

Simplify by adding and subtracting.

Subtract $1$ from $5$ .

$4 + 3 - 7$

Add $4$ and $3$ .

$7 - 7$

Subtract $7$ from $7$ .

$0$

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.

$\frac{5 x^{4} + x^{3} + 3 x^{2} - 7}{x + 1}$

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.

$- 1$	$5$	$1$	$3$	$0$	$- 7$
	\xa0				\xa0

The first number in the dividend $(5)$ is put into the first position of the result area (below the horizontal line).

$- 1$	$5$	$1$	$3$	$0$	$- 7$
	\xa0				\xa0
	$5$

Multiply the newest entry in the result $(5)$ by the divisor $(- 1)$ and place the result of $(- 5)$ under the next term in the dividend $(1)$ .

$- 1$	$5$	$1$	$3$	$0$	$- 7$
	\xa0	$- 5$			\xa0
	$5$

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

$- 1$	$5$	$1$	$3$	$0$	$- 7$
	\xa0	$- 5$			\xa0
	$5$	$- 4$

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 $(3)$ .

$- 1$	$5$	$1$	$3$	$0$	$- 7$
	\xa0	$- 5$	$4$		\xa0
	$5$	$- 4$

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

$- 1$	$5$	$1$	$3$	$0$	$- 7$
	\xa0	$- 5$	$4$		\xa0
	$5$	$- 4$	$7$

Multiply the newest entry in the result $(7)$ by the divisor $(- 1)$ and place the result of $(- 7)$ under the next term in the dividend $(0)$ .

$- 1$	$5$	$1$	$3$	$0$	$- 7$
	\xa0	$- 5$	$4$	$- 7$	\xa0
	$5$	$- 4$	$7$

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

$- 1$	$5$	$1$	$3$	$0$	$- 7$
	\xa0	$- 5$	$4$	$- 7$	\xa0
	$5$	$- 4$	$7$	$- 7$

Multiply the newest entry in the result $(- 7)$ by the divisor $(- 1)$ and place the result of $(7)$ under the next term in the dividend $(- 7)$ .

$- 1$	$5$	$1$	$3$	$0$	$- 7$
	\xa0	$- 5$	$4$	$- 7$	$7$
	$5$	$- 4$	$7$	$- 7$

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

$- 1$	$5$	$1$	$3$	$0$	$- 7$
	\xa0	$- 5$	$4$	$- 7$	$7$
	$5$	$- 4$	$7$	$- 7$	$0$

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

$5 x^{3} + - 4 x^{2} + (7) x - 7$

Simplify the quotient polynomial.

$5 x^{3} - 4 x^{2} + 7 x - 7$

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

$x \approx 0.92421413$

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

$(x + 1) (x - 0.92421413)$

These are the roots (zeros) of the polynomial $5 x^{4} + x^{3} + 3 x^{2} - 7$ .

$x = - 1, 0.92421413$

Step 3

Find the Roots (Zeros) 5x^4+x^3+3x^2-7

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