Algebra Examples

$f (x) = x^{3} + 3 x^{2} - 7 x - 21$

Step 1

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.

$p = \pm 1, \pm 3, \pm 7, \pm 21$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 3, \pm 7, \pm 21$

Step 3

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.

${(- 3)}^{3} + 3 {(- 3)}^{2} - 7 \cdot - 3 - 21$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - 3$ is a root of the polynomial.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

$- 27 + 3 {(- 3)}^{2} - 7 \cdot - 3 - 21$

Step 4.1.2

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

$- 27 + 3 \cdot 9 - 7 \cdot - 3 - 21$

Step 4.1.3

Multiply $3$ by $9$ .

$- 27 + 27 - 7 \cdot - 3 - 21$

Step 4.1.4

Multiply $- 7$ by $- 3$ .

$- 27 + 27 + 21 - 21$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Add $- 27$ and $27$ .

$0 + 21 - 21$

Step 4.2.2

Add $0$ and $21$ .

$21 - 21$

Step 4.2.3

Subtract $21$ from $21$ .

$0$

Step 5

Since $- 3$ is a known root, divide the polynomial by $x + 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} + 3 x^{2} - 7 x - 21}{x + 3}$

Step 6

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

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Step 6.1

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

$- 3$	$1$	$3$	$- 7$	$- 21$

Step 6.2

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

$- 3$	$1$	$3$	$- 7$	$- 21$

	$1$

Step 6.3

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

$- 3$	$1$	$3$	$- 7$	$- 21$
		$- 3$
	$1$

Step 6.4

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

$- 3$	$1$	$3$	$- 7$	$- 21$
		$- 3$
	$1$	$0$

Step 6.5

Multiply the newest entry in the result $(0)$ by the divisor $(- 3)$ and place the result of $(0)$ under the next term in the dividend $(- 7)$ .

$- 3$	$1$	$3$	$- 7$	$- 21$
		$- 3$	$0$
	$1$	$0$

Step 6.6

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

$- 3$	$1$	$3$	$- 7$	$- 21$
		$- 3$	$0$
	$1$	$0$	$- 7$

Step 6.7

Multiply the newest entry in the result $(- 7)$ by the divisor $(- 3)$ and place the result of $(21)$ under the next term in the dividend $(- 21)$ .

$- 3$	$1$	$3$	$- 7$	$- 21$
		$- 3$	$0$	$21$
	$1$	$0$	$- 7$

Step 6.8

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

$- 3$	$1$	$3$	$- 7$	$- 21$
		$- 3$	$0$	$21$
	$1$	$0$	$- 7$	$0$

Step 6.9

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

$1 x^{2} + 0 x - 7$

Step 6.10

Simplify the quotient polynomial.

$x^{2} - 7$

Step 7

Solve the equation to find any remaining roots.

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Step 7.1

Add $7$ to both sides of the equation.

$x^{2} = 7$

Step 7.2

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

$x = \pm \sqrt{7}$

Step 7.3

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

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Step 7.3.1

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

$x = \sqrt{7}$

Step 7.3.2

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

$x = - \sqrt{7}$

Step 7.3.3

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

$x = \sqrt{7}, - \sqrt{7}$

Step 8

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

$(x + 3) (x - \sqrt{7}) (x + \sqrt{7})$

Step 9

These are the roots (zeros) of the polynomial $x^{3} + 3 x^{2} - 7 x - 21$ .

$x = - 3, \sqrt{7}, - \sqrt{7}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = - 3, \sqrt{7}, - \sqrt{7}$

Decimal Form:

$x = - 3, 2.64575131 \dots, - 2.64575131 \dots$

Step 11