Precalculus Examples

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

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.

$p = \pm 1, \pm 3$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 3$

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.

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

Step 4

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

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Step 4.1

Simplify each term.

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Step 4.1.1

One to any power is one.

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

Step 4.1.2

One to any power is one.

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

Step 4.1.3

Multiply $3$ by $1$ .

$1 + 3 - (1) - 3$

Step 4.1.4

Multiply $- 1$ by $1$ .

$1 + 3 - 1 - 3$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Add $1$ and $3$ .

$4 - 1 - 3$

Step 4.2.2

Subtract $1$ from $4$ .

$3 - 3$

Step 4.2.3

Subtract $3$ from $3$ .

$0$

Step 5

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.

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

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.

$1$	$1$	$3$	$- 1$	$- 3$

Step 6.2

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

$1$	$1$	$3$	$- 1$	$- 3$

	$1$

Step 6.3

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 $(3)$ .

$1$	$1$	$3$	$- 1$	$- 3$
		$1$
	$1$

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.

$1$	$1$	$3$	$- 1$	$- 3$
		$1$
	$1$	$4$

Step 6.5

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 $(- 1)$ .

$1$	$1$	$3$	$- 1$	$- 3$
		$1$	$4$
	$1$	$4$

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.

$1$	$1$	$3$	$- 1$	$- 3$
		$1$	$4$
	$1$	$4$	$3$

Step 6.7

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 $(- 3)$ .

$1$	$1$	$3$	$- 1$	$- 3$
		$1$	$4$	$3$
	$1$	$4$	$3$

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.

$1$	$1$	$3$	$- 1$	$- 3$
		$1$	$4$	$3$
	$1$	$4$	$3$	$0$

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.

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

Step 6.10

Simplify the quotient polynomial.

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

Step 7

Factor $x^{2} + 4 x + 3$ using the AC method.

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Step 7.1

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 $3$ and whose sum is $4$ .

$1, 3$

Step 7.2

Write the factored form using these integers.

$(x - 1) + (x + 1) (x + 3)$

$(x - 1) (x + 1) (x + 3)$

Step 8

Factor the left side of the equation.

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Step 8.1

Factor out the greatest common factor from each group.

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Step 8.1.1

Group the first two terms and the last two terms.

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

Step 8.1.2

Factor out the greatest common factor (GCF) from each group.

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

Step 8.2

Factor the polynomial by factoring out the greatest common factor, $x + 3$ .

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

Step 8.3

Rewrite $1$ as $1^{2}$ .

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

Step 8.4

Factor.

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Step 8.4.1

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 $b = 1$ .

$(x + 3) ((x + 1) (x - 1)) = 0$

Step 8.4.2

Remove unnecessary parentheses.

$(x + 3) (x + 1) (x - 1) = 0$

Step 9

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

$x + 3 = 0$

$x + 1 = 0$

$x - 1 = 0$

Step 10

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 10.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 10.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 11

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 11.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 11.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 12

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 12.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 12.2

Add $1$ to both sides of the equation.

$x = 1$

Step 13

The final solution is all the values that make $(x + 3) (x + 1) (x - 1) = 0$ true.

$x = - 3, - 1, 1$

Step 14