Algebra Examples

$x^{4} + 14 x^{3} + 14 x^{2} + 14 x + 13 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Regroup terms.

$14 x^{3} + 14 x^{2} + x^{4} + 14 x + 13 = 0$

Step 1.2

Factor $14 x^{2}$ out of $14 x^{3} + 14 x^{2}$ .

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Step 1.2.1

Factor $14 x^{2}$ out of $14 x^{3}$ .

$14 x^{2} (x) + 14 x^{2} + x^{4} + 14 x + 13 = 0$

Step 1.2.2

Factor $14 x^{2}$ out of $14 x^{2}$ .

$14 x^{2} (x) + 14 x^{2} (1) + x^{4} + 14 x + 13 = 0$

Step 1.2.3

Factor $14 x^{2}$ out of $14 x^{2} (x) + 14 x^{2} (1)$ .

$14 x^{2} (x + 1) + x^{4} + 14 x + 13 = 0$

Step 1.3

Factor $x^{4} + 14 x + 13$ using the rational roots test.

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Step 1.3.1

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.

$p = \pm 1, \pm 13$

$q = \pm 1$

Step 1.3.2

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

$\pm 1, \pm 13$

Step 1.3.3

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

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Step 1.3.3.1

Substitute $- 1$ into the polynomial.

${(- 1)}^{4} + 14 \cdot - 1 + 13$

Step 1.3.3.2

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

$1 + 14 \cdot - 1 + 13$

Step 1.3.3.3

Multiply $14$ by $- 1$ .

$1 - 14 + 13$

Step 1.3.3.4

Subtract $14$ from $1$ .

$- 13 + 13$

Step 1.3.3.5

Add $- 13$ and $13$ .

$0$

Step 1.3.4

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.

$\frac{x^{4} + 14 x + 13}{x + 1}$

Step 1.3.5

Divide $x^{4} + 14 x + 13$ by $x + 1$ .

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Step 1.3.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$14 x$

$13$

Step 1.3.5.2

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x$ .

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$14 x$

$13$

Step 1.3.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$14 x$

$13$

$x^{4}$

$x^{3}$

Step 1.3.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} + x^{3}$

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$14 x$

$13$

$x^{4}$

$x^{3}$

Step 1.3.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$14 x$

$13$

$x^{4}$

$x^{3}$

Step 1.3.5.6

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$14 x$

$13$

$x^{4}$

$x^{3}$

$0 x^{2}$

Step 1.3.5.7

Divide the highest order term in the dividend $- x^{3}$ by the highest order term in divisor $x$ .

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$14 x$

$13$

$x^{4}$

$x^{3}$

$0 x^{2}$

Step 1.3.5.8

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$14 x$

$13$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

Step 1.3.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- x^{3} - x^{2}$

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$14 x$

$13$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

Step 1.3.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$14 x$

$13$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

Step 1.3.5.11

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$14 x$

$13$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$14 x$

Step 1.3.5.12

Divide the highest order term in the dividend $x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$14 x$

$13$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$14 x$

Step 1.3.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$14 x$

$13$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$14 x$

$x^{2}$

$x$

Step 1.3.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $x^{2} + x$

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$14 x$

$13$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$14 x$

$x^{2}$

$x$

Step 1.3.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$14 x$

$13$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$14 x$

$x^{2}$

$x$

$13 x$

Step 1.3.5.16

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$14 x$

$13$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$14 x$

$x^{2}$

$x$

$13 x$

$13$

Step 1.3.5.17

Divide the highest order term in the dividend $13 x$ by the highest order term in divisor $x$ .

$x^{3}$

$x^{2}$

$x$

$13$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$14 x$

$13$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$14 x$

$x^{2}$

$x$

$13 x$

$13$

Step 1.3.5.18

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$x$

$13$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$14 x$

$13$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$14 x$

$x^{2}$

$x$

$13 x$

$13$

$13 x$

$13$

Step 1.3.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $13 x + 13$

$x^{3}$

$x^{2}$

$x$

$13$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$14 x$

$13$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$14 x$

$x^{2}$

$x$

$13 x$

$13$

$13 x$

$13$

Step 1.3.5.20

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$x^{2}$

$x$

$13$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$14 x$

$13$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$14 x$

$x^{2}$

$x$

$13 x$

$13$

$13 x$

$13$

$0$

Step 1.3.5.21

Since the remander is $0$ , the final answer is the quotient.

$x^{3} - x^{2} + x + 13$

Step 1.3.6

Write $x^{4} + 14 x + 13$ as a set of factors.

$14 x^{2} (x + 1) + (x + 1) (x^{3} - x^{2} + x + 13) = 0$

Step 1.4

Factor $x + 1$ out of $14 x^{2} (x + 1) + (x + 1) (x^{3} - x^{2} + x + 13)$ .

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Step 1.4.1

Factor $x + 1$ out of $14 x^{2} (x + 1)$ .

$(x + 1) (14 x^{2}) + (x + 1) (x^{3} - x^{2} + x + 13) = 0$

Step 1.4.2

Factor $x + 1$ out of $(x + 1) (14 x^{2}) + (x + 1) (x^{3} - x^{2} + x + 13)$ .

$(x + 1) (14 x^{2} + x^{3} - x^{2} + x + 13) = 0$

Step 1.5

Subtract $x^{2}$ from $14 x^{2}$ .

$(x + 1) (x^{3} + 13 x^{2} + x + 13) = 0$

Step 1.6

Factor.

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Step 1.6.1

Rewrite $x^{3} + 13 x^{2} + x + 13$ in a factored form.

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Step 1.6.1.1

Factor out the greatest common factor from each group.

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Step 1.6.1.1.1

Group the first two terms and the last two terms.

$(x + 1) ((x^{3} + 13 x^{2}) + x + 13) = 0$

Step 1.6.1.1.2

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

$(x + 1) (x^{2} (x + 13) + 1 (x + 13)) = 0$

Step 1.6.1.2

Factor the polynomial by factoring out the greatest common factor, $x + 13$ .

$(x + 1) ((x + 13) (x^{2} + 1)) = 0$

Step 1.6.2

Remove unnecessary parentheses.

$(x + 1) (x + 13) (x^{2} + 1) = 0$

Step 2

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

$x + 1 = 0$

$x + 13 = 0$

$x^{2} + 1 = 0$

Step 3

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

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Step 3.1

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

$x + 1 = 0$

Step 3.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 4

Set $x + 13$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x + 13$ equal to $0$ .

$x + 13 = 0$

Step 4.2

Subtract $13$ from both sides of the equation.

$x = - 13$

Step 5

Set $x^{2} + 1$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x^{2} + 1$ equal to $0$ .

$x^{2} + 1 = 0$

Step 5.2

Solve $x^{2} + 1 = 0$ for $x$ .

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Step 5.2.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 5.2.2

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

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

Step 5.2.3

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

$x = \pm i$

Step 5.2.4

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

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Step 5.2.4.1

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

$x = i$

Step 5.2.4.2

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

$x = - i$

Step 5.2.4.3

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

$x = i, - i$

Step 6

The final solution is all the values that make $(x + 1) (x + 13) (x^{2} + 1) = 0$ true.

$x = - 1, - 13, i, - i$