Precalculus Examples

$x^{4} - 2 x^{3} + 10 x^{2} - 50 x - 375 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Regroup terms.

$- 2 x^{3} + 10 x^{2} + x^{4} - 50 x - 375 = 0$

Step 1.2

Factor $- 2 x^{2}$ out of $- 2 x^{3} + 10 x^{2}$ .

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Step 1.2.1

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

$- 2 x^{2} x + 10 x^{2} + x^{4} - 50 x - 375 = 0$

Step 1.2.2

Factor $- 2 x^{2}$ out of $10 x^{2}$ .

$- 2 x^{2} x - 2 x^{2} \cdot - 5 + x^{4} - 50 x - 375 = 0$

Step 1.2.3

Factor $- 2 x^{2}$ out of $- 2 x^{2} (x) - 2 x^{2} (- 5)$ .

$- 2 x^{2} (x - 5) + x^{4} - 50 x - 375 = 0$

Step 1.3

Factor $x^{4} - 50 x - 375$ 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.

$p = \pm 1, \pm 375, \pm 3, \pm 125, \pm 5, \pm 75, \pm 15, \pm 25$

$q = \pm 1$

Step 1.3.2

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

$\pm 1, \pm 375, \pm 3, \pm 125, \pm 5, \pm 75, \pm 15, \pm 25$

Step 1.3.3

Substitute $5$ and simplify the expression. In this case, the expression is equal to $0$ so $5$ is a root of the polynomial.

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Step 1.3.3.1

Substitute $5$ into the polynomial.

$5^{4} - 50 \cdot 5 - 375$

Step 1.3.3.2

Raise $5$ to the power of $4$ .

$625 - 50 \cdot 5 - 375$

Step 1.3.3.3

Multiply $- 50$ by $5$ .

$625 - 250 - 375$

Step 1.3.3.4

Subtract $250$ from $625$ .

$375 - 375$

Step 1.3.3.5

Subtract $375$ from $375$ .

$0$

Step 1.3.4

Since $5$ is a known root, divide the polynomial by $x - 5$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} - 50 x - 375}{x - 5}$

Step 1.3.5

Divide $x^{4} - 50 x - 375$ by $x - 5$ .

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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 $0$ .

$x$

$5$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$50 x$

$375$

Step 1.3.5.2

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

$x^{3}$

$x$

$5$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$50 x$

$375$

Step 1.3.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$5$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$50 x$

$375$

$x^{4}$

$5 x^{3}$

Step 1.3.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} - 5 x^{3}$

$x^{3}$

$x$

$5$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$50 x$

$375$

$x^{4}$

$5 x^{3}$

Step 1.3.5.5

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

$x^{3}$

$x$

$5$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$50 x$

$375$

$x^{4}$

$5 x^{3}$

Step 1.3.5.6

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

$x^{3}$

$x$

$5$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$50 x$

$375$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

Step 1.3.5.7

Divide the highest order term in the dividend $5 x^{3}$ by the highest order term in divisor $x$ .

$x^{3}$

$5 x^{2}$

$x$

$5$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$50 x$

$375$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

Step 1.3.5.8

Multiply the new quotient term by the divisor.

$x^{3}$

$5 x^{2}$

$x$

$5$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$50 x$

$375$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$25 x^{2}$

Step 1.3.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{3} - 25 x^{2}$

$x^{3}$

$5 x^{2}$

$x$

$5$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$50 x$

$375$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$25 x^{2}$

Step 1.3.5.10

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

$x^{3}$

$5 x^{2}$

$x$

$5$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$50 x$

$375$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$25 x^{2}$

Step 1.3.5.11

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

$x^{3}$

$5 x^{2}$

$x$

$5$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$50 x$

$375$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$25 x^{2}$

$50 x$

Step 1.3.5.12

Divide the highest order term in the dividend $25 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$5 x^{2}$

$25 x$

$x$

$5$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$50 x$

$375$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$25 x^{2}$

$50 x$

Step 1.3.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$5 x^{2}$

$25 x$

$x$

$5$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$50 x$

$375$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$25 x^{2}$

$50 x$

$25 x^{2}$

$125 x$

Step 1.3.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $25 x^{2} - 125 x$

$x^{3}$

$5 x^{2}$

$25 x$

$x$

$5$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$50 x$

$375$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$25 x^{2}$

$50 x$

$25 x^{2}$

$125 x$

Step 1.3.5.15

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

$x^{3}$

$5 x^{2}$

$25 x$

$x$

$5$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$50 x$

$375$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$25 x^{2}$

$50 x$

$25 x^{2}$

$125 x$

$75 x$

Step 1.3.5.16

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

$x^{3}$

$5 x^{2}$

$25 x$

$x$

$5$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$50 x$

$375$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$25 x^{2}$

$50 x$

$25 x^{2}$

$125 x$

$75 x$

$375$

Step 1.3.5.17

Divide the highest order term in the dividend $75 x$ by the highest order term in divisor $x$ .

$x^{3}$

$5 x^{2}$

$25 x$

$75$

$x$

$5$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$50 x$

$375$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$25 x^{2}$

$50 x$

$25 x^{2}$

$125 x$

$75 x$

$375$

Step 1.3.5.18

Multiply the new quotient term by the divisor.

$x^{3}$

$5 x^{2}$

$25 x$

$75$

$x$

$5$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$50 x$

$375$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$25 x^{2}$

$50 x$

$25 x^{2}$

$125 x$

$75 x$

$375$

$75 x$

$375$

Step 1.3.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $75 x - 375$

$x^{3}$

$5 x^{2}$

$25 x$

$75$

$x$

$5$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$50 x$

$375$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$25 x^{2}$

$50 x$

$25 x^{2}$

$125 x$

$75 x$

$375$

$75 x$

$375$

Step 1.3.5.20

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

$x^{3}$

$5 x^{2}$

$25 x$

$75$

$x$

$5$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$50 x$

$375$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$25 x^{2}$

$50 x$

$25 x^{2}$

$125 x$

$75 x$

$375$

$75 x$

$375$

$0$

Step 1.3.5.21

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

$x^{3} + 5 x^{2} + 25 x + 75$

Step 1.3.6

Write $x^{4} - 50 x - 375$ as a set of factors.

$- 2 x^{2} (x - 5) + (x - 5) (x^{3} + 5 x^{2} + 25 x + 75) = 0$

Step 1.4

Factor $x - 5$ out of $- 2 x^{2} (x - 5) + (x - 5) (x^{3} + 5 x^{2} + 25 x + 75)$ .

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Step 1.4.1

Factor $x - 5$ out of $- 2 x^{2} (x - 5)$ .

$(x - 5) (- 2 x^{2}) + (x - 5) (x^{3} + 5 x^{2} + 25 x + 75) = 0$

Step 1.4.2

Factor $x - 5$ out of $(x - 5) (- 2 x^{2}) + (x - 5) (x^{3} + 5 x^{2} + 25 x + 75)$ .

$(x - 5) (- 2 x^{2} + x^{3} + 5 x^{2} + 25 x + 75) = 0$

Step 1.5

Add $- 2 x^{2}$ and $5 x^{2}$ .

$(x - 5) (x^{3} + 3 x^{2} + 25 x + 75) = 0$

Step 1.6

Factor.

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Step 1.6.1

Rewrite $x^{3} + 3 x^{2} + 25 x + 75$ 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.

$(x - 5) ((x^{3} + 3 x^{2}) + 25 x + 75) = 0$

Step 1.6.1.1.2

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

$(x - 5) (x^{2} (x + 3) + 25 (x + 3)) = 0$

Step 1.6.1.2

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

$(x - 5) ((x + 3) (x^{2} + 25)) = 0$

Step 1.6.2

Remove unnecessary parentheses.

$(x - 5) (x + 3) (x^{2} + 25) = 0$

Step 2

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

$x - 5 = 0$

$x + 3 = 0$

$x^{2} + 25 = 0$

Step 3

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 3.2

Add $5$ to both sides of the equation.

$x = 5$

Step 4

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

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Step 4.1

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

$x + 3 = 0$

Step 4.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 5

Set $x^{2} + 25$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x^{2} + 25$ equal to $0$ .

$x^{2} + 25 = 0$

Step 5.2

Solve $x^{2} + 25 = 0$ for $x$ .

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Step 5.2.1

Subtract $25$ from both sides of the equation.

$x^{2} = - 25$

Step 5.2.2

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

$x = \pm \sqrt{- 25}$

Step 5.2.3

Simplify $\pm \sqrt{- 25}$ .

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Step 5.2.3.1

Rewrite $- 25$ as $- 1 (25)$ .

$x = \pm \sqrt{- 1 (25)}$

Step 5.2.3.2

Rewrite $\sqrt{- 1 (25)}$ as $\sqrt{- 1} \cdot \sqrt{25}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{25}$

Step 5.2.3.3

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

$x = \pm i \cdot \sqrt{25}$

Step 5.2.3.4

Rewrite $25$ as $5^{2}$ .

$x = \pm i \cdot \sqrt{5^{2}}$

Step 5.2.3.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm i \cdot 5$

Step 5.2.3.6

Move $5$ to the left of $i$ .

$x = \pm 5 i$

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.

$x = 5 i$

Step 5.2.4.2

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

$x = - 5 i$

Step 5.2.4.3

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

$x = 5 i, - 5 i$

Step 6

The final solution is all the values that make $(x - 5) (x + 3) (x^{2} + 25) = 0$ true.

$x = 5, - 3, 5 i, - 5 i$