Algebra Examples

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

Step 1

Factor the left side of the equation.

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Step 1.1

Regroup terms.

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

Step 1.2

Factor $x^{2}$ out of $x^{5} + x^{2}$ .

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Step 1.2.1

Factor $x^{2}$ out of $x^{5}$ .

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

Step 1.2.2

Multiply by $1$ .

$x^{2} x^{3} + x^{2} \cdot 1 - 3 x^{4} + x^{3} + 4 = 0$

Step 1.2.3

Factor $x^{2}$ out of $x^{2} x^{3} + x^{2} \cdot 1$ .

$x^{2} (x^{3} + 1) - 3 x^{4} + x^{3} + 4 = 0$

Step 1.3

Rewrite $1$ as $1^{3}$ .

$x^{2} (x^{3} + 1^{3}) - 3 x^{4} + x^{3} + 4 = 0$

Step 1.4

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 1$ .

$x^{2} ((x + 1) (x^{2} - x \cdot 1 + 1^{2})) - 3 x^{4} + x^{3} + 4 = 0$

Step 1.5

Factor.

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Step 1.5.1

Simplify.

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Step 1.5.1.1

Multiply $- 1$ by $1$ .

$x^{2} ((x + 1) (x^{2} - x + 1^{2})) - 3 x^{4} + x^{3} + 4 = 0$

Step 1.5.1.2

One to any power is one.

$x^{2} ((x + 1) (x^{2} - x + 1)) - 3 x^{4} + x^{3} + 4 = 0$

Step 1.5.2

Remove unnecessary parentheses.

$x^{2} (x + 1) (x^{2} - x + 1) - 3 x^{4} + x^{3} + 4 = 0$

Step 1.6

Factor $- 3 x^{4} + x^{3} + 4$ using the rational roots test.

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Step 1.6.1

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.

$p = \pm 1, \pm 4, \pm 2$

$q = \pm 1, \pm 3$

Step 1.6.2

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

$\pm 1, \pm 0.\overline{3}, \pm 4, \pm 1.\overline{3}, \pm 2, \pm 0.\overline{6}$

Step 1.6.3

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 1.6.3.1

Substitute $- 1$ into the polynomial.

$- 3 {(- 1)}^{4} + {(- 1)}^{3} + 4$

Step 1.6.3.2

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

$- 3 \cdot 1 + {(- 1)}^{3} + 4$

Step 1.6.3.3

Multiply $- 3$ by $1$ .

$- 3 + {(- 1)}^{3} + 4$

Step 1.6.3.4

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

$- 3 - 1 + 4$

Step 1.6.3.5

Subtract $1$ from $- 3$ .

$- 4 + 4$

Step 1.6.3.6

Add $- 4$ and $4$ .

$0$

Step 1.6.4

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.

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

Step 1.6.5

Divide $- 3 x^{4} + x^{3} + 4$ by $x + 1$ .

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Step 1.6.5.1

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

$x$

$1$

$3 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$4$

Step 1.6.5.2

Divide the highest order term in the dividend $- 3 x^{4}$ by the highest order term in divisor $x$ .

$3 x^{3}$

$x$

$1$

$3 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$4$

Step 1.6.5.3

Multiply the new quotient term by the divisor.

$3 x^{3}$

$x$

$1$

$3 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$4$

$3 x^{4}$

$3 x^{3}$

Step 1.6.5.4

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

$3 x^{3}$

$x$

$1$

$3 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$4$

$3 x^{4}$

$3 x^{3}$

Step 1.6.5.5

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

$3 x^{3}$

$x$

$1$

$3 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$4$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

Step 1.6.5.6

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

$3 x^{3}$

$x$

$1$

$3 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$4$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$0 x^{2}$

Step 1.6.5.7

Divide the highest order term in the dividend $4 x^{3}$ by the highest order term in divisor $x$ .

$3 x^{3}$

$4 x^{2}$

$x$

$1$

$3 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$4$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$0 x^{2}$

Step 1.6.5.8

Multiply the new quotient term by the divisor.

$3 x^{3}$

$4 x^{2}$

$x$

$1$

$3 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$4$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

Step 1.6.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{3} + 4 x^{2}$

$3 x^{3}$

$4 x^{2}$

$x$

$1$

$3 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$4$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

Step 1.6.5.10

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

$3 x^{3}$

$4 x^{2}$

$x$

$1$

$3 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$4$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

Step 1.6.5.11

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

$3 x^{3}$

$4 x^{2}$

$x$

$1$

$3 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$4$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$0 x$

Step 1.6.5.12

Divide the highest order term in the dividend $- 4 x^{2}$ by the highest order term in divisor $x$ .

$3 x^{3}$

$4 x^{2}$

$4 x$

$x$

$1$

$3 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$4$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$0 x$

Step 1.6.5.13

Multiply the new quotient term by the divisor.

$3 x^{3}$

$4 x^{2}$

$4 x$

$x$

$1$

$3 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$4$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x$

Step 1.6.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{2} - 4 x$

$3 x^{3}$

$4 x^{2}$

$4 x$

$x$

$1$

$3 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$4$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x$

Step 1.6.5.15

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

$3 x^{3}$

$4 x^{2}$

$4 x$

$x$

$1$

$3 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$4$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x$

Step 1.6.5.16

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

$3 x^{3}$

$4 x^{2}$

$4 x$

$x$

$1$

$3 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$4$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x$

$4$

Step 1.6.5.17

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

$3 x^{3}$

$4 x^{2}$

$4 x$

$4$

$x$

$1$

$3 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$4$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x$

$4$

Step 1.6.5.18

Multiply the new quotient term by the divisor.

$3 x^{3}$

$4 x^{2}$

$4 x$

$4$

$x$

$1$

$3 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$4$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x$

$4$

$4 x$

$4$

Step 1.6.5.19

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

$3 x^{3}$

$4 x^{2}$

$4 x$

$4$

$x$

$1$

$3 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$4$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x$

$4$

$4 x$

$4$

Step 1.6.5.20

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

$3 x^{3}$

$4 x^{2}$

$4 x$

$4$

$x$

$1$

$3 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$4$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x$

$4$

$4 x$

$4$

$0$

Step 1.6.5.21

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

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

Step 1.6.6

Write $- 3 x^{4} + x^{3} + 4$ as a set of factors.

$x^{2} (x + 1) (x^{2} - x + 1) + (x + 1) (- 3 x^{3} + 4 x^{2} - 4 x + 4) = 0$

Step 1.7

Factor $x + 1$ out of $x^{2} (x + 1) (x^{2} - x + 1) + (x + 1) (- 3 x^{3} + 4 x^{2} - 4 x + 4)$ .

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Step 1.7.1

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

$(x + 1) (x^{2} (x^{2} - x + 1)) + (x + 1) (- 3 x^{3} + 4 x^{2} - 4 x + 4) = 0$

Step 1.7.2

Factor $x + 1$ out of $(x + 1) (x^{2} (x^{2} - x + 1)) + (x + 1) (- 3 x^{3} + 4 x^{2} - 4 x + 4)$ .

$(x + 1) (x^{2} (x^{2} - x + 1) - 3 x^{3} + 4 x^{2} - 4 x + 4) = 0$

Step 1.8

Apply the distributive property.

$(x + 1) (x^{2} x^{2} + x^{2} (- x) + x^{2} \cdot 1 - 3 x^{3} + 4 x^{2} - 4 x + 4) = 0$

Step 1.9

Simplify.

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Step 1.9.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 1.9.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x + 1) (x^{2 + 2} + x^{2} (- x) + x^{2} \cdot 1 - 3 x^{3} + 4 x^{2} - 4 x + 4) = 0$

Step 1.9.1.2

Add $2$ and $2$ .

$(x + 1) (x^{4} + x^{2} (- x) + x^{2} \cdot 1 - 3 x^{3} + 4 x^{2} - 4 x + 4) = 0$

Step 1.9.2

Rewrite using the commutative property of multiplication.

$(x + 1) (x^{4} - x^{2} x + x^{2} \cdot 1 - 3 x^{3} + 4 x^{2} - 4 x + 4) = 0$

Step 1.9.3

Multiply $x^{2}$ by $1$ .

$(x + 1) (x^{4} - x^{2} x + x^{2} - 3 x^{3} + 4 x^{2} - 4 x + 4) = 0$

Step 1.10

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.10.1

Move $x$ .

$(x + 1) (x^{4} - (x \cdot x^{2}) + x^{2} - 3 x^{3} + 4 x^{2} - 4 x + 4) = 0$

Step 1.10.2

Multiply $x$ by $x^{2}$ .

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Step 1.10.2.1

Raise $x$ to the power of $1$ .

$(x + 1) (x^{4} - (x \cdot x^{2}) + x^{2} - 3 x^{3} + 4 x^{2} - 4 x + 4) = 0$

Step 1.10.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x + 1) (x^{4} - x^{1 + 2} + x^{2} - 3 x^{3} + 4 x^{2} - 4 x + 4) = 0$

Step 1.10.3

Add $1$ and $2$ .

$(x + 1) (x^{4} - x^{3} + x^{2} - 3 x^{3} + 4 x^{2} - 4 x + 4) = 0$

Step 1.11

Subtract $3 x^{3}$ from $- x^{3}$ .

$(x + 1) (x^{4} - 4 x^{3} + x^{2} + 4 x^{2} - 4 x + 4) = 0$

Step 1.12

Add $x^{2}$ and $4 x^{2}$ .

$(x + 1) (x^{4} - 4 x^{3} + 5 x^{2} - 4 x + 4) = 0$

Step 2

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

$x + 1 = 0$

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

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 $x^{4} - 4 x^{3} + 5 x^{2} - 4 x + 4$ equal to $0$ and solve for $x$ .

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Step 4.1

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

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

Step 4.2

Solve $x^{4} - 4 x^{3} + 5 x^{2} - 4 x + 4 = 0$ for $x$ .

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Step 4.2.1

Factor the left side of the equation.

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Step 4.2.1.1

Regroup terms.

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

Step 4.2.1.2

Factor $- 4 x$ out of $- 4 x^{3} - 4 x$ .

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Step 4.2.1.2.1

Factor $- 4 x$ out of $- 4 x^{3}$ .

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

Step 4.2.1.2.2

Factor $- 4 x$ out of $- 4 x$ .

$- 4 x \cdot x^{2} - 4 x \cdot 1 + x^{4} + 5 x^{2} + 4 = 0$

Step 4.2.1.2.3

Factor $- 4 x$ out of $- 4 x (x^{2}) - 4 x (1)$ .

$- 4 x (x^{2} + 1) + x^{4} + 5 x^{2} + 4 = 0$

Step 4.2.1.3

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$- 4 x (x^{2} + 1) + {(x^{2})}^{2} + 5 x^{2} + 4 = 0$

Step 4.2.1.4

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$- 4 x (x^{2} + 1) + u^{2} + 5 u + 4 = 0$

Step 4.2.1.5

Factor $u^{2} + 5 u + 4$ using the AC method.

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Step 4.2.1.5.1

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 $4$ and whose sum is $5$ .

$1, 4$

Step 4.2.1.5.2

Write the factored form using these integers.

$- 4 x (x^{2} + 1) + (u + 1) (u + 4) = 0$

Step 4.2.1.6

Replace all occurrences of $u$ with $x^{2}$ .

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

Step 4.2.1.7

Factor $x^{2} + 1$ out of $- 4 x (x^{2} + 1) + (x^{2} + 1) (x^{2} + 4)$ .

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Step 4.2.1.7.1

Factor $x^{2} + 1$ out of $- 4 x (x^{2} + 1)$ .

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

Step 4.2.1.7.2

Factor $x^{2} + 1$ out of $(x^{2} + 1) (- 4 x) + (x^{2} + 1) (x^{2} + 4)$ .

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

Step 4.2.1.8

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$(x^{2} + 1) (- 4 u + u^{2} + 4) = 0$

Step 4.2.1.9

Factor using the perfect square rule.

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Step 4.2.1.9.1

Rearrange terms.

$(x^{2} + 1) (u^{2} - 4 u + 4) = 0$

Step 4.2.1.9.2

Rewrite $4$ as $2^{2}$ .

$(x^{2} + 1) (u^{2} - 4 u + 2^{2}) = 0$

Step 4.2.1.9.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 u = 2 \cdot u \cdot 2$

Step 4.2.1.9.4

Rewrite the polynomial.

$(x^{2} + 1) (u^{2} - 2 \cdot u \cdot 2 + 2^{2}) = 0$

Step 4.2.1.9.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = 2$ .

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

Step 4.2.1.10

Replace all occurrences of $u$ with $x$ .

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

Step 4.2.2

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

$x^{2} + 1 = 0$

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

Step 4.2.3

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

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Step 4.2.3.1

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

$x^{2} + 1 = 0$

Step 4.2.3.2

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

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Step 4.2.3.2.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 4.2.3.2.2

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

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

Step 4.2.3.2.3

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

$x = \pm i$

Step 4.2.3.2.4

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

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Step 4.2.3.2.4.1

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

$x = i$

Step 4.2.3.2.4.2

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

$x = - i$

Step 4.2.3.2.4.3

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

$x = i, - i$

Step 4.2.4

Set ${(x - 2)}^{2}$ equal to $0$ and solve for $x$ .

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Step 4.2.4.1

Set ${(x - 2)}^{2}$ equal to $0$ .

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

Step 4.2.4.2

Solve ${(x - 2)}^{2} = 0$ for $x$ .

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Step 4.2.4.2.1

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 4.2.4.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 4.2.5

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

$x = i, - i, 2$

Step 5

The final solution is all the values that make $(x + 1) (x^{4} - 4 x^{3} + 5 x^{2} - 4 x + 4) = 0$ true.

$x = - 1, i, - i, 2$

Step 6