Algebra Examples

$P (x) = x^{4} - 6 x^{3} + 4 x^{2} + 15 x + 4$

Step 1

Set $x^{4} - 6 x^{3} + 4 x^{2} + 15 x + 4$ equal to $0$ .

$x^{4} - 6 x^{3} + 4 x^{2} + 15 x + 4 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor $x^{4} - 6 x^{3} + 4 x^{2} + 15 x + 4$ using the rational roots test.

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Step 2.1.1.1

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.

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

$q = \pm 1$

Step 2.1.1.2

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

$\pm 1, \pm 4, \pm 2$

Step 2.1.1.3

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 2.1.1.3.1

Substitute $- 1$ into the polynomial.

${(- 1)}^{4} - 6 {(- 1)}^{3} + 4 {(- 1)}^{2} + 15 \cdot - 1 + 4$

Step 2.1.1.3.2

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

$1 - 6 {(- 1)}^{3} + 4 {(- 1)}^{2} + 15 \cdot - 1 + 4$

Step 2.1.1.3.3

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

$1 - 6 \cdot - 1 + 4 {(- 1)}^{2} + 15 \cdot - 1 + 4$

Step 2.1.1.3.4

Multiply $- 6$ by $- 1$ .

$1 + 6 + 4 {(- 1)}^{2} + 15 \cdot - 1 + 4$

Step 2.1.1.3.5

Add $1$ and $6$ .

$7 + 4 {(- 1)}^{2} + 15 \cdot - 1 + 4$

Step 2.1.1.3.6

Raise $- 1$ to the power of $2$ .

$7 + 4 \cdot 1 + 15 \cdot - 1 + 4$

Step 2.1.1.3.7

Multiply $4$ by $1$ .

$7 + 4 + 15 \cdot - 1 + 4$

Step 2.1.1.3.8

Add $7$ and $4$ .

$11 + 15 \cdot - 1 + 4$

Step 2.1.1.3.9

Multiply $15$ by $- 1$ .

$11 - 15 + 4$

Step 2.1.1.3.10

Subtract $15$ from $11$ .

$- 4 + 4$

Step 2.1.1.3.11

Add $- 4$ and $4$ .

$0$

Step 2.1.1.4

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.

$\frac{x^{4} - 6 x^{3} + 4 x^{2} + 15 x + 4}{x + 1}$

Step 2.1.1.5

Divide $x^{4} - 6 x^{3} + 4 x^{2} + 15 x + 4$ by $x + 1$ .

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Step 2.1.1.5.1

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

$x$

$1$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$15 x$

$4$

Step 2.1.1.5.2

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

$x^{3}$

$x$

$1$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$15 x$

$4$

Step 2.1.1.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$1$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$15 x$

$4$

$x^{4}$

$x^{3}$

Step 2.1.1.5.4

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

$x^{3}$

$x$

$1$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$15 x$

$4$

$x^{4}$

$x^{3}$

Step 2.1.1.5.5

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

$x^{3}$

$x$

$1$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$15 x$

$4$

$x^{4}$

$x^{3}$

$7 x^{3}$

Step 2.1.1.5.6

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

$x^{3}$

$x$

$1$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$15 x$

$4$

$x^{4}$

$x^{3}$

$7 x^{3}$

$4 x^{2}$

Step 2.1.1.5.7

Divide the highest order term in the dividend $- 7 x^{3}$ by the highest order term in divisor $x$ .

$x^{3}$

$7 x^{2}$

$x$

$1$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$15 x$

$4$

$x^{4}$

$x^{3}$

$7 x^{3}$

$4 x^{2}$

Step 2.1.1.5.8

Multiply the new quotient term by the divisor.

$x^{3}$

$7 x^{2}$

$x$

$1$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$15 x$

$4$

$x^{4}$

$x^{3}$

$7 x^{3}$

$4 x^{2}$

$7 x^{3}$

$7 x^{2}$

Step 2.1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 7 x^{3} - 7 x^{2}$

$x^{3}$

$7 x^{2}$

$x$

$1$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$15 x$

$4$

$x^{4}$

$x^{3}$

$7 x^{3}$

$4 x^{2}$

$7 x^{3}$

$7 x^{2}$

Step 2.1.1.5.10

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

$x^{3}$

$7 x^{2}$

$x$

$1$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$15 x$

$4$

$x^{4}$

$x^{3}$

$7 x^{3}$

$4 x^{2}$

$7 x^{3}$

$7 x^{2}$

$11 x^{2}$

Step 2.1.1.5.11

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

$x^{3}$

$7 x^{2}$

$x$

$1$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$15 x$

$4$

$x^{4}$

$x^{3}$

$7 x^{3}$

$4 x^{2}$

$7 x^{3}$

$7 x^{2}$

$11 x^{2}$

$15 x$

Step 2.1.1.5.12

Divide the highest order term in the dividend $11 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$7 x^{2}$

$11 x$

$x$

$1$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$15 x$

$4$

$x^{4}$

$x^{3}$

$7 x^{3}$

$4 x^{2}$

$7 x^{3}$

$7 x^{2}$

$11 x^{2}$

$15 x$

Step 2.1.1.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$7 x^{2}$

$11 x$

$x$

$1$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$15 x$

$4$

$x^{4}$

$x^{3}$

$7 x^{3}$

$4 x^{2}$

$7 x^{3}$

$7 x^{2}$

$11 x^{2}$

$15 x$

$11 x^{2}$

$11 x$

Step 2.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $11 x^{2} + 11 x$

$x^{3}$

$7 x^{2}$

$11 x$

$x$

$1$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$15 x$

$4$

$x^{4}$

$x^{3}$

$7 x^{3}$

$4 x^{2}$

$7 x^{3}$

$7 x^{2}$

$11 x^{2}$

$15 x$

$11 x^{2}$

$11 x$

Step 2.1.1.5.15

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

$x^{3}$

$7 x^{2}$

$11 x$

$x$

$1$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$15 x$

$4$

$x^{4}$

$x^{3}$

$7 x^{3}$

$4 x^{2}$

$7 x^{3}$

$7 x^{2}$

$11 x^{2}$

$15 x$

$11 x^{2}$

$11 x$

$4 x$

Step 2.1.1.5.16

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

$x^{3}$

$7 x^{2}$

$11 x$

$x$

$1$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$15 x$

$4$

$x^{4}$

$x^{3}$

$7 x^{3}$

$4 x^{2}$

$7 x^{3}$

$7 x^{2}$

$11 x^{2}$

$15 x$

$11 x^{2}$

$11 x$

$4 x$

$4$

Step 2.1.1.5.17

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

$x^{3}$

$7 x^{2}$

$11 x$

$4$

$x$

$1$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$15 x$

$4$

$x^{4}$

$x^{3}$

$7 x^{3}$

$4 x^{2}$

$7 x^{3}$

$7 x^{2}$

$11 x^{2}$

$15 x$

$11 x^{2}$

$11 x$

$4 x$

$4$

Step 2.1.1.5.18

Multiply the new quotient term by the divisor.

$x^{3}$

$7 x^{2}$

$11 x$

$4$

$x$

$1$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$15 x$

$4$

$x^{4}$

$x^{3}$

$7 x^{3}$

$4 x^{2}$

$7 x^{3}$

$7 x^{2}$

$11 x^{2}$

$15 x$

$11 x^{2}$

$11 x$

$4 x$

$4$

$4 x$

$4$

Step 2.1.1.5.19

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

$x^{3}$

$7 x^{2}$

$11 x$

$4$

$x$

$1$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$15 x$

$4$

$x^{4}$

$x^{3}$

$7 x^{3}$

$4 x^{2}$

$7 x^{3}$

$7 x^{2}$

$11 x^{2}$

$15 x$

$11 x^{2}$

$11 x$

$4 x$

$4$

$4 x$

$4$

Step 2.1.1.5.20

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

$x^{3}$

$7 x^{2}$

$11 x$

$4$

$x$

$1$

$x^{4}$

$6 x^{3}$

$4 x^{2}$

$15 x$

$4$

$x^{4}$

$x^{3}$

$7 x^{3}$

$4 x^{2}$

$7 x^{3}$

$7 x^{2}$

$11 x^{2}$

$15 x$

$11 x^{2}$

$11 x$

$4 x$

$4$

$4 x$

$4$

$0$

Step 2.1.1.5.21

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

$x^{3} - 7 x^{2} + 11 x + 4$

Step 2.1.1.6

Write $x^{4} - 6 x^{3} + 4 x^{2} + 15 x + 4$ as a set of factors.

$(x + 1) (x^{3} - 7 x^{2} + 11 x + 4) = 0$

Step 2.1.2

Factor $x^{3} - 7 x^{2} + 11 x + 4$ using the rational roots test.

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Step 2.1.2.1

Factor $x^{3} - 7 x^{2} + 11 x + 4$ using the rational roots test.

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Step 2.1.2.1.1

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.

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

$q = \pm 1$

Step 2.1.2.1.2

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

$\pm 1, \pm 4, \pm 2$

Step 2.1.2.1.3

Substitute $4$ and simplify the expression. In this case, the expression is equal to $0$ so $4$ is a root of the polynomial.

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Step 2.1.2.1.3.1

Substitute $4$ into the polynomial.

$4^{3} - 7 \cdot 4^{2} + 11 \cdot 4 + 4$

Step 2.1.2.1.3.2

Raise $4$ to the power of $3$ .

$64 - 7 \cdot 4^{2} + 11 \cdot 4 + 4$

Step 2.1.2.1.3.3

Raise $4$ to the power of $2$ .

$64 - 7 \cdot 16 + 11 \cdot 4 + 4$

Step 2.1.2.1.3.4

Multiply $- 7$ by $16$ .

$64 - 112 + 11 \cdot 4 + 4$

Step 2.1.2.1.3.5

Subtract $112$ from $64$ .

$- 48 + 11 \cdot 4 + 4$

Step 2.1.2.1.3.6

Multiply $11$ by $4$ .

$- 48 + 44 + 4$

Step 2.1.2.1.3.7

Add $- 48$ and $44$ .

$- 4 + 4$

Step 2.1.2.1.3.8

Add $- 4$ and $4$ .

$0$

Step 2.1.2.1.4

Since $4$ is a known root, divide the polynomial by $x - 4$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 7 x^{2} + 11 x + 4}{x - 4}$

Step 2.1.2.1.5

Divide $x^{3} - 7 x^{2} + 11 x + 4$ by $x - 4$ .

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Step 2.1.2.1.5.1

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

$x$

$4$

$x^{3}$

$7 x^{2}$

$11 x$

$4$

Step 2.1.2.1.5.2

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

					$x^{2}$
	$x$	-	$4$		$x^{3}$	-	$7 x^{2}$	+	$11 x$	+	$4$

Step 2.1.2.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$4$		$x^{3}$	-	$7 x^{2}$	+	$11 x$	+	$4$
			+	$x^{3}$	-	$4 x^{2}$

Step 2.1.2.1.5.4

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

				$x^{2}$
$x$	-	$4$		$x^{3}$	-	$7 x^{2}$	+	$11 x$	+	$4$
			-	$x^{3}$	+	$4 x^{2}$

Step 2.1.2.1.5.5

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

				$x^{2}$
$x$	-	$4$		$x^{3}$	-	$7 x^{2}$	+	$11 x$	+	$4$
			-	$x^{3}$	+	$4 x^{2}$
					-	$3 x^{2}$

Step 2.1.2.1.5.6

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

				$x^{2}$
$x$	-	$4$		$x^{3}$	-	$7 x^{2}$	+	$11 x$	+	$4$
			-	$x^{3}$	+	$4 x^{2}$
					-	$3 x^{2}$	+	$11 x$

Step 2.1.2.1.5.7

Divide the highest order term in the dividend $- 3 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$3 x$
$x$	-	$4$		$x^{3}$	-	$7 x^{2}$	+	$11 x$	+	$4$
			-	$x^{3}$	+	$4 x^{2}$
					-	$3 x^{2}$	+	$11 x$

Step 2.1.2.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$3 x$
$x$	-	$4$		$x^{3}$	-	$7 x^{2}$	+	$11 x$	+	$4$
			-	$x^{3}$	+	$4 x^{2}$
					-	$3 x^{2}$	+	$11 x$
					-	$3 x^{2}$	+	$12 x$

Step 2.1.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x^{2} + 12 x$

				$x^{2}$	-	$3 x$
$x$	-	$4$		$x^{3}$	-	$7 x^{2}$	+	$11 x$	+	$4$
			-	$x^{3}$	+	$4 x^{2}$
					-	$3 x^{2}$	+	$11 x$
					+	$3 x^{2}$	-	$12 x$

Step 2.1.2.1.5.10

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

				$x^{2}$	-	$3 x$
$x$	-	$4$		$x^{3}$	-	$7 x^{2}$	+	$11 x$	+	$4$
			-	$x^{3}$	+	$4 x^{2}$
					-	$3 x^{2}$	+	$11 x$
					+	$3 x^{2}$	-	$12 x$
							-	$x$

Step 2.1.2.1.5.11

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

				$x^{2}$	-	$3 x$
$x$	-	$4$		$x^{3}$	-	$7 x^{2}$	+	$11 x$	+	$4$
			-	$x^{3}$	+	$4 x^{2}$
					-	$3 x^{2}$	+	$11 x$
					+	$3 x^{2}$	-	$12 x$
							-	$x$	+	$4$

Step 2.1.2.1.5.12

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

				$x^{2}$	-	$3 x$	-	$1$
$x$	-	$4$		$x^{3}$	-	$7 x^{2}$	+	$11 x$	+	$4$
			-	$x^{3}$	+	$4 x^{2}$
					-	$3 x^{2}$	+	$11 x$
					+	$3 x^{2}$	-	$12 x$
							-	$x$	+	$4$

Step 2.1.2.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$3 x$	-	$1$
$x$	-	$4$		$x^{3}$	-	$7 x^{2}$	+	$11 x$	+	$4$
			-	$x^{3}$	+	$4 x^{2}$
					-	$3 x^{2}$	+	$11 x$
					+	$3 x^{2}$	-	$12 x$
							-	$x$	+	$4$
							-	$x$	+	$4$

Step 2.1.2.1.5.14

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

				$x^{2}$	-	$3 x$	-	$1$
$x$	-	$4$		$x^{3}$	-	$7 x^{2}$	+	$11 x$	+	$4$
			-	$x^{3}$	+	$4 x^{2}$
					-	$3 x^{2}$	+	$11 x$
					+	$3 x^{2}$	-	$12 x$
							-	$x$	+	$4$
							+	$x$	-	$4$

Step 2.1.2.1.5.15

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

				$x^{2}$	-	$3 x$	-	$1$
$x$	-	$4$		$x^{3}$	-	$7 x^{2}$	+	$11 x$	+	$4$
			-	$x^{3}$	+	$4 x^{2}$
					-	$3 x^{2}$	+	$11 x$
					+	$3 x^{2}$	-	$12 x$
							-	$x$	+	$4$
							+	$x$	-	$4$
										$0$

Step 2.1.2.1.5.16

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

$x^{2} - 3 x - 1$

Step 2.1.2.1.6

Write $x^{3} - 7 x^{2} + 11 x + 4$ as a set of factors.

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

Step 2.1.2.2

Remove unnecessary parentheses.

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

Step 2.2

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

$x + 1 = 0$

$x - 4 = 0$

$x^{2} - 3 x - 1 = 0$

Step 2.3

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

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Step 2.3.1

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

$x + 1 = 0$

Step 2.3.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.4

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 2.4.2

Add $4$ to both sides of the equation.

$x = 4$

Step 2.5

Set $x^{2} - 3 x - 1$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x^{2} - 3 x - 1$ equal to $0$ .

$x^{2} - 3 x - 1 = 0$

Step 2.5.2

Solve $x^{2} - 3 x - 1 = 0$ for $x$ .

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Step 2.5.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.5.2.2

Substitute the values $a = 1$ , $b = - 3$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (1 \cdot - 1)}}{2 \cdot 1}$

Step 2.5.2.3

Simplify.

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Step 2.5.2.3.1

Simplify the numerator.

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Step 2.5.2.3.1.1

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 2.5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot - 1$ .

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Step 2.5.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 1}}{2 \cdot 1}$

Step 2.5.2.3.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{3 \pm \sqrt{9 + 4}}{2 \cdot 1}$

Step 2.5.2.3.1.3

Add $9$ and $4$ .

$x = \frac{3 \pm \sqrt{13}}{2 \cdot 1}$

Step 2.5.2.3.2

Multiply $2$ by $1$ .

$x = \frac{3 \pm \sqrt{13}}{2}$

Step 2.5.2.4

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{13}}{2}, \frac{3 - \sqrt{13}}{2}$

Step 2.6

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

$x = - 1, 4, \frac{3 + \sqrt{13}}{2}, \frac{3 - \sqrt{13}}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = - 1, 4, \frac{3 + \sqrt{13}}{2}, \frac{3 - \sqrt{13}}{2}$

Decimal Form:

$x = - 1, 4, 3.30277563 \dots, - 0.30277563 \dots$

Step 4