Algebra Examples

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

Step 1

Factor $x^{3} - 3 x^{2} + 3 x - 1$ using the rational roots test.

Tap for more steps...

Step 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.

$p = \pm 1$

$q = \pm 1$

Step 1.2

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

$\pm 1$

Step 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.

Tap for more steps...

Step 1.3.1

Substitute $1$ into the polynomial.

$1^{3} - 3 \cdot 1^{2} + 3 \cdot 1 - 1$

Step 1.3.2

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

$1 - 3 \cdot 1^{2} + 3 \cdot 1 - 1$

Step 1.3.3

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

$1 - 3 \cdot 1 + 3 \cdot 1 - 1$

Step 1.3.4

Multiply $- 3$ by $1$ .

$1 - 3 + 3 \cdot 1 - 1$

Step 1.3.5

Subtract $3$ from $1$ .

$- 2 + 3 \cdot 1 - 1$

Step 1.3.6

Multiply $3$ by $1$ .

$- 2 + 3 - 1$

Step 1.3.7

Add $- 2$ and $3$ .

$1 - 1$

Step 1.3.8

Subtract $1$ from $1$ .

$0$

Step 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.

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

Step 1.5

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

Tap for more steps...

Step 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 $0$ .

$x$

$1$

$x^{3}$

$3 x^{2}$

$3 x$

$1$

Step 1.5.2

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

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

Step 1.5.3

Multiply the new quotient term by the divisor.

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

Step 1.5.4

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

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

Step 1.5.5

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

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

Step 1.5.6

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

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

Step 1.5.7

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

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

Step 1.5.8

Multiply the new quotient term by the divisor.

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

Step 1.5.9

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

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

Step 1.5.10

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

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

Step 1.5.11

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

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

Step 1.5.12

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

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

Step 1.5.13

Multiply the new quotient term by the divisor.

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

Step 1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $x - 1$

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

Step 1.5.15

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

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

Step 1.5.16

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

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

Step 1.6

Write $x^{3} - 3 x^{2} + 3 x - 1$ as a set of factors.

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

Step 2

Factor using the perfect square rule.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

$2 x = 2 \cdot x \cdot 1$

Step 2.3

Rewrite the polynomial.

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

Step 2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

$(x - 1) {(x - 1)}^{2}$

Step 3

Combine like factors.

Tap for more steps...

Step 3.1

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

${(x - 1)}^{1} {(x - 1)}^{2}$

Step 3.2

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

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

Step 3.3

Add $1$ and $2$ .

${(x - 1)}^{3}$