Write as a Set of Linear Factors 36x^4-156x^3+229x^2-64x+5

$36 x^{4} - 156 x^{3} + 229 x^{2} - 64 x + 5$

Step 1

Factor $36 x^{4} - 156 x^{3} + 229 x^{2} - 64 x + 5$ using the rational roots test.

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.

$p = \pm 1, \pm 5 q = \pm 1, \pm 36, \pm 2, \pm 18, \pm 3, \pm 12, \pm 4, \pm 9, \pm 6$

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

$\pm 1, \pm 0.02\overline{7}, \pm 0.5, \pm 0.0\overline{5}, \pm 0.\overline{3}, \pm 0.08\overline{3}, \pm 0.25, \pm 0.\overline{1}, \pm 0.1\overline{6}, \pm 5, \pm 0.13\overline{8}, \pm 2.5, \pm 0.2\overline{7}, \pm 1.\overline{6}, \pm 0.41\overline{6}, \pm 1.25, \pm 0.\overline{5}, \pm 0.8\overline{3}$

Substitute $0.1\overline{6}$ and simplify the expression. In this case, the expression is equal to $0$ so $0.1\overline{6}$ is a root of the polynomial.

Substitute $0.1\overline{6}$ into the polynomial.

$36 \cdot {0.1\overline{6}}^{4} - 156 \cdot {0.1\overline{6}}^{3} + 229 \cdot {0.1\overline{6}}^{2} - 64 \cdot 0.1\overline{6} + 5$

Raise $0.1\overline{6}$ to the power of $4$ .

$36 \cdot 0.0007716 - 156 \cdot {0.1\overline{6}}^{3} + 229 \cdot {0.1\overline{6}}^{2} - 64 \cdot 0.1\overline{6} + 5$

Multiply $36$ by $0.0007716$ .

$0.02\overline{7} - 156 \cdot {0.1\overline{6}}^{3} + 229 \cdot {0.1\overline{6}}^{2} - 64 \cdot 0.1\overline{6} + 5$

Raise $0.1\overline{6}$ to the power of $3$ .

$0.02\overline{7} - 156 \cdot 0.004\overline{629} + 229 \cdot {0.1\overline{6}}^{2} - 64 \cdot 0.1\overline{6} + 5$

Multiply $- 156$ by $0.004\overline{629}$ .

$0.02\overline{7} - 0.7\overline{2} + 229 \cdot {0.1\overline{6}}^{2} - 64 \cdot 0.1\overline{6} + 5$

Subtract $0.7\overline{2}$ from $0.02\overline{7}$ .

$- 0.69\overline{4} + 229 \cdot {0.1\overline{6}}^{2} - 64 \cdot 0.1\overline{6} + 5$

Raise $0.1\overline{6}$ to the power of $2$ .

$- 0.69\overline{4} + 229 \cdot 0.02\overline{7} - 64 \cdot 0.1\overline{6} + 5$

Multiply $229$ by $0.02\overline{7}$ .

$- 0.69\overline{4} + 6.36111111 - 64 \cdot 0.1\overline{6} + 5$

Add $- 0.69\overline{4}$ and $6.36111111$ .

$5.66666666 - 64 \cdot 0.1\overline{6} + 5$

Multiply $- 64$ by $0.1\overline{6}$ .

$5.66666666 - 10.\overline{6} + 5$

Subtract $10.\overline{6}$ from $5.66666666$ .

$- 5 + 5$

Add $- 5$ and $5$ .

$0$

Since $0.1\overline{6}$ is a known root, divide the polynomial by $6 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{36 x^{4} - 156 x^{3} + 229 x^{2} - 64 x + 5}{6 x - 1}$

Divide $36 x^{4} - 156 x^{3} + 229 x^{2} - 64 x + 5$ by $6 x - 1$ .

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

$6 x$

$1$

$36 x^{4}$

$156 x^{3}$

$229 x^{2}$

$64 x$

$5$

Divide the highest order term in the dividend $36 x^{4}$ by the highest order term in divisor $6 x$ .

$6 x^{3}$

$6 x$

$1$

$36 x^{4}$

$156 x^{3}$

$229 x^{2}$

$64 x$

$5$

Multiply the new quotient term by the divisor.

$6 x^{3}$

$6 x$

$1$

$36 x^{4}$

$156 x^{3}$

$229 x^{2}$

$64 x$

$5$

$36 x^{4}$

$6 x^{3}$

The expression needs to be subtracted from the dividend, so change all the signs in $36 x^{4} - 6 x^{3}$

$6 x^{3}$

$6 x$

$1$

$36 x^{4}$

$156 x^{3}$

$229 x^{2}$

$64 x$

$5$

$36 x^{4}$

$6 x^{3}$

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

$6 x^{3}$

$6 x$

$1$

$36 x^{4}$

$156 x^{3}$

$229 x^{2}$

$64 x$

$5$

$36 x^{4}$

$6 x^{3}$

$150 x^{3}$

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

$6 x^{3}$

$6 x$

$1$

$36 x^{4}$

$156 x^{3}$

$229 x^{2}$

$64 x$

$5$

$36 x^{4}$

$6 x^{3}$

$150 x^{3}$

$229 x^{2}$

Divide the highest order term in the dividend $- 150 x^{3}$ by the highest order term in divisor $6 x$ .

$6 x^{3}$

$25 x^{2}$

$6 x$

$1$

$36 x^{4}$

$156 x^{3}$

$229 x^{2}$

$64 x$

$5$

$36 x^{4}$

$6 x^{3}$

$150 x^{3}$

$229 x^{2}$

Multiply the new quotient term by the divisor.

$6 x^{3}$

$25 x^{2}$

$6 x$

$1$

$36 x^{4}$

$156 x^{3}$

$229 x^{2}$

$64 x$

$5$

$36 x^{4}$

$6 x^{3}$

$150 x^{3}$

$229 x^{2}$

$150 x^{3}$

$25 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $- 150 x^{3} + 25 x^{2}$

$6 x^{3}$

$25 x^{2}$

$6 x$

$1$

$36 x^{4}$

$156 x^{3}$

$229 x^{2}$

$64 x$

$5$

$36 x^{4}$

$6 x^{3}$

$150 x^{3}$

$229 x^{2}$

$150 x^{3}$

$25 x^{2}$

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

$6 x^{3}$

$25 x^{2}$

$6 x$

$1$

$36 x^{4}$

$156 x^{3}$

$229 x^{2}$

$64 x$

$5$

$36 x^{4}$

$6 x^{3}$

$150 x^{3}$

$229 x^{2}$

$150 x^{3}$

$25 x^{2}$

$204 x^{2}$

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

$6 x^{3}$

$25 x^{2}$

$6 x$

$1$

$36 x^{4}$

$156 x^{3}$

$229 x^{2}$

$64 x$

$5$

$36 x^{4}$

$6 x^{3}$

$150 x^{3}$

$229 x^{2}$

$150 x^{3}$

$25 x^{2}$

$204 x^{2}$

$64 x$

Divide the highest order term in the dividend $204 x^{2}$ by the highest order term in divisor $6 x$ .

$6 x^{3}$

$25 x^{2}$

$34 x$

$6 x$

$1$

$36 x^{4}$

$156 x^{3}$

$229 x^{2}$

$64 x$

$5$

$36 x^{4}$

$6 x^{3}$

$150 x^{3}$

$229 x^{2}$

$150 x^{3}$

$25 x^{2}$

$204 x^{2}$

$64 x$

Multiply the new quotient term by the divisor.

$6 x^{3}$

$25 x^{2}$

$34 x$

$6 x$

$1$

$36 x^{4}$

$156 x^{3}$

$229 x^{2}$

$64 x$

$5$

$36 x^{4}$

$6 x^{3}$

$150 x^{3}$

$229 x^{2}$

$150 x^{3}$

$25 x^{2}$

$204 x^{2}$

$64 x$

$204 x^{2}$

$34 x$

The expression needs to be subtracted from the dividend, so change all the signs in $204 x^{2} - 34 x$

$6 x^{3}$

$25 x^{2}$

$34 x$

$6 x$

$1$

$36 x^{4}$

$156 x^{3}$

$229 x^{2}$

$64 x$

$5$

$36 x^{4}$

$6 x^{3}$

$150 x^{3}$

$229 x^{2}$

$150 x^{3}$

$25 x^{2}$

$204 x^{2}$

$64 x$

$204 x^{2}$

$34 x$

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

$6 x^{3}$

$25 x^{2}$

$34 x$

$6 x$

$1$

$36 x^{4}$

$156 x^{3}$

$229 x^{2}$

$64 x$

$5$

$36 x^{4}$

$6 x^{3}$

$150 x^{3}$

$229 x^{2}$

$150 x^{3}$

$25 x^{2}$

$204 x^{2}$

$64 x$

$204 x^{2}$

$34 x$

$30 x$

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

$6 x^{3}$

$25 x^{2}$

$34 x$

$6 x$

$1$

$36 x^{4}$

$156 x^{3}$

$229 x^{2}$

$64 x$

$5$

$36 x^{4}$

$6 x^{3}$

$150 x^{3}$

$229 x^{2}$

$150 x^{3}$

$25 x^{2}$

$204 x^{2}$

$64 x$

$204 x^{2}$

$34 x$

$30 x$

$5$

Divide the highest order term in the dividend $- 30 x$ by the highest order term in divisor $6 x$ .

$6 x^{3}$

$25 x^{2}$

$34 x$

$5$

$6 x$

$1$

$36 x^{4}$

$156 x^{3}$

$229 x^{2}$

$64 x$

$5$

$36 x^{4}$

$6 x^{3}$

$150 x^{3}$

$229 x^{2}$

$150 x^{3}$

$25 x^{2}$

$204 x^{2}$

$64 x$

$204 x^{2}$

$34 x$

$30 x$

$5$

Multiply the new quotient term by the divisor.

$6 x^{3}$

$25 x^{2}$

$34 x$

$5$

$6 x$

$1$

$36 x^{4}$

$156 x^{3}$

$229 x^{2}$

$64 x$

$5$

$36 x^{4}$

$6 x^{3}$

$150 x^{3}$

$229 x^{2}$

$150 x^{3}$

$25 x^{2}$

$204 x^{2}$

$64 x$

$204 x^{2}$

$34 x$

$30 x$

$5$

$30 x$

$5$

The expression needs to be subtracted from the dividend, so change all the signs in $- 30 x + 5$

$6 x^{3}$

$25 x^{2}$

$34 x$

$5$

$6 x$

$1$

$36 x^{4}$

$156 x^{3}$

$229 x^{2}$

$64 x$

$5$

$36 x^{4}$

$6 x^{3}$

$150 x^{3}$

$229 x^{2}$

$150 x^{3}$

$25 x^{2}$

$204 x^{2}$

$64 x$

$204 x^{2}$

$34 x$

$30 x$

$5$

$30 x$

$5$

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

$6 x^{3}$

$25 x^{2}$

$34 x$

$5$

$6 x$

$1$

$36 x^{4}$

$156 x^{3}$

$229 x^{2}$

$64 x$

$5$

$36 x^{4}$

$6 x^{3}$

$150 x^{3}$

$229 x^{2}$

$150 x^{3}$

$25 x^{2}$

$204 x^{2}$

$64 x$

$204 x^{2}$

$34 x$

$30 x$

$5$

$30 x$

$5$

$0$

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

$6 x^{3} - 25 x^{2} + 34 x - 5$

Write $36 x^{4} - 156 x^{3} + 229 x^{2} - 64 x + 5$ as a set of factors.

$(6 x - 1) (6 x^{3} - 25 x^{2} + 34 x - 5)$

Step 2

Factor $6 x^{3} - 25 x^{2} + 34 x - 5$ using the rational roots test.

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.

$p = \pm 1, \pm 5 q = \pm 1, \pm 6, \pm 2, \pm 3$

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

$\pm 1, \pm 0.1\overline{6}, \pm 0.5, \pm 0.\overline{3}, \pm 5, \pm 0.8\overline{3}, \pm 2.5, \pm 1.\overline{6}$

Substitute $0.1\overline{6}$ and simplify the expression. In this case, the expression is equal to $0$ so $0.1\overline{6}$ is a root of the polynomial.

Substitute $0.1\overline{6}$ into the polynomial.

$6 \cdot {0.1\overline{6}}^{3} - 25 \cdot {0.1\overline{6}}^{2} + 34 \cdot 0.1\overline{6} - 5$

Raise $0.1\overline{6}$ to the power of $3$ .

$6 \cdot 0.004\overline{629} - 25 \cdot {0.1\overline{6}}^{2} + 34 \cdot 0.1\overline{6} - 5$

Multiply $6$ by $0.004\overline{629}$ .

$0.02\overline{7} - 25 \cdot {0.1\overline{6}}^{2} + 34 \cdot 0.1\overline{6} - 5$

Raise $0.1\overline{6}$ to the power of $2$ .

$0.02\overline{7} - 25 \cdot 0.02\overline{7} + 34 \cdot 0.1\overline{6} - 5$

Multiply $- 25$ by $0.02\overline{7}$ .

$0.02\overline{7} - 0.69\overline{4} + 34 \cdot 0.1\overline{6} - 5$

Subtract $0.69\overline{4}$ from $0.02\overline{7}$ .

$- 0.66666666 + 34 \cdot 0.1\overline{6} - 5$

Multiply $34$ by $0.1\overline{6}$ .

$- 0.66666666 + 5.66666666 - 5$

Add $- 0.66666666$ and $5.66666666$ .

$5 - 5$

Subtract $5$ from $5$ .

$0$

Since $0.1\overline{6}$ is a known root, divide the polynomial by $6 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{6 x^{3} - 25 x^{2} + 34 x - 5}{6 x - 1}$

Divide $6 x^{3} - 25 x^{2} + 34 x - 5$ by $6 x - 1$ .

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

$6 x$

$1$

$6 x^{3}$

$25 x^{2}$

$34 x$

$5$

Divide the highest order term in the dividend $6 x^{3}$ by the highest order term in divisor $6 x$ .

					$x^{2}$
	$6 x$	-	$1$		$6 x^{3}$	-	$25 x^{2}$	+	$34 x$	-	$5$

Multiply the new quotient term by the divisor.

				$x^{2}$
$6 x$	-	$1$		$6 x^{3}$	-	$25 x^{2}$	+	$34 x$	-	$5$
			+	$6 x^{3}$	-	$x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{3} - x^{2}$

				$x^{2}$
$6 x$	-	$1$		$6 x^{3}$	-	$25 x^{2}$	+	$34 x$	-	$5$
			-	$6 x^{3}$	+	$x^{2}$

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

				$x^{2}$
$6 x$	-	$1$		$6 x^{3}$	-	$25 x^{2}$	+	$34 x$	-	$5$
			-	$6 x^{3}$	+	$x^{2}$
					-	$24 x^{2}$

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

				$x^{2}$
$6 x$	-	$1$		$6 x^{3}$	-	$25 x^{2}$	+	$34 x$	-	$5$
			-	$6 x^{3}$	+	$x^{2}$
					-	$24 x^{2}$	+	$34 x$

Divide the highest order term in the dividend $- 24 x^{2}$ by the highest order term in divisor $6 x$ .

				$x^{2}$	-	$4 x$
$6 x$	-	$1$		$6 x^{3}$	-	$25 x^{2}$	+	$34 x$	-	$5$
			-	$6 x^{3}$	+	$x^{2}$
					-	$24 x^{2}$	+	$34 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$
$6 x$	-	$1$		$6 x^{3}$	-	$25 x^{2}$	+	$34 x$	-	$5$
			-	$6 x^{3}$	+	$x^{2}$
					-	$24 x^{2}$	+	$34 x$
					-	$24 x^{2}$	+	$4 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 24 x^{2} + 4 x$

				$x^{2}$	-	$4 x$
$6 x$	-	$1$		$6 x^{3}$	-	$25 x^{2}$	+	$34 x$	-	$5$
			-	$6 x^{3}$	+	$x^{2}$
					-	$24 x^{2}$	+	$34 x$
					+	$24 x^{2}$	-	$4 x$

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

				$x^{2}$	-	$4 x$
$6 x$	-	$1$		$6 x^{3}$	-	$25 x^{2}$	+	$34 x$	-	$5$
			-	$6 x^{3}$	+	$x^{2}$
					-	$24 x^{2}$	+	$34 x$
					+	$24 x^{2}$	-	$4 x$
							+	$30 x$

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

				$x^{2}$	-	$4 x$
$6 x$	-	$1$		$6 x^{3}$	-	$25 x^{2}$	+	$34 x$	-	$5$
			-	$6 x^{3}$	+	$x^{2}$
					-	$24 x^{2}$	+	$34 x$
					+	$24 x^{2}$	-	$4 x$
							+	$30 x$	-	$5$

Divide the highest order term in the dividend $30 x$ by the highest order term in divisor $6 x$ .

				$x^{2}$	-	$4 x$	+	$5$
$6 x$	-	$1$		$6 x^{3}$	-	$25 x^{2}$	+	$34 x$	-	$5$
			-	$6 x^{3}$	+	$x^{2}$
					-	$24 x^{2}$	+	$34 x$
					+	$24 x^{2}$	-	$4 x$
							+	$30 x$	-	$5$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$	+	$5$
$6 x$	-	$1$		$6 x^{3}$	-	$25 x^{2}$	+	$34 x$	-	$5$
			-	$6 x^{3}$	+	$x^{2}$
					-	$24 x^{2}$	+	$34 x$
					+	$24 x^{2}$	-	$4 x$
							+	$30 x$	-	$5$
							+	$30 x$	-	$5$

The expression needs to be subtracted from the dividend, so change all the signs in $30 x - 5$

				$x^{2}$	-	$4 x$	+	$5$
$6 x$	-	$1$		$6 x^{3}$	-	$25 x^{2}$	+	$34 x$	-	$5$
			-	$6 x^{3}$	+	$x^{2}$
					-	$24 x^{2}$	+	$34 x$
					+	$24 x^{2}$	-	$4 x$
							+	$30 x$	-	$5$
							-	$30 x$	+	$5$

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

				$x^{2}$	-	$4 x$	+	$5$
$6 x$	-	$1$		$6 x^{3}$	-	$25 x^{2}$	+	$34 x$	-	$5$
			-	$6 x^{3}$	+	$x^{2}$
					-	$24 x^{2}$	+	$34 x$
					+	$24 x^{2}$	-	$4 x$
							+	$30 x$	-	$5$
							-	$30 x$	+	$5$
										$0$

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

$x^{2} - 4 x + 5$

Write $6 x^{3} - 25 x^{2} + 34 x - 5$ as a set of factors.

$(6 x - 1) ((6 x - 1) (x^{2} - 4 x + 5))$

Remove unnecessary parentheses.

$(6 x - 1) (6 x - 1) (x^{2} - 4 x + 5)$

Step 3

Combine like factors.

Raise $6 x - 1$ to the power of $1$ .

${(6 x - 1)}^{1} (6 x - 1) (x^{2} - 4 x + 5)$

Raise $6 x - 1$ to the power of $1$ .

${(6 x - 1)}^{1} {(6 x - 1)}^{1} (x^{2} - 4 x + 5)$

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

${(6 x - 1)}^{1 + 1} (x^{2} - 4 x + 5)$

Add $1$ and $1$ .

${(6 x - 1)}^{2} (x^{2} - 4 x + 5)$

Write as a Set of Linear Factors 36x^4-156x^3+229x^2-64x+5

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