Write as a Set of Linear Factors 16x^4-72x^3+113x^2-44x+5

$16 x^{4} - 72 x^{3} + 113 x^{2} - 44 x + 5$

Step 1

Factor $16 x^{4} - 72 x^{3} + 113 x^{2} - 44 x + 5$ 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.

$p = \pm 1, \pm 5 q = \pm 1, \pm 16, \pm 2, \pm 8, \pm 4$

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

$\pm 1, \pm 0.0625, \pm 0.5, \pm 0.125, \pm 0.25, \pm 5, \pm 0.3125, \pm 2.5, \pm 0.625, \pm 1.25$

Substitute $0.25$ and simplify the expression. In this case, the expression is equal to $0$ so $0.25$ is a root of the polynomial.

Substitute $0.25$ into the polynomial.

$16 \cdot {0.25}^{4} - 72 \cdot {0.25}^{3} + 113 \cdot {0.25}^{2} - 44 \cdot 0.25 + 5$

Raise $0.25$ to the power of $4$ .

$16 \cdot 0.00390625 - 72 \cdot {0.25}^{3} + 113 \cdot {0.25}^{2} - 44 \cdot 0.25 + 5$

Multiply $16$ by $0.00390625$ .

$0.0625 - 72 \cdot {0.25}^{3} + 113 \cdot {0.25}^{2} - 44 \cdot 0.25 + 5$

Raise $0.25$ to the power of $3$ .

$0.0625 - 72 \cdot 0.015625 + 113 \cdot {0.25}^{2} - 44 \cdot 0.25 + 5$

Multiply $- 72$ by $0.015625$ .

$0.0625 - 1.125 + 113 \cdot {0.25}^{2} - 44 \cdot 0.25 + 5$

Subtract $1.125$ from $0.0625$ .

$- 1.0625 + 113 \cdot {0.25}^{2} - 44 \cdot 0.25 + 5$

Raise $0.25$ to the power of $2$ .

$- 1.0625 + 113 \cdot 0.0625 - 44 \cdot 0.25 + 5$

Multiply $113$ by $0.0625$ .

$- 1.0625 + 7.0625 - 44 \cdot 0.25 + 5$

Add $- 1.0625$ and $7.0625$ .

$6 - 44 \cdot 0.25 + 5$

Multiply $- 44$ by $0.25$ .

$6 - 11 + 5$

Subtract $11$ from $6$ .

$- 5 + 5$

Add $- 5$ and $5$ .

$0$

Since $0.25$ is a known root, divide the polynomial by $4 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{16 x^{4} - 72 x^{3} + 113 x^{2} - 44 x + 5}{4 x - 1}$

Divide $16 x^{4} - 72 x^{3} + 113 x^{2} - 44 x + 5$ by $4 x - 1$ .

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

$4 x$

$1$

$16 x^{4}$

$72 x^{3}$

$113 x^{2}$

$44 x$

$5$

Divide the highest order term in the dividend $16 x^{4}$ by the highest order term in divisor $4 x$ .

$4 x^{3}$

$4 x$

$1$

$16 x^{4}$

$72 x^{3}$

$113 x^{2}$

$44 x$

$5$

Multiply the new quotient term by the divisor.

$4 x^{3}$

$4 x$

$1$

$16 x^{4}$

$72 x^{3}$

$113 x^{2}$

$44 x$

$5$

$16 x^{4}$

$4 x^{3}$

The expression needs to be subtracted from the dividend, so change all the signs in $16 x^{4} - 4 x^{3}$

$4 x^{3}$

$4 x$

$1$

$16 x^{4}$

$72 x^{3}$

$113 x^{2}$

$44 x$

$5$

$16 x^{4}$

$4 x^{3}$

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

$4 x^{3}$

$4 x$

$1$

$16 x^{4}$

$72 x^{3}$

$113 x^{2}$

$44 x$

$5$

$16 x^{4}$

$4 x^{3}$

$68 x^{3}$

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

$4 x^{3}$

$4 x$

$1$

$16 x^{4}$

$72 x^{3}$

$113 x^{2}$

$44 x$

$5$

$16 x^{4}$

$4 x^{3}$

$68 x^{3}$

$113 x^{2}$

Divide the highest order term in the dividend $- 68 x^{3}$ by the highest order term in divisor $4 x$ .

$4 x^{3}$

$17 x^{2}$

$4 x$

$1$

$16 x^{4}$

$72 x^{3}$

$113 x^{2}$

$44 x$

$5$

$16 x^{4}$

$4 x^{3}$

$68 x^{3}$

$113 x^{2}$

Multiply the new quotient term by the divisor.

$4 x^{3}$

$17 x^{2}$

$4 x$

$1$

$16 x^{4}$

$72 x^{3}$

$113 x^{2}$

$44 x$

$5$

$16 x^{4}$

$4 x^{3}$

$68 x^{3}$

$113 x^{2}$

$68 x^{3}$

$17 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $- 68 x^{3} + 17 x^{2}$

$4 x^{3}$

$17 x^{2}$

$4 x$

$1$

$16 x^{4}$

$72 x^{3}$

$113 x^{2}$

$44 x$

$5$

$16 x^{4}$

$4 x^{3}$

$68 x^{3}$

$113 x^{2}$

$68 x^{3}$

$17 x^{2}$

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

$4 x^{3}$

$17 x^{2}$

$4 x$

$1$

$16 x^{4}$

$72 x^{3}$

$113 x^{2}$

$44 x$

$5$

$16 x^{4}$

$4 x^{3}$

$68 x^{3}$

$113 x^{2}$

$68 x^{3}$

$17 x^{2}$

$96 x^{2}$

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

$4 x^{3}$

$17 x^{2}$

$4 x$

$1$

$16 x^{4}$

$72 x^{3}$

$113 x^{2}$

$44 x$

$5$

$16 x^{4}$

$4 x^{3}$

$68 x^{3}$

$113 x^{2}$

$68 x^{3}$

$17 x^{2}$

$96 x^{2}$

$44 x$

Divide the highest order term in the dividend $96 x^{2}$ by the highest order term in divisor $4 x$ .

$4 x^{3}$

$17 x^{2}$

$24 x$

$4 x$

$1$

$16 x^{4}$

$72 x^{3}$

$113 x^{2}$

$44 x$

$5$

$16 x^{4}$

$4 x^{3}$

$68 x^{3}$

$113 x^{2}$

$68 x^{3}$

$17 x^{2}$

$96 x^{2}$

$44 x$

Multiply the new quotient term by the divisor.

$4 x^{3}$

$17 x^{2}$

$24 x$

$4 x$

$1$

$16 x^{4}$

$72 x^{3}$

$113 x^{2}$

$44 x$

$5$

$16 x^{4}$

$4 x^{3}$

$68 x^{3}$

$113 x^{2}$

$68 x^{3}$

$17 x^{2}$

$96 x^{2}$

$44 x$

$96 x^{2}$

$24 x$

The expression needs to be subtracted from the dividend, so change all the signs in $96 x^{2} - 24 x$

$4 x^{3}$

$17 x^{2}$

$24 x$

$4 x$

$1$

$16 x^{4}$

$72 x^{3}$

$113 x^{2}$

$44 x$

$5$

$16 x^{4}$

$4 x^{3}$

$68 x^{3}$

$113 x^{2}$

$68 x^{3}$

$17 x^{2}$

$96 x^{2}$

$44 x$

$96 x^{2}$

$24 x$

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

$4 x^{3}$

$17 x^{2}$

$24 x$

$4 x$

$1$

$16 x^{4}$

$72 x^{3}$

$113 x^{2}$

$44 x$

$5$

$16 x^{4}$

$4 x^{3}$

$68 x^{3}$

$113 x^{2}$

$68 x^{3}$

$17 x^{2}$

$96 x^{2}$

$44 x$

$96 x^{2}$

$24 x$

$20 x$

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

$4 x^{3}$

$17 x^{2}$

$24 x$

$4 x$

$1$

$16 x^{4}$

$72 x^{3}$

$113 x^{2}$

$44 x$

$5$

$16 x^{4}$

$4 x^{3}$

$68 x^{3}$

$113 x^{2}$

$68 x^{3}$

$17 x^{2}$

$96 x^{2}$

$44 x$

$96 x^{2}$

$24 x$

$20 x$

$5$

Divide the highest order term in the dividend $- 20 x$ by the highest order term in divisor $4 x$ .

$4 x^{3}$

$17 x^{2}$

$24 x$

$5$

$4 x$

$1$

$16 x^{4}$

$72 x^{3}$

$113 x^{2}$

$44 x$

$5$

$16 x^{4}$

$4 x^{3}$

$68 x^{3}$

$113 x^{2}$

$68 x^{3}$

$17 x^{2}$

$96 x^{2}$

$44 x$

$96 x^{2}$

$24 x$

$20 x$

$5$

Multiply the new quotient term by the divisor.

$4 x^{3}$

$17 x^{2}$

$24 x$

$5$

$4 x$

$1$

$16 x^{4}$

$72 x^{3}$

$113 x^{2}$

$44 x$

$5$

$16 x^{4}$

$4 x^{3}$

$68 x^{3}$

$113 x^{2}$

$68 x^{3}$

$17 x^{2}$

$96 x^{2}$

$44 x$

$96 x^{2}$

$24 x$

$20 x$

$5$

$20 x$

$5$

The expression needs to be subtracted from the dividend, so change all the signs in $- 20 x + 5$

$4 x^{3}$

$17 x^{2}$

$24 x$

$5$

$4 x$

$1$

$16 x^{4}$

$72 x^{3}$

$113 x^{2}$

$44 x$

$5$

$16 x^{4}$

$4 x^{3}$

$68 x^{3}$

$113 x^{2}$

$68 x^{3}$

$17 x^{2}$

$96 x^{2}$

$44 x$

$96 x^{2}$

$24 x$

$20 x$

$5$

$20 x$

$5$

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

$4 x^{3}$

$17 x^{2}$

$24 x$

$5$

$4 x$

$1$

$16 x^{4}$

$72 x^{3}$

$113 x^{2}$

$44 x$

$5$

$16 x^{4}$

$4 x^{3}$

$68 x^{3}$

$113 x^{2}$

$68 x^{3}$

$17 x^{2}$

$96 x^{2}$

$44 x$

$96 x^{2}$

$24 x$

$20 x$

$5$

$20 x$

$5$

$0$

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

$4 x^{3} - 17 x^{2} + 24 x - 5$

Write $16 x^{4} - 72 x^{3} + 113 x^{2} - 44 x + 5$ as a set of factors.

$(4 x - 1) (4 x^{3} - 17 x^{2} + 24 x - 5)$

Step 2

Factor $4 x^{3} - 17 x^{2} + 24 x - 5$ 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.

$p = \pm 1, \pm 5 q = \pm 1, \pm 4, \pm 2$

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 5, \pm 1.25, \pm 2.5$

Substitute $0.25$ and simplify the expression. In this case, the expression is equal to $0$ so $0.25$ is a root of the polynomial.

Substitute $0.25$ into the polynomial.

$4 \cdot {0.25}^{3} - 17 \cdot {0.25}^{2} + 24 \cdot 0.25 - 5$

Raise $0.25$ to the power of $3$ .

$4 \cdot 0.015625 - 17 \cdot {0.25}^{2} + 24 \cdot 0.25 - 5$

Multiply $4$ by $0.015625$ .

$0.0625 - 17 \cdot {0.25}^{2} + 24 \cdot 0.25 - 5$

Raise $0.25$ to the power of $2$ .

$0.0625 - 17 \cdot 0.0625 + 24 \cdot 0.25 - 5$

Multiply $- 17$ by $0.0625$ .

$0.0625 - 1.0625 + 24 \cdot 0.25 - 5$

Subtract $1.0625$ from $0.0625$ .

$- 1 + 24 \cdot 0.25 - 5$

Multiply $24$ by $0.25$ .

$- 1 + 6 - 5$

Add $- 1$ and $6$ .

$5 - 5$

Subtract $5$ from $5$ .

$0$

Since $0.25$ is a known root, divide the polynomial by $4 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{3} - 17 x^{2} + 24 x - 5}{4 x - 1}$

Divide $4 x^{3} - 17 x^{2} + 24 x - 5$ by $4 x - 1$ .

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

$4 x$

$1$

$4 x^{3}$

$17 x^{2}$

$24 x$

$5$

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

					$x^{2}$
	$4 x$	-	$1$		$4 x^{3}$	-	$17 x^{2}$	+	$24 x$	-	$5$

Multiply the new quotient term by the divisor.

				$x^{2}$
$4 x$	-	$1$		$4 x^{3}$	-	$17 x^{2}$	+	$24 x$	-	$5$
			+	$4 x^{3}$	-	$x^{2}$

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

				$x^{2}$
$4 x$	-	$1$		$4 x^{3}$	-	$17 x^{2}$	+	$24 x$	-	$5$
			-	$4 x^{3}$	+	$x^{2}$

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

				$x^{2}$
$4 x$	-	$1$		$4 x^{3}$	-	$17 x^{2}$	+	$24 x$	-	$5$
			-	$4 x^{3}$	+	$x^{2}$
					-	$16 x^{2}$

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

				$x^{2}$
$4 x$	-	$1$		$4 x^{3}$	-	$17 x^{2}$	+	$24 x$	-	$5$
			-	$4 x^{3}$	+	$x^{2}$
					-	$16 x^{2}$	+	$24 x$

Divide the highest order term in the dividend $- 16 x^{2}$ by the highest order term in divisor $4 x$ .

				$x^{2}$	-	$4 x$
$4 x$	-	$1$		$4 x^{3}$	-	$17 x^{2}$	+	$24 x$	-	$5$
			-	$4 x^{3}$	+	$x^{2}$
					-	$16 x^{2}$	+	$24 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$
$4 x$	-	$1$		$4 x^{3}$	-	$17 x^{2}$	+	$24 x$	-	$5$
			-	$4 x^{3}$	+	$x^{2}$
					-	$16 x^{2}$	+	$24 x$
					-	$16 x^{2}$	+	$4 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 16 x^{2} + 4 x$

				$x^{2}$	-	$4 x$
$4 x$	-	$1$		$4 x^{3}$	-	$17 x^{2}$	+	$24 x$	-	$5$
			-	$4 x^{3}$	+	$x^{2}$
					-	$16 x^{2}$	+	$24 x$
					+	$16 x^{2}$	-	$4 x$

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

				$x^{2}$	-	$4 x$
$4 x$	-	$1$		$4 x^{3}$	-	$17 x^{2}$	+	$24 x$	-	$5$
			-	$4 x^{3}$	+	$x^{2}$
					-	$16 x^{2}$	+	$24 x$
					+	$16 x^{2}$	-	$4 x$
							+	$20 x$

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

				$x^{2}$	-	$4 x$
$4 x$	-	$1$		$4 x^{3}$	-	$17 x^{2}$	+	$24 x$	-	$5$
			-	$4 x^{3}$	+	$x^{2}$
					-	$16 x^{2}$	+	$24 x$
					+	$16 x^{2}$	-	$4 x$
							+	$20 x$	-	$5$

Divide the highest order term in the dividend $20 x$ by the highest order term in divisor $4 x$ .

				$x^{2}$	-	$4 x$	+	$5$
$4 x$	-	$1$		$4 x^{3}$	-	$17 x^{2}$	+	$24 x$	-	$5$
			-	$4 x^{3}$	+	$x^{2}$
					-	$16 x^{2}$	+	$24 x$
					+	$16 x^{2}$	-	$4 x$
							+	$20 x$	-	$5$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$	+	$5$
$4 x$	-	$1$		$4 x^{3}$	-	$17 x^{2}$	+	$24 x$	-	$5$
			-	$4 x^{3}$	+	$x^{2}$
					-	$16 x^{2}$	+	$24 x$
					+	$16 x^{2}$	-	$4 x$
							+	$20 x$	-	$5$
							+	$20 x$	-	$5$

The expression needs to be subtracted from the dividend, so change all the signs in $20 x - 5$

				$x^{2}$	-	$4 x$	+	$5$
$4 x$	-	$1$		$4 x^{3}$	-	$17 x^{2}$	+	$24 x$	-	$5$
			-	$4 x^{3}$	+	$x^{2}$
					-	$16 x^{2}$	+	$24 x$
					+	$16 x^{2}$	-	$4 x$
							+	$20 x$	-	$5$
							-	$20 x$	+	$5$

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

				$x^{2}$	-	$4 x$	+	$5$
$4 x$	-	$1$		$4 x^{3}$	-	$17 x^{2}$	+	$24 x$	-	$5$
			-	$4 x^{3}$	+	$x^{2}$
					-	$16 x^{2}$	+	$24 x$
					+	$16 x^{2}$	-	$4 x$
							+	$20 x$	-	$5$
							-	$20 x$	+	$5$
										$0$

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

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

Write $4 x^{3} - 17 x^{2} + 24 x - 5$ as a set of factors.

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

Remove unnecessary parentheses.

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

Step 3

Combine like factors.

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

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

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

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

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

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

Add $1$ and $1$ .

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

Write as a Set of Linear Factors 16x^4-72x^3+113x^2-44x+5

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