Factor f(x)=x^4-7x^3+9x^2+27x-54

$f (x) = x^{4} - 7 x^{3} + 9 x^{2} + 27 x - 54$

Step 1

Factor $x^{4} - 7 x^{3} + 9 x^{2} + 27 x - 54$ 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.

$p = \pm 1, \pm 54, \pm 2, \pm 27, \pm 3, \pm 18, \pm 6, \pm 9 q = \pm 1$

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

$\pm 1, \pm 54, \pm 2, \pm 27, \pm 3, \pm 18, \pm 6, \pm 9$

Substitute $- 2$ and simplify the expression. In this case, the expression is equal to $0$ so $- 2$ is a root of the polynomial.

Substitute $- 2$ into the polynomial.

${(- 2)}^{4} - 7 {(- 2)}^{3} + 9 {(- 2)}^{2} + 27 \cdot - 2 - 54$

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

$16 - 7 {(- 2)}^{3} + 9 {(- 2)}^{2} + 27 \cdot - 2 - 54$

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

$16 - 7 \cdot - 8 + 9 {(- 2)}^{2} + 27 \cdot - 2 - 54$

Multiply $- 7$ by $- 8$ .

$16 + 56 + 9 {(- 2)}^{2} + 27 \cdot - 2 - 54$

Add $16$ and $56$ .

$72 + 9 {(- 2)}^{2} + 27 \cdot - 2 - 54$

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

$72 + 9 \cdot 4 + 27 \cdot - 2 - 54$

Multiply $9$ by $4$ .

$72 + 36 + 27 \cdot - 2 - 54$

Add $72$ and $36$ .

$108 + 27 \cdot - 2 - 54$

Multiply $27$ by $- 2$ .

$108 - 54 - 54$

Subtract $54$ from $108$ .

$54 - 54$

Subtract $54$ from $54$ .

$0$

Since $- 2$ is a known root, divide the polynomial by $x + 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} - 7 x^{3} + 9 x^{2} + 27 x - 54}{x + 2}$

Divide $x^{4} - 7 x^{3} + 9 x^{2} + 27 x - 54$ by $x + 2$ .

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

$x$

$2$

$x^{4}$

$7 x^{3}$

$9 x^{2}$

$27 x$

$54$

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

$x^{3}$

$x$

$2$

$x^{4}$

$7 x^{3}$

$9 x^{2}$

$27 x$

$54$

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$2$

$x^{4}$

$7 x^{3}$

$9 x^{2}$

$27 x$

$54$

$x^{4}$

$2 x^{3}$

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

$x^{3}$

$x$

$2$

$x^{4}$

$7 x^{3}$

$9 x^{2}$

$27 x$

$54$

$x^{4}$

$2 x^{3}$

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

$x^{3}$

$x$

$2$

$x^{4}$

$7 x^{3}$

$9 x^{2}$

$27 x$

$54$

$x^{4}$

$2 x^{3}$

$9 x^{3}$

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

$x^{3}$

$x$

$2$

$x^{4}$

$7 x^{3}$

$9 x^{2}$

$27 x$

$54$

$x^{4}$

$2 x^{3}$

$9 x^{3}$

$9 x^{2}$

Divide the highest order term in the dividend $- 9 x^{3}$ by the highest order term in divisor $x$ .

$x^{3}$

$9 x^{2}$

$x$

$2$

$x^{4}$

$7 x^{3}$

$9 x^{2}$

$27 x$

$54$

$x^{4}$

$2 x^{3}$

$9 x^{3}$

$9 x^{2}$

Multiply the new quotient term by the divisor.

$x^{3}$

$9 x^{2}$

$x$

$2$

$x^{4}$

$7 x^{3}$

$9 x^{2}$

$27 x$

$54$

$x^{4}$

$2 x^{3}$

$9 x^{3}$

$9 x^{2}$

$9 x^{3}$

$18 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $- 9 x^{3} - 18 x^{2}$

$x^{3}$

$9 x^{2}$

$x$

$2$

$x^{4}$

$7 x^{3}$

$9 x^{2}$

$27 x$

$54$

$x^{4}$

$2 x^{3}$

$9 x^{3}$

$9 x^{2}$

$9 x^{3}$

$18 x^{2}$

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

$x^{3}$

$9 x^{2}$

$x$

$2$

$x^{4}$

$7 x^{3}$

$9 x^{2}$

$27 x$

$54$

$x^{4}$

$2 x^{3}$

$9 x^{3}$

$9 x^{2}$

$9 x^{3}$

$18 x^{2}$

$27 x^{2}$

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

$x^{3}$

$9 x^{2}$

$x$

$2$

$x^{4}$

$7 x^{3}$

$9 x^{2}$

$27 x$

$54$

$x^{4}$

$2 x^{3}$

$9 x^{3}$

$9 x^{2}$

$9 x^{3}$

$18 x^{2}$

$27 x^{2}$

$27 x$

Divide the highest order term in the dividend $27 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$9 x^{2}$

$27 x$

$x$

$2$

$x^{4}$

$7 x^{3}$

$9 x^{2}$

$27 x$

$54$

$x^{4}$

$2 x^{3}$

$9 x^{3}$

$9 x^{2}$

$9 x^{3}$

$18 x^{2}$

$27 x^{2}$

$27 x$

Multiply the new quotient term by the divisor.

$x^{3}$

$9 x^{2}$

$27 x$

$x$

$2$

$x^{4}$

$7 x^{3}$

$9 x^{2}$

$27 x$

$54$

$x^{4}$

$2 x^{3}$

$9 x^{3}$

$9 x^{2}$

$9 x^{3}$

$18 x^{2}$

$27 x^{2}$

$27 x$

$27 x^{2}$

$54 x$

The expression needs to be subtracted from the dividend, so change all the signs in $27 x^{2} + 54 x$

$x^{3}$

$9 x^{2}$

$27 x$

$x$

$2$

$x^{4}$

$7 x^{3}$

$9 x^{2}$

$27 x$

$54$

$x^{4}$

$2 x^{3}$

$9 x^{3}$

$9 x^{2}$

$9 x^{3}$

$18 x^{2}$

$27 x^{2}$

$27 x$

$27 x^{2}$

$54 x$

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

$x^{3}$

$9 x^{2}$

$27 x$

$x$

$2$

$x^{4}$

$7 x^{3}$

$9 x^{2}$

$27 x$

$54$

$x^{4}$

$2 x^{3}$

$9 x^{3}$

$9 x^{2}$

$9 x^{3}$

$18 x^{2}$

$27 x^{2}$

$27 x$

$27 x^{2}$

$54 x$

$27 x$

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

$x^{3}$

$9 x^{2}$

$27 x$

$x$

$2$

$x^{4}$

$7 x^{3}$

$9 x^{2}$

$27 x$

$54$

$x^{4}$

$2 x^{3}$

$9 x^{3}$

$9 x^{2}$

$9 x^{3}$

$18 x^{2}$

$27 x^{2}$

$27 x$

$27 x^{2}$

$54 x$

$27 x$

$54$

Divide the highest order term in the dividend $- 27 x$ by the highest order term in divisor $x$ .

$x^{3}$

$9 x^{2}$

$27 x$

$27$

$x$

$2$

$x^{4}$

$7 x^{3}$

$9 x^{2}$

$27 x$

$54$

$x^{4}$

$2 x^{3}$

$9 x^{3}$

$9 x^{2}$

$9 x^{3}$

$18 x^{2}$

$27 x^{2}$

$27 x$

$27 x^{2}$

$54 x$

$27 x$

$54$

Multiply the new quotient term by the divisor.

$x^{3}$

$9 x^{2}$

$27 x$

$27$

$x$

$2$

$x^{4}$

$7 x^{3}$

$9 x^{2}$

$27 x$

$54$

$x^{4}$

$2 x^{3}$

$9 x^{3}$

$9 x^{2}$

$9 x^{3}$

$18 x^{2}$

$27 x^{2}$

$27 x$

$27 x^{2}$

$54 x$

$27 x$

$54$

$27 x$

$54$

The expression needs to be subtracted from the dividend, so change all the signs in $- 27 x - 54$

$x^{3}$

$9 x^{2}$

$27 x$

$27$

$x$

$2$

$x^{4}$

$7 x^{3}$

$9 x^{2}$

$27 x$

$54$

$x^{4}$

$2 x^{3}$

$9 x^{3}$

$9 x^{2}$

$9 x^{3}$

$18 x^{2}$

$27 x^{2}$

$27 x$

$27 x^{2}$

$54 x$

$27 x$

$54$

$27 x$

$54$

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

$x^{3}$

$9 x^{2}$

$27 x$

$27$

$x$

$2$

$x^{4}$

$7 x^{3}$

$9 x^{2}$

$27 x$

$54$

$x^{4}$

$2 x^{3}$

$9 x^{3}$

$9 x^{2}$

$9 x^{3}$

$18 x^{2}$

$27 x^{2}$

$27 x$

$27 x^{2}$

$54 x$

$27 x$

$54$

$27 x$

$54$

$0$

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

$x^{3} - 9 x^{2} + 27 x - 27$

Write $x^{4} - 7 x^{3} + 9 x^{2} + 27 x - 54$ as a set of factors.

$f (x) = (x + 2) (x^{3} - 9 x^{2} + 27 x - 27)$

Step 2

Factor $x^{3} - 9 x^{2} + 27 x - 27$ 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.

$p = \pm 1, \pm 27, \pm 3, \pm 9 q = \pm 1$

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

$\pm 1, \pm 27, \pm 3, \pm 9$

Substitute $3$ and simplify the expression. In this case, the expression is equal to $0$ so $3$ is a root of the polynomial.

Substitute $3$ into the polynomial.

$3^{3} - 9 \cdot 3^{2} + 27 \cdot 3 - 27$

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

$27 - 9 \cdot 3^{2} + 27 \cdot 3 - 27$

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

$27 - 9 \cdot 9 + 27 \cdot 3 - 27$

Multiply $- 9$ by $9$ .

$27 - 81 + 27 \cdot 3 - 27$

Subtract $81$ from $27$ .

$- 54 + 27 \cdot 3 - 27$

Multiply $27$ by $3$ .

$- 54 + 81 - 27$

Add $- 54$ and $81$ .

$27 - 27$

Subtract $27$ from $27$ .

$0$

Since $3$ is a known root, divide the polynomial by $x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 9 x^{2} + 27 x - 27}{x - 3}$

Divide $x^{3} - 9 x^{2} + 27 x - 27$ by $x - 3$ .

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

$x$

$3$

$x^{3}$

$9 x^{2}$

$27 x$

$27$

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

					$x^{2}$
	$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$27 x$	-	$27$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$27 x$	-	$27$
			+	$x^{3}$	-	$3 x^{2}$

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$27 x$	-	$27$
			-	$x^{3}$	+	$3 x^{2}$

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$27 x$	-	$27$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$27 x$	-	$27$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$27 x$

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

				$x^{2}$	-	$6 x$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$27 x$	-	$27$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$27 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$6 x$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$27 x$	-	$27$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$27 x$
					-	$6 x^{2}$	+	$18 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{2} + 18 x$

				$x^{2}$	-	$6 x$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$27 x$	-	$27$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$27 x$
					+	$6 x^{2}$	-	$18 x$

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

				$x^{2}$	-	$6 x$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$27 x$	-	$27$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$27 x$
					+	$6 x^{2}$	-	$18 x$
							+	$9 x$

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

				$x^{2}$	-	$6 x$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$27 x$	-	$27$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$27 x$
					+	$6 x^{2}$	-	$18 x$
							+	$9 x$	-	$27$

Divide the highest order term in the dividend $9 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$6 x$	+	$9$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$27 x$	-	$27$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$27 x$
					+	$6 x^{2}$	-	$18 x$
							+	$9 x$	-	$27$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$6 x$	+	$9$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$27 x$	-	$27$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$27 x$
					+	$6 x^{2}$	-	$18 x$
							+	$9 x$	-	$27$
							+	$9 x$	-	$27$

The expression needs to be subtracted from the dividend, so change all the signs in $9 x - 27$

				$x^{2}$	-	$6 x$	+	$9$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$27 x$	-	$27$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$27 x$
					+	$6 x^{2}$	-	$18 x$
							+	$9 x$	-	$27$
							-	$9 x$	+	$27$

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

				$x^{2}$	-	$6 x$	+	$9$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$27 x$	-	$27$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$27 x$
					+	$6 x^{2}$	-	$18 x$
							+	$9 x$	-	$27$
							-	$9 x$	+	$27$
										$0$

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

$x^{2} - 6 x + 9$

Write $x^{3} - 9 x^{2} + 27 x - 27$ as a set of factors.

$f (x) = (x + 2) ((x - 3) (x^{2} - 6 x + 9))$

Step 3

Factor using the perfect square rule.

Rewrite $9$ as $3^{2}$ .

$f (x) = (x + 2) ((x - 3) (x^{2} - 6 x + 3^{2}))$

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

$6 x = 2 \cdot x \cdot 3$

Rewrite the polynomial.

$f (x) = (x + 2) ((x - 3) (x^{2} - 2 \cdot x \cdot 3 + 3^{2}))$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 3$ .

$f (x) = (x + 2) ((x - 3) {(x - 3)}^{2})$

Step 4

Combine like factors.

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

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

Add $1$ and $2$ .

$f (x) = (x + 2) {(x - 3)}^{3}$

Factor f(x)=x^4-7x^3+9x^2+27x-54

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