Factor 3x^4-14x^3-3x^2+54x-40

$3 x^{4} - 14 x^{3} - 3 x^{2} + 54 x - 40$

Step 1

Regroup terms.

$3 x^{4} - 3 x^{2} - 14 x^{3} + 54 x - 40$

Step 2

Factor $3 x^{2}$ out of $3 x^{4} - 3 x^{2}$ .

Factor $3 x^{2}$ out of $3 x^{4}$ .

$3 x^{2} (x^{2}) - 3 x^{2} - 14 x^{3} + 54 x - 40$

Factor $3 x^{2}$ out of $- 3 x^{2}$ .

$3 x^{2} (x^{2}) + 3 x^{2} (- 1) - 14 x^{3} + 54 x - 40$

Factor $3 x^{2}$ out of $3 x^{2} (x^{2}) + 3 x^{2} (- 1)$ .

$3 x^{2} (x^{2} - 1) - 14 x^{3} + 54 x - 40$

Step 3

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

$3 x^{2} (x^{2} - 1^{2}) - 14 x^{3} + 54 x - 40$

Step 4

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$3 x^{2} ((x + 1) (x - 1)) - 14 x^{3} + 54 x - 40$

Remove unnecessary parentheses.

$3 x^{2} (x + 1) (x - 1) - 14 x^{3} + 54 x - 40$

Step 5

Factor $2$ out of $- 14 x^{3} + 54 x - 40$ .

Factor $2$ out of $- 14 x^{3}$ .

$3 x^{2} (x + 1) (x - 1) + 2 (- 7 x^{3}) + 54 x - 40$

Factor $2$ out of $54 x$ .

$3 x^{2} (x + 1) (x - 1) + 2 (- 7 x^{3}) + 2 (27 x) - 40$

Factor $2$ out of $- 40$ .

$3 x^{2} (x + 1) (x - 1) + 2 (- 7 x^{3}) + 2 (27 x) + 2 (- 20)$

Factor $2$ out of $2 (- 7 x^{3}) + 2 (27 x)$ .

$3 x^{2} (x + 1) (x - 1) + 2 (- 7 x^{3} + 27 x) + 2 (- 20)$

Factor $2$ out of $2 (- 7 x^{3} + 27 x) + 2 (- 20)$ .

$3 x^{2} (x + 1) (x - 1) + 2 (- 7 x^{3} + 27 x - 20)$

Step 6

Factor.

Factor $- 7 x^{3} + 27 x - 20$ 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.

$p = \pm 1, \pm 20, \pm 2, \pm 10, \pm 4, \pm 5 q = \pm 1, \pm 7$

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

$\pm 1, \pm 0.\overline{142857}, \pm 20, \pm 2.\overline{857142}, \pm 2, \pm 0.\overline{285714}, \pm 10, \pm 1.\overline{428571}, \pm 4, \pm 0.\overline{571428}, \pm 5, \pm 0.\overline{714285}$

Substitute $1$ and simplify the expression. In this case, the expression is equal to $0$ so $1$ is a root of the polynomial.

Substitute $1$ into the polynomial.

$- 7 \cdot 1^{3} + 27 \cdot 1 - 20$

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

$- 7 \cdot 1 + 27 \cdot 1 - 20$

Multiply $- 7$ by $1$ .

$- 7 + 27 \cdot 1 - 20$

Multiply $27$ by $1$ .

$- 7 + 27 - 20$

Add $- 7$ and $27$ .

$20 - 20$

Subtract $20$ from $20$ .

$0$

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.

$\frac{- 7 x^{3} + 27 x - 20}{x - 1}$

Divide $- 7 x^{3} + 27 x - 20$ by $x - 1$ .

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

$x$

$1$

$7 x^{3}$

$0 x^{2}$

$27 x$

$20$

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

				-	$7 x^{2}$
	$x$	-	$1$	-	$7 x^{3}$	+	$0 x^{2}$	+	$27 x$	-	$20$

Multiply the new quotient term by the divisor.

			-	$7 x^{2}$
$x$	-	$1$	-	$7 x^{3}$	+	$0 x^{2}$	+	$27 x$	-	$20$
			-	$7 x^{3}$	+	$7 x^{2}$

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

			-	$7 x^{2}$
$x$	-	$1$	-	$7 x^{3}$	+	$0 x^{2}$	+	$27 x$	-	$20$
			+	$7 x^{3}$	-	$7 x^{2}$

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

			-	$7 x^{2}$
$x$	-	$1$	-	$7 x^{3}$	+	$0 x^{2}$	+	$27 x$	-	$20$
			+	$7 x^{3}$	-	$7 x^{2}$
					-	$7 x^{2}$

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

			-	$7 x^{2}$
$x$	-	$1$	-	$7 x^{3}$	+	$0 x^{2}$	+	$27 x$	-	$20$
			+	$7 x^{3}$	-	$7 x^{2}$
					-	$7 x^{2}$	+	$27 x$

Divide the highest order term in the dividend $- 7 x^{2}$ by the highest order term in divisor $x$ .

			-	$7 x^{2}$	-	$7 x$
$x$	-	$1$	-	$7 x^{3}$	+	$0 x^{2}$	+	$27 x$	-	$20$
			+	$7 x^{3}$	-	$7 x^{2}$
					-	$7 x^{2}$	+	$27 x$

Multiply the new quotient term by the divisor.

			-	$7 x^{2}$	-	$7 x$
$x$	-	$1$	-	$7 x^{3}$	+	$0 x^{2}$	+	$27 x$	-	$20$
			+	$7 x^{3}$	-	$7 x^{2}$
					-	$7 x^{2}$	+	$27 x$
					-	$7 x^{2}$	+	$7 x$

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

			-	$7 x^{2}$	-	$7 x$
$x$	-	$1$	-	$7 x^{3}$	+	$0 x^{2}$	+	$27 x$	-	$20$
			+	$7 x^{3}$	-	$7 x^{2}$
					-	$7 x^{2}$	+	$27 x$
					+	$7 x^{2}$	-	$7 x$

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

			-	$7 x^{2}$	-	$7 x$
$x$	-	$1$	-	$7 x^{3}$	+	$0 x^{2}$	+	$27 x$	-	$20$
			+	$7 x^{3}$	-	$7 x^{2}$
					-	$7 x^{2}$	+	$27 x$
					+	$7 x^{2}$	-	$7 x$
							+	$20 x$

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

			-	$7 x^{2}$	-	$7 x$
$x$	-	$1$	-	$7 x^{3}$	+	$0 x^{2}$	+	$27 x$	-	$20$
			+	$7 x^{3}$	-	$7 x^{2}$
					-	$7 x^{2}$	+	$27 x$
					+	$7 x^{2}$	-	$7 x$
							+	$20 x$	-	$20$

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

			-	$7 x^{2}$	-	$7 x$	+	$20$
$x$	-	$1$	-	$7 x^{3}$	+	$0 x^{2}$	+	$27 x$	-	$20$
			+	$7 x^{3}$	-	$7 x^{2}$
					-	$7 x^{2}$	+	$27 x$
					+	$7 x^{2}$	-	$7 x$
							+	$20 x$	-	$20$

Multiply the new quotient term by the divisor.

			-	$7 x^{2}$	-	$7 x$	+	$20$
$x$	-	$1$	-	$7 x^{3}$	+	$0 x^{2}$	+	$27 x$	-	$20$
			+	$7 x^{3}$	-	$7 x^{2}$
					-	$7 x^{2}$	+	$27 x$
					+	$7 x^{2}$	-	$7 x$
							+	$20 x$	-	$20$
							+	$20 x$	-	$20$

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

			-	$7 x^{2}$	-	$7 x$	+	$20$
$x$	-	$1$	-	$7 x^{3}$	+	$0 x^{2}$	+	$27 x$	-	$20$
			+	$7 x^{3}$	-	$7 x^{2}$
					-	$7 x^{2}$	+	$27 x$
					+	$7 x^{2}$	-	$7 x$
							+	$20 x$	-	$20$
							-	$20 x$	+	$20$

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

			-	$7 x^{2}$	-	$7 x$	+	$20$
$x$	-	$1$	-	$7 x^{3}$	+	$0 x^{2}$	+	$27 x$	-	$20$
			+	$7 x^{3}$	-	$7 x^{2}$
					-	$7 x^{2}$	+	$27 x$
					+	$7 x^{2}$	-	$7 x$
							+	$20 x$	-	$20$
							-	$20 x$	+	$20$
										$0$

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

$- 7 x^{2} - 7 x + 20$

Write $- 7 x^{3} + 27 x - 20$ as a set of factors.

$3 x^{2} (x + 1) (x - 1) + 2 ((x - 1) (- 7 x^{2} - 7 x + 20))$

Remove unnecessary parentheses.

$3 x^{2} (x + 1) (x - 1) + 2 (x - 1) (- 7 x^{2} - 7 x + 20)$

Step 7

Factor $x - 1$ out of $3 x^{2} (x + 1) (x - 1) + 2 (x - 1) (- 7 x^{2} - 7 x + 20)$ .

Factor $x - 1$ out of $3 x^{2} (x + 1) (x - 1)$ .

$(x - 1) (3 x^{2} (x + 1)) + 2 (x - 1) (- 7 x^{2} - 7 x + 20)$

Factor $x - 1$ out of $2 (x - 1) (- 7 x^{2} - 7 x + 20)$ .

$(x - 1) (3 x^{2} (x + 1)) + (x - 1) (2 (- 7 x^{2} - 7 x + 20))$

Factor $x - 1$ out of $(x - 1) (3 x^{2} (x + 1)) + (x - 1) (2 (- 7 x^{2} - 7 x + 20))$ .

$(x - 1) (3 x^{2} (x + 1) + 2 (- 7 x^{2} - 7 x + 20))$

Step 8

Apply the distributive property.

$(x - 1) (3 x^{2} x + 3 x^{2} \cdot 1 + 2 (- 7 x^{2} - 7 x + 20))$

Step 9

Multiply $x^{2}$ by $x$ by adding the exponents.

Move $x$ .

$(x - 1) (3 (x \cdot x^{2}) + 3 x^{2} \cdot 1 + 2 (- 7 x^{2} - 7 x + 20))$

Multiply $x$ by $x^{2}$ .

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

$(x - 1) (3 (x^{1} x^{2}) + 3 x^{2} \cdot 1 + 2 (- 7 x^{2} - 7 x + 20))$

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

$(x - 1) (3 x^{1 + 2} + 3 x^{2} \cdot 1 + 2 (- 7 x^{2} - 7 x + 20))$

Add $1$ and $2$ .

$(x - 1) (3 x^{3} + 3 x^{2} \cdot 1 + 2 (- 7 x^{2} - 7 x + 20))$

Step 10

Multiply $3$ by $1$ .

$(x - 1) (3 x^{3} + 3 x^{2} + 2 (- 7 x^{2} - 7 x + 20))$

Step 11

Apply the distributive property.

$(x - 1) (3 x^{3} + 3 x^{2} + 2 (- 7 x^{2}) + 2 (- 7 x) + 2 \cdot 20)$

Step 12

Simplify.

Multiply $- 7$ by $2$ .

$(x - 1) (3 x^{3} + 3 x^{2} - 14 x^{2} + 2 (- 7 x) + 2 \cdot 20)$

Multiply $- 7$ by $2$ .

$(x - 1) (3 x^{3} + 3 x^{2} - 14 x^{2} - 14 x + 2 \cdot 20)$

Multiply $2$ by $20$ .

$(x - 1) (3 x^{3} + 3 x^{2} - 14 x^{2} - 14 x + 40)$

Step 13

Subtract $14 x^{2}$ from $3 x^{2}$ .

$(x - 1) (3 x^{3} - 11 x^{2} - 14 x + 40)$

Step 14

Factor.

Rewrite $3 x^{3} - 11 x^{2} - 14 x + 40$ in a factored form.

Factor $3 x^{3} - 11 x^{2} - 14 x + 40$ 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.

$p = \pm 1, \pm 40, \pm 2, \pm 20, \pm 4, \pm 10, \pm 5, \pm 8 q = \pm 1, \pm 3$

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

$\pm 1, \pm 0.\overline{3}, \pm 40, \pm 13.\overline{3}, \pm 2, \pm 0.\overline{6}, \pm 20, \pm 6.\overline{6}, \pm 4, \pm 1.\overline{3}, \pm 10, \pm 3.\overline{3}, \pm 5, \pm 1.\overline{6}, \pm 8, \pm 2.\overline{6}$

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.

$3 {(- 2)}^{3} - 11 {(- 2)}^{2} - 14 \cdot - 2 + 40$

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

$3 \cdot - 8 - 11 {(- 2)}^{2} - 14 \cdot - 2 + 40$

Multiply $3$ by $- 8$ .

$- 24 - 11 {(- 2)}^{2} - 14 \cdot - 2 + 40$

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

$- 24 - 11 \cdot 4 - 14 \cdot - 2 + 40$

Multiply $- 11$ by $4$ .

$- 24 - 44 - 14 \cdot - 2 + 40$

Subtract $44$ from $- 24$ .

$- 68 - 14 \cdot - 2 + 40$

Multiply $- 14$ by $- 2$ .

$- 68 + 28 + 40$

Add $- 68$ and $28$ .

$- 40 + 40$

Add $- 40$ and $40$ .

$0$

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.

$\frac{3 x^{3} - 11 x^{2} - 14 x + 40}{x + 2}$

Divide $3 x^{3} - 11 x^{2} - 14 x + 40$ 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 $0$ .

$x$

$2$

$3 x^{3}$

$11 x^{2}$

$14 x$

$40$

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

					$3 x^{2}$
	$x$	+	$2$		$3 x^{3}$	-	$11 x^{2}$	-	$14 x$	+	$40$

Multiply the new quotient term by the divisor.

				$3 x^{2}$
$x$	+	$2$		$3 x^{3}$	-	$11 x^{2}$	-	$14 x$	+	$40$
			+	$3 x^{3}$	+	$6 x^{2}$

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

				$3 x^{2}$
$x$	+	$2$		$3 x^{3}$	-	$11 x^{2}$	-	$14 x$	+	$40$
			-	$3 x^{3}$	-	$6 x^{2}$

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

				$3 x^{2}$
$x$	+	$2$		$3 x^{3}$	-	$11 x^{2}$	-	$14 x$	+	$40$
			-	$3 x^{3}$	-	$6 x^{2}$
					-	$17 x^{2}$

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

				$3 x^{2}$
$x$	+	$2$		$3 x^{3}$	-	$11 x^{2}$	-	$14 x$	+	$40$
			-	$3 x^{3}$	-	$6 x^{2}$
					-	$17 x^{2}$	-	$14 x$

Divide the highest order term in the dividend $- 17 x^{2}$ by the highest order term in divisor $x$ .

				$3 x^{2}$	-	$17 x$
$x$	+	$2$		$3 x^{3}$	-	$11 x^{2}$	-	$14 x$	+	$40$
			-	$3 x^{3}$	-	$6 x^{2}$
					-	$17 x^{2}$	-	$14 x$

Multiply the new quotient term by the divisor.

				$3 x^{2}$	-	$17 x$
$x$	+	$2$		$3 x^{3}$	-	$11 x^{2}$	-	$14 x$	+	$40$
			-	$3 x^{3}$	-	$6 x^{2}$
					-	$17 x^{2}$	-	$14 x$
					-	$17 x^{2}$	-	$34 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 17 x^{2} - 34 x$

				$3 x^{2}$	-	$17 x$
$x$	+	$2$		$3 x^{3}$	-	$11 x^{2}$	-	$14 x$	+	$40$
			-	$3 x^{3}$	-	$6 x^{2}$
					-	$17 x^{2}$	-	$14 x$
					+	$17 x^{2}$	+	$34 x$

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

				$3 x^{2}$	-	$17 x$
$x$	+	$2$		$3 x^{3}$	-	$11 x^{2}$	-	$14 x$	+	$40$
			-	$3 x^{3}$	-	$6 x^{2}$
					-	$17 x^{2}$	-	$14 x$
					+	$17 x^{2}$	+	$34 x$
							+	$20 x$

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

				$3 x^{2}$	-	$17 x$
$x$	+	$2$		$3 x^{3}$	-	$11 x^{2}$	-	$14 x$	+	$40$
			-	$3 x^{3}$	-	$6 x^{2}$
					-	$17 x^{2}$	-	$14 x$
					+	$17 x^{2}$	+	$34 x$
							+	$20 x$	+	$40$

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

				$3 x^{2}$	-	$17 x$	+	$20$
$x$	+	$2$		$3 x^{3}$	-	$11 x^{2}$	-	$14 x$	+	$40$
			-	$3 x^{3}$	-	$6 x^{2}$
					-	$17 x^{2}$	-	$14 x$
					+	$17 x^{2}$	+	$34 x$
							+	$20 x$	+	$40$

Multiply the new quotient term by the divisor.

				$3 x^{2}$	-	$17 x$	+	$20$
$x$	+	$2$		$3 x^{3}$	-	$11 x^{2}$	-	$14 x$	+	$40$
			-	$3 x^{3}$	-	$6 x^{2}$
					-	$17 x^{2}$	-	$14 x$
					+	$17 x^{2}$	+	$34 x$
							+	$20 x$	+	$40$
							+	$20 x$	+	$40$

The expression needs to be subtracted from the dividend, so change all the signs in $20 x + 40$

				$3 x^{2}$	-	$17 x$	+	$20$
$x$	+	$2$		$3 x^{3}$	-	$11 x^{2}$	-	$14 x$	+	$40$
			-	$3 x^{3}$	-	$6 x^{2}$
					-	$17 x^{2}$	-	$14 x$
					+	$17 x^{2}$	+	$34 x$
							+	$20 x$	+	$40$
							-	$20 x$	-	$40$

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

				$3 x^{2}$	-	$17 x$	+	$20$
$x$	+	$2$		$3 x^{3}$	-	$11 x^{2}$	-	$14 x$	+	$40$
			-	$3 x^{3}$	-	$6 x^{2}$
					-	$17 x^{2}$	-	$14 x$
					+	$17 x^{2}$	+	$34 x$
							+	$20 x$	+	$40$
							-	$20 x$	-	$40$
										$0$

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

$3 x^{2} - 17 x + 20$

Write $3 x^{3} - 11 x^{2} - 14 x + 40$ as a set of factors.

$(x - 1) ((x + 2) (3 x^{2} - 17 x + 20))$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 20 = 60$ and whose sum is $b = - 17$ .

Factor $- 17$ out of $- 17 x$ .

$(x - 1) ((x + 2) (3 x^{2} - 17 (x) + 20))$

Rewrite $- 17$ as $- 5$ plus $- 12$

$(x - 1) ((x + 2) (3 x^{2} + (- 5 - 12) x + 20))$

Apply the distributive property.

$(x - 1) ((x + 2) (3 x^{2} - 5 x - 12 x + 20))$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x - 1) ((x + 2) ((3 x^{2} - 5 x) - 12 x + 20))$

Factor out the greatest common factor (GCF) from each group.

$(x - 1) ((x + 2) (x (3 x - 5) - 4 (3 x - 5)))$

Factor the polynomial by factoring out the greatest common factor, $3 x - 5$ .

$(x - 1) ((x + 2) ((3 x - 5) (x - 4)))$

Remove unnecessary parentheses.

$(x - 1) ((x + 2) (3 x - 5) (x - 4))$

Remove unnecessary parentheses.

$(x - 1) (x + 2) (3 x - 5) (x - 4)$

Factor 3x^4-14x^3-3x^2+54x-40

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