Algebra Examples

$f (x) = 3 x^{4} + 3 x^{3} - x^{2} - 12 x - 12$

Step 1

Find every combination of $\pm \frac{p}{q}$ .

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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.

$p = \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$

$q = \pm 1, \pm 3$

Step 1.2

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

$\pm 1, \pm \frac{1}{3}, \pm 2, \pm \frac{2}{3}, \pm 3, \pm 4, \pm \frac{4}{3}, \pm 6, \pm 12$

Step 2

Apply synthetic division on $\frac{3 x^{4} + 3 x^{3} - x^{2} - 12 x - 12}{x - 2}$ when $x = 2$ .

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Step 2.1

Place the numbers representing the divisor and the dividend into a division-like configuration.

$2$	$3$	$3$	$- 1$	$- 12$	$- 12$

Step 2.2

The first number in the dividend $(3)$ is put into the first position of the result area (below the horizontal line).

$2$	$3$	$3$	$- 1$	$- 12$	$- 12$

	$3$

Step 2.3

Multiply the newest entry in the result $(3)$ by the divisor $(2)$ and place the result of $(6)$ under the next term in the dividend $(3)$ .

$2$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$6$
	$3$

Step 2.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$2$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$6$
	$3$	$9$

Step 2.5

Multiply the newest entry in the result $(9)$ by the divisor $(2)$ and place the result of $(18)$ under the next term in the dividend $(- 1)$ .

$2$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$6$	$18$
	$3$	$9$

Step 2.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$2$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$6$	$18$
	$3$	$9$	$17$

Step 2.7

Multiply the newest entry in the result $(17)$ by the divisor $(2)$ and place the result of $(34)$ under the next term in the dividend $(- 12)$ .

$2$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$6$	$18$	$34$
	$3$	$9$	$17$

Step 2.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$2$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$6$	$18$	$34$
	$3$	$9$	$17$	$22$

Step 2.9

Multiply the newest entry in the result $(22)$ by the divisor $(2)$ and place the result of $(44)$ under the next term in the dividend $(- 12)$ .

$2$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$6$	$18$	$34$	$44$
	$3$	$9$	$17$	$22$

Step 2.10

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$2$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$6$	$18$	$34$	$44$
	$3$	$9$	$17$	$22$	$32$

Step 2.11

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$3 x^{3} + 9 x^{2} + (17) x + 22 + \frac{32}{x - 2}$

Step 2.12

Simplify the quotient polynomial.

$3 x^{3} + 9 x^{2} + 17 x + 22 + \frac{32}{x - 2}$

Step 3

Since $2 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $2$ is an upper bound for the real roots of the function.

Upper Bound: $2$

Step 4

Apply synthetic division on $\frac{3 x^{4} + 3 x^{3} - x^{2} - 12 x - 12}{x + 2}$ when $x = - 2$ .

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Step 4.1

Place the numbers representing the divisor and the dividend into a division-like configuration.

$- 2$	$3$	$3$	$- 1$	$- 12$	$- 12$

Step 4.2

The first number in the dividend $(3)$ is put into the first position of the result area (below the horizontal line).

$- 2$	$3$	$3$	$- 1$	$- 12$	$- 12$

	$3$

Step 4.3

Multiply the newest entry in the result $(3)$ by the divisor $(- 2)$ and place the result of $(- 6)$ under the next term in the dividend $(3)$ .

$- 2$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 6$
	$3$

Step 4.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 2$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 6$
	$3$	$- 3$

Step 4.5

Multiply the newest entry in the result $(- 3)$ by the divisor $(- 2)$ and place the result of $(6)$ under the next term in the dividend $(- 1)$ .

$- 2$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 6$	$6$
	$3$	$- 3$

Step 4.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 2$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 6$	$6$
	$3$	$- 3$	$5$

Step 4.7

Multiply the newest entry in the result $(5)$ by the divisor $(- 2)$ and place the result of $(- 10)$ under the next term in the dividend $(- 12)$ .

$- 2$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 6$	$6$	$- 10$
	$3$	$- 3$	$5$

Step 4.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 2$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 6$	$6$	$- 10$
	$3$	$- 3$	$5$	$- 22$

Step 4.9

Multiply the newest entry in the result $(- 22)$ by the divisor $(- 2)$ and place the result of $(44)$ under the next term in the dividend $(- 12)$ .

$- 2$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 6$	$6$	$- 10$	$44$
	$3$	$- 3$	$5$	$- 22$

Step 4.10

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 2$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 6$	$6$	$- 10$	$44$
	$3$	$- 3$	$5$	$- 22$	$32$

Step 4.11

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$3 x^{3} + - 3 x^{2} + (5) x - 22 + \frac{32}{x + 2}$

Step 4.12

Simplify the quotient polynomial.

$3 x^{3} - 3 x^{2} + 5 x - 22 + \frac{32}{x + 2}$

Step 5

Since $- 2 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 2$ is a lower bound for the real roots of the function.

Lower Bound: $- 2$

Step 6

Apply synthetic division on $\frac{3 x^{4} + 3 x^{3} - x^{2} - 12 x - 12}{x - 3}$ when $x = 3$ .

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Step 6.1

Place the numbers representing the divisor and the dividend into a division-like configuration.

$3$	$3$	$3$	$- 1$	$- 12$	$- 12$

Step 6.2

The first number in the dividend $(3)$ is put into the first position of the result area (below the horizontal line).

$3$	$3$	$3$	$- 1$	$- 12$	$- 12$

	$3$

Step 6.3

Multiply the newest entry in the result $(3)$ by the divisor $(3)$ and place the result of $(9)$ under the next term in the dividend $(3)$ .

$3$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$9$
	$3$

Step 6.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$3$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$9$
	$3$	$12$

Step 6.5

Multiply the newest entry in the result $(12)$ by the divisor $(3)$ and place the result of $(36)$ under the next term in the dividend $(- 1)$ .

$3$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$9$	$36$
	$3$	$12$

Step 6.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$3$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$9$	$36$
	$3$	$12$	$35$

Step 6.7

Multiply the newest entry in the result $(35)$ by the divisor $(3)$ and place the result of $(105)$ under the next term in the dividend $(- 12)$ .

$3$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$9$	$36$	$105$
	$3$	$12$	$35$

Step 6.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$3$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$9$	$36$	$105$
	$3$	$12$	$35$	$93$

Step 6.9

Multiply the newest entry in the result $(93)$ by the divisor $(3)$ and place the result of $(279)$ under the next term in the dividend $(- 12)$ .

$3$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$9$	$36$	$105$	$279$
	$3$	$12$	$35$	$93$

Step 6.10

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$3$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$9$	$36$	$105$	$279$
	$3$	$12$	$35$	$93$	$267$

Step 6.11

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$3 x^{3} + 12 x^{2} + (35) x + 93 + \frac{267}{x - 3}$

Step 6.12

Simplify the quotient polynomial.

$3 x^{3} + 12 x^{2} + 35 x + 93 + \frac{267}{x - 3}$

Step 7

Since $3 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $3$ is an upper bound for the real roots of the function.

Upper Bound: $3$

Step 8

Apply synthetic division on $\frac{3 x^{4} + 3 x^{3} - x^{2} - 12 x - 12}{x + 3}$ when $x = - 3$ .

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Step 8.1

Place the numbers representing the divisor and the dividend into a division-like configuration.

$- 3$	$3$	$3$	$- 1$	$- 12$	$- 12$

Step 8.2

The first number in the dividend $(3)$ is put into the first position of the result area (below the horizontal line).

$- 3$	$3$	$3$	$- 1$	$- 12$	$- 12$

	$3$

Step 8.3

Multiply the newest entry in the result $(3)$ by the divisor $(- 3)$ and place the result of $(- 9)$ under the next term in the dividend $(3)$ .

$- 3$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 9$
	$3$

Step 8.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 3$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 9$
	$3$	$- 6$

Step 8.5

Multiply the newest entry in the result $(- 6)$ by the divisor $(- 3)$ and place the result of $(18)$ under the next term in the dividend $(- 1)$ .

$- 3$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 9$	$18$
	$3$	$- 6$

Step 8.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 3$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 9$	$18$
	$3$	$- 6$	$17$

Step 8.7

Multiply the newest entry in the result $(17)$ by the divisor $(- 3)$ and place the result of $(- 51)$ under the next term in the dividend $(- 12)$ .

$- 3$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 9$	$18$	$- 51$
	$3$	$- 6$	$17$

Step 8.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 3$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 9$	$18$	$- 51$
	$3$	$- 6$	$17$	$- 63$

Step 8.9

Multiply the newest entry in the result $(- 63)$ by the divisor $(- 3)$ and place the result of $(189)$ under the next term in the dividend $(- 12)$ .

$- 3$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 9$	$18$	$- 51$	$189$
	$3$	$- 6$	$17$	$- 63$

Step 8.10

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 3$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 9$	$18$	$- 51$	$189$
	$3$	$- 6$	$17$	$- 63$	$177$

Step 8.11

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$3 x^{3} + - 6 x^{2} + (17) x - 63 + \frac{177}{x + 3}$

Step 8.12

Simplify the quotient polynomial.

$3 x^{3} - 6 x^{2} + 17 x - 63 + \frac{177}{x + 3}$

Step 9

Since $- 3 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 3$ is a lower bound for the real roots of the function.

Lower Bound: $- 3$

Step 10

Apply synthetic division on $\frac{3 x^{4} + 3 x^{3} - x^{2} - 12 x - 12}{x - 4}$ when $x = 4$ .

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Step 10.1

Place the numbers representing the divisor and the dividend into a division-like configuration.

$4$	$3$	$3$	$- 1$	$- 12$	$- 12$

Step 10.2

The first number in the dividend $(3)$ is put into the first position of the result area (below the horizontal line).

$4$	$3$	$3$	$- 1$	$- 12$	$- 12$

	$3$

Step 10.3

Multiply the newest entry in the result $(3)$ by the divisor $(4)$ and place the result of $(12)$ under the next term in the dividend $(3)$ .

$4$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$12$
	$3$

Step 10.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$4$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$12$
	$3$	$15$

Step 10.5

Multiply the newest entry in the result $(15)$ by the divisor $(4)$ and place the result of $(60)$ under the next term in the dividend $(- 1)$ .

$4$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$12$	$60$
	$3$	$15$

Step 10.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$4$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$12$	$60$
	$3$	$15$	$59$

Step 10.7

Multiply the newest entry in the result $(59)$ by the divisor $(4)$ and place the result of $(236)$ under the next term in the dividend $(- 12)$ .

$4$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$12$	$60$	$236$
	$3$	$15$	$59$

Step 10.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$4$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$12$	$60$	$236$
	$3$	$15$	$59$	$224$

Step 10.9

Multiply the newest entry in the result $(224)$ by the divisor $(4)$ and place the result of $(896)$ under the next term in the dividend $(- 12)$ .

$4$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$12$	$60$	$236$	$896$
	$3$	$15$	$59$	$224$

Step 10.10

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$4$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$12$	$60$	$236$	$896$
	$3$	$15$	$59$	$224$	$884$

Step 10.11

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$3 x^{3} + 15 x^{2} + (59) x + 224 + \frac{884}{x - 4}$

Step 10.12

Simplify the quotient polynomial.

$3 x^{3} + 15 x^{2} + 59 x + 224 + \frac{884}{x - 4}$

Step 11

Since $4 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $4$ is an upper bound for the real roots of the function.

Upper Bound: $4$

Step 12

Apply synthetic division on $\frac{3 x^{4} + 3 x^{3} - x^{2} - 12 x - 12}{x + 4}$ when $x = - 4$ .

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Step 12.1

Place the numbers representing the divisor and the dividend into a division-like configuration.

$- 4$	$3$	$3$	$- 1$	$- 12$	$- 12$

Step 12.2

The first number in the dividend $(3)$ is put into the first position of the result area (below the horizontal line).

$- 4$	$3$	$3$	$- 1$	$- 12$	$- 12$

	$3$

Step 12.3

Multiply the newest entry in the result $(3)$ by the divisor $(- 4)$ and place the result of $(- 12)$ under the next term in the dividend $(3)$ .

$- 4$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 12$
	$3$

Step 12.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 4$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 12$
	$3$	$- 9$

Step 12.5

Multiply the newest entry in the result $(- 9)$ by the divisor $(- 4)$ and place the result of $(36)$ under the next term in the dividend $(- 1)$ .

$- 4$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 12$	$36$
	$3$	$- 9$

Step 12.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 4$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 12$	$36$
	$3$	$- 9$	$35$

Step 12.7

Multiply the newest entry in the result $(35)$ by the divisor $(- 4)$ and place the result of $(- 140)$ under the next term in the dividend $(- 12)$ .

$- 4$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 12$	$36$	$- 140$
	$3$	$- 9$	$35$

Step 12.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 4$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 12$	$36$	$- 140$
	$3$	$- 9$	$35$	$- 152$

Step 12.9

Multiply the newest entry in the result $(- 152)$ by the divisor $(- 4)$ and place the result of $(608)$ under the next term in the dividend $(- 12)$ .

$- 4$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 12$	$36$	$- 140$	$608$
	$3$	$- 9$	$35$	$- 152$

Step 12.10

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 4$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 12$	$36$	$- 140$	$608$
	$3$	$- 9$	$35$	$- 152$	$596$

Step 12.11

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$3 x^{3} + - 9 x^{2} + (35) x - 152 + \frac{596}{x + 4}$

Step 12.12

Simplify the quotient polynomial.

$3 x^{3} - 9 x^{2} + 35 x - 152 + \frac{596}{x + 4}$

Step 13

Since $- 4 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 4$ is a lower bound for the real roots of the function.

Lower Bound: $- 4$

Step 14

Apply synthetic division on $\frac{3 x^{4} + 3 x^{3} - x^{2} - 12 x - 12}{x + \frac{4}{3}}$ when $x = - \frac{4}{3}$ .

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Step 14.1

Place the numbers representing the divisor and the dividend into a division-like configuration.

$- \frac{4}{3}$	$3$	$3$	$- 1$	$- 12$	$- 12$

Step 14.2

The first number in the dividend $(3)$ is put into the first position of the result area (below the horizontal line).

$- \frac{4}{3}$	$3$	$3$	$- 1$	$- 12$	$- 12$

	$3$

Step 14.3

Multiply the newest entry in the result $(3)$ by the divisor $(- \frac{4}{3})$ and place the result of $(- 4)$ under the next term in the dividend $(3)$ .

$- \frac{4}{3}$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 4$
	$3$

Step 14.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- \frac{4}{3}$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 4$
	$3$	$- 1$

Step 14.5

Multiply the newest entry in the result $(- 1)$ by the divisor $(- \frac{4}{3})$ and place the result of $(\frac{4}{3})$ under the next term in the dividend $(- 1)$ .

$- \frac{4}{3}$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 4$	$\frac{4}{3}$
	$3$	$- 1$

Step 14.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- \frac{4}{3}$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 4$	$\frac{4}{3}$
	$3$	$- 1$	$\frac{1}{3}$

Step 14.7

Multiply the newest entry in the result $(\frac{1}{3})$ by the divisor $(- \frac{4}{3})$ and place the result of $(- \frac{4}{9})$ under the next term in the dividend $(- 12)$ .

$- \frac{4}{3}$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 4$	$\frac{4}{3}$	$- \frac{4}{9}$
	$3$	$- 1$	$\frac{1}{3}$

Step 14.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- \frac{4}{3}$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 4$	$\frac{4}{3}$	$- \frac{4}{9}$
	$3$	$- 1$	$\frac{1}{3}$	$- \frac{112}{9}$

Step 14.9

Multiply the newest entry in the result $(- \frac{112}{9})$ by the divisor $(- \frac{4}{3})$ and place the result of $(\frac{448}{27})$ under the next term in the dividend $(- 12)$ .

$- \frac{4}{3}$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 4$	$\frac{4}{3}$	$- \frac{4}{9}$	$\frac{448}{27}$
	$3$	$- 1$	$\frac{1}{3}$	$- \frac{112}{9}$

Step 14.10

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- \frac{4}{3}$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 4$	$\frac{4}{3}$	$- \frac{4}{9}$	$\frac{448}{27}$
	$3$	$- 1$	$\frac{1}{3}$	$- \frac{112}{9}$	$\frac{124}{27}$

Step 14.11

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$3 x^{3} + - 1 x^{2} + (\frac{1}{3}) x - \frac{112}{9} + \frac{\frac{124}{27}}{x + \frac{4}{3}}$

Step 14.12

Simplify the quotient polynomial.

$3 x^{3} - x^{2} + \frac{x}{3} - \frac{112}{9} + \frac{124}{9 (3 x + 4)}$

Step 15

Since $- \frac{4}{3} < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- \frac{4}{3}$ is a lower bound for the real roots of the function.

Lower Bound: $- \frac{4}{3}$

Step 16

Apply synthetic division on $\frac{3 x^{4} + 3 x^{3} - x^{2} - 12 x - 12}{x - 6}$ when $x = 6$ .

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Step 16.1

Place the numbers representing the divisor and the dividend into a division-like configuration.

$6$	$3$	$3$	$- 1$	$- 12$	$- 12$

Step 16.2

The first number in the dividend $(3)$ is put into the first position of the result area (below the horizontal line).

$6$	$3$	$3$	$- 1$	$- 12$	$- 12$

	$3$

Step 16.3

Multiply the newest entry in the result $(3)$ by the divisor $(6)$ and place the result of $(18)$ under the next term in the dividend $(3)$ .

$6$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$18$
	$3$

Step 16.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$6$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$18$
	$3$	$21$

Step 16.5

Multiply the newest entry in the result $(21)$ by the divisor $(6)$ and place the result of $(126)$ under the next term in the dividend $(- 1)$ .

$6$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$18$	$126$
	$3$	$21$

Step 16.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$6$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$18$	$126$
	$3$	$21$	$125$

Step 16.7

Multiply the newest entry in the result $(125)$ by the divisor $(6)$ and place the result of $(750)$ under the next term in the dividend $(- 12)$ .

$6$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$18$	$126$	$750$
	$3$	$21$	$125$

Step 16.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$6$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$18$	$126$	$750$
	$3$	$21$	$125$	$738$

Step 16.9

Multiply the newest entry in the result $(738)$ by the divisor $(6)$ and place the result of $(4428)$ under the next term in the dividend $(- 12)$ .

$6$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$18$	$126$	$750$	$4428$
	$3$	$21$	$125$	$738$

Step 16.10

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$6$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$18$	$126$	$750$	$4428$
	$3$	$21$	$125$	$738$	$4416$

Step 16.11

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$3 x^{3} + 21 x^{2} + (125) x + 738 + \frac{4416}{x - 6}$

Step 16.12

Simplify the quotient polynomial.

$3 x^{3} + 21 x^{2} + 125 x + 738 + \frac{4416}{x - 6}$

Step 17

Since $6 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $6$ is an upper bound for the real roots of the function.

Upper Bound: $6$

Step 18

Apply synthetic division on $\frac{3 x^{4} + 3 x^{3} - x^{2} - 12 x - 12}{x + 6}$ when $x = - 6$ .

Tap for more steps...

Step 18.1

Place the numbers representing the divisor and the dividend into a division-like configuration.

$- 6$	$3$	$3$	$- 1$	$- 12$	$- 12$

Step 18.2

The first number in the dividend $(3)$ is put into the first position of the result area (below the horizontal line).

$- 6$	$3$	$3$	$- 1$	$- 12$	$- 12$

	$3$

Step 18.3

Multiply the newest entry in the result $(3)$ by the divisor $(- 6)$ and place the result of $(- 18)$ under the next term in the dividend $(3)$ .

$- 6$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 18$
	$3$

Step 18.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 6$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 18$
	$3$	$- 15$

Step 18.5

Multiply the newest entry in the result $(- 15)$ by the divisor $(- 6)$ and place the result of $(90)$ under the next term in the dividend $(- 1)$ .

$- 6$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 18$	$90$
	$3$	$- 15$

Step 18.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 6$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 18$	$90$
	$3$	$- 15$	$89$

Step 18.7

Multiply the newest entry in the result $(89)$ by the divisor $(- 6)$ and place the result of $(- 534)$ under the next term in the dividend $(- 12)$ .

$- 6$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 18$	$90$	$- 534$
	$3$	$- 15$	$89$

Step 18.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 6$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 18$	$90$	$- 534$
	$3$	$- 15$	$89$	$- 546$

Step 18.9

Multiply the newest entry in the result $(- 546)$ by the divisor $(- 6)$ and place the result of $(3276)$ under the next term in the dividend $(- 12)$ .

$- 6$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 18$	$90$	$- 534$	$3276$
	$3$	$- 15$	$89$	$- 546$

Step 18.10

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 6$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 18$	$90$	$- 534$	$3276$
	$3$	$- 15$	$89$	$- 546$	$3264$

Step 18.11

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$3 x^{3} + - 15 x^{2} + (89) x - 546 + \frac{3264}{x + 6}$

Step 18.12

Simplify the quotient polynomial.

$3 x^{3} - 15 x^{2} + 89 x - 546 + \frac{3264}{x + 6}$

Step 19

Since $- 6 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 6$ is a lower bound for the real roots of the function.

Lower Bound: $- 6$

Step 20

Apply synthetic division on $\frac{3 x^{4} + 3 x^{3} - x^{2} - 12 x - 12}{x - 12}$ when $x = 12$ .

Tap for more steps...

Step 20.1

Place the numbers representing the divisor and the dividend into a division-like configuration.

$12$	$3$	$3$	$- 1$	$- 12$	$- 12$

Step 20.2

The first number in the dividend $(3)$ is put into the first position of the result area (below the horizontal line).

$12$	$3$	$3$	$- 1$	$- 12$	$- 12$

	$3$

Step 20.3

Multiply the newest entry in the result $(3)$ by the divisor $(12)$ and place the result of $(36)$ under the next term in the dividend $(3)$ .

$12$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$36$
	$3$

Step 20.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$12$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$36$
	$3$	$39$

Step 20.5

Multiply the newest entry in the result $(39)$ by the divisor $(12)$ and place the result of $(468)$ under the next term in the dividend $(- 1)$ .

$12$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$36$	$468$
	$3$	$39$

Step 20.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$12$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$36$	$468$
	$3$	$39$	$467$

Step 20.7

Multiply the newest entry in the result $(467)$ by the divisor $(12)$ and place the result of $(5604)$ under the next term in the dividend $(- 12)$ .

$12$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$36$	$468$	$5604$
	$3$	$39$	$467$

Step 20.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$12$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$36$	$468$	$5604$
	$3$	$39$	$467$	$5592$

Step 20.9

Multiply the newest entry in the result $(5592)$ by the divisor $(12)$ and place the result of $(67104)$ under the next term in the dividend $(- 12)$ .

$12$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$36$	$468$	$5604$	$67104$
	$3$	$39$	$467$	$5592$

Step 20.10

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$12$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$36$	$468$	$5604$	$67104$
	$3$	$39$	$467$	$5592$	$67092$

Step 20.11

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$3 x^{3} + 39 x^{2} + (467) x + 5592 + \frac{67092}{x - 12}$

Step 20.12

Simplify the quotient polynomial.

$3 x^{3} + 39 x^{2} + 467 x + 5592 + \frac{67092}{x - 12}$

Step 21

Since $12 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $12$ is an upper bound for the real roots of the function.

Upper Bound: $12$

Step 22

Apply synthetic division on $\frac{3 x^{4} + 3 x^{3} - x^{2} - 12 x - 12}{x + 12}$ when $x = - 12$ .

Tap for more steps...

Step 22.1

Place the numbers representing the divisor and the dividend into a division-like configuration.

$- 12$	$3$	$3$	$- 1$	$- 12$	$- 12$

Step 22.2

The first number in the dividend $(3)$ is put into the first position of the result area (below the horizontal line).

$- 12$	$3$	$3$	$- 1$	$- 12$	$- 12$

	$3$

Step 22.3

Multiply the newest entry in the result $(3)$ by the divisor $(- 12)$ and place the result of $(- 36)$ under the next term in the dividend $(3)$ .

$- 12$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 36$
	$3$

Step 22.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 12$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 36$
	$3$	$- 33$

Step 22.5

Multiply the newest entry in the result $(- 33)$ by the divisor $(- 12)$ and place the result of $(396)$ under the next term in the dividend $(- 1)$ .

$- 12$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 36$	$396$
	$3$	$- 33$

Step 22.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 12$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 36$	$396$
	$3$	$- 33$	$395$

Step 22.7

Multiply the newest entry in the result $(395)$ by the divisor $(- 12)$ and place the result of $(- 4740)$ under the next term in the dividend $(- 12)$ .

$- 12$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 36$	$396$	$- 4740$
	$3$	$- 33$	$395$

Step 22.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 12$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 36$	$396$	$- 4740$
	$3$	$- 33$	$395$	$- 4752$

Step 22.9

Multiply the newest entry in the result $(- 4752)$ by the divisor $(- 12)$ and place the result of $(57024)$ under the next term in the dividend $(- 12)$ .

$- 12$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 36$	$396$	$- 4740$	$57024$
	$3$	$- 33$	$395$	$- 4752$

Step 22.10

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 12$	$3$	$3$	$- 1$	$- 12$	$- 12$
		$- 36$	$396$	$- 4740$	$57024$
	$3$	$- 33$	$395$	$- 4752$	$57012$

Step 22.11

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$3 x^{3} + - 33 x^{2} + (395) x - 4752 + \frac{57012}{x + 12}$

Step 22.12

Simplify the quotient polynomial.

$3 x^{3} - 33 x^{2} + 395 x - 4752 + \frac{57012}{x + 12}$

Step 23

Since $- 12 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 12$ is a lower bound for the real roots of the function.

Lower Bound: $- 12$

Step 24

Determine the upper and lower bounds.

Upper Bounds: $2, 3, 4, 6, 12$

Lower Bounds: $- 2, - 3, - 4, - \frac{4}{3}, - 6, - 12$

Step 25