Precalculus Examples

$f (x) = x^{4} - 4 x^{3} + 16 x - 16$

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.

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

$q = \pm 1$

Step 1.2

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

$\pm 1, \pm 2, \pm 4, \pm 8, \pm 16$

Step 2

Apply synthetic division on $\frac{x^{4} - 4 x^{3} + 16 x - 16}{x + 2}$ when $x = - 2$ .

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Step 2.1

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

$- 2$	$1$	$- 4$	$0$	$16$	$- 16$

Step 2.2

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$- 2$	$1$	$- 4$	$0$	$16$	$- 16$

	$1$

Step 2.3

Multiply the newest entry in the result $(1)$ by the divisor $(- 2)$ and place the result of $(- 2)$ under the next term in the dividend $(- 4)$ .

$- 2$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 2$
	$1$

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.

$- 2$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 2$
	$1$	$- 6$

Step 2.5

Multiply the newest entry in the result $(- 6)$ by the divisor $(- 2)$ and place the result of $(12)$ under the next term in the dividend $(0)$ .

$- 2$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 2$	$12$
	$1$	$- 6$

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.

$- 2$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 2$	$12$
	$1$	$- 6$	$12$

Step 2.7

Multiply the newest entry in the result $(12)$ by the divisor $(- 2)$ and place the result of $(- 24)$ under the next term in the dividend $(16)$ .

$- 2$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 2$	$12$	$- 24$
	$1$	$- 6$	$12$

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.

$- 2$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 2$	$12$	$- 24$
	$1$	$- 6$	$12$	$- 8$

Step 2.9

Multiply the newest entry in the result $(- 8)$ by the divisor $(- 2)$ and place the result of $(16)$ under the next term in the dividend $(- 16)$ .

$- 2$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 2$	$12$	$- 24$	$16$
	$1$	$- 6$	$12$	$- 8$

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.

$- 2$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 2$	$12$	$- 24$	$16$
	$1$	$- 6$	$12$	$- 8$	$0$

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.

$1 x^{3} + - 6 x^{2} + (12) x - 8$

Step 2.12

Simplify the quotient polynomial.

$x^{3} - 6 x^{2} + 12 x - 8$

Step 3

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 4

Apply synthetic division on $\frac{x^{4} - 4 x^{3} + 16 x - 16}{x - 4}$ when $x = 4$ .

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Step 4.1

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

$4$	$1$	$- 4$	$0$	$16$	$- 16$

Step 4.2

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$4$	$1$	$- 4$	$0$	$16$	$- 16$

	$1$

Step 4.3

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

$4$	$1$	$- 4$	$0$	$16$	$- 16$
		$4$
	$1$

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.

$4$	$1$	$- 4$	$0$	$16$	$- 16$
		$4$
	$1$	$0$

Step 4.5

Multiply the newest entry in the result $(0)$ by the divisor $(4)$ and place the result of $(0)$ under the next term in the dividend $(0)$ .

$4$	$1$	$- 4$	$0$	$16$	$- 16$
		$4$	$0$
	$1$	$0$

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.

$4$	$1$	$- 4$	$0$	$16$	$- 16$
		$4$	$0$
	$1$	$0$	$0$

Step 4.7

Multiply the newest entry in the result $(0)$ by the divisor $(4)$ and place the result of $(0)$ under the next term in the dividend $(16)$ .

$4$	$1$	$- 4$	$0$	$16$	$- 16$
		$4$	$0$	$0$
	$1$	$0$	$0$

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.

$4$	$1$	$- 4$	$0$	$16$	$- 16$
		$4$	$0$	$0$
	$1$	$0$	$0$	$16$

Step 4.9

Multiply the newest entry in the result $(16)$ by the divisor $(4)$ and place the result of $(64)$ under the next term in the dividend $(- 16)$ .

$4$	$1$	$- 4$	$0$	$16$	$- 16$
		$4$	$0$	$0$	$64$
	$1$	$0$	$0$	$16$

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.

$4$	$1$	$- 4$	$0$	$16$	$- 16$
		$4$	$0$	$0$	$64$
	$1$	$0$	$0$	$16$	$48$

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.

$1 x^{3} + 0 x^{2} + (0) x + 16 + \frac{48}{x - 4}$

Step 4.12

Simplify the quotient polynomial.

$x^{3} + 16 + \frac{48}{x - 4}$

Step 5

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 6

Apply synthetic division on $\frac{x^{4} - 4 x^{3} + 16 x - 16}{x + 4}$ when $x = - 4$ .

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Step 6.1

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

$- 4$	$1$	$- 4$	$0$	$16$	$- 16$

Step 6.2

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$- 4$	$1$	$- 4$	$0$	$16$	$- 16$

	$1$

Step 6.3

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

$- 4$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 4$
	$1$

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.

$- 4$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 4$
	$1$	$- 8$

Step 6.5

Multiply the newest entry in the result $(- 8)$ by the divisor $(- 4)$ and place the result of $(32)$ under the next term in the dividend $(0)$ .

$- 4$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 4$	$32$
	$1$	$- 8$

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.

$- 4$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 4$	$32$
	$1$	$- 8$	$32$

Step 6.7

Multiply the newest entry in the result $(32)$ by the divisor $(- 4)$ and place the result of $(- 128)$ under the next term in the dividend $(16)$ .

$- 4$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 4$	$32$	$- 128$
	$1$	$- 8$	$32$

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.

$- 4$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 4$	$32$	$- 128$
	$1$	$- 8$	$32$	$- 112$

Step 6.9

Multiply the newest entry in the result $(- 112)$ by the divisor $(- 4)$ and place the result of $(448)$ under the next term in the dividend $(- 16)$ .

$- 4$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 4$	$32$	$- 128$	$448$
	$1$	$- 8$	$32$	$- 112$

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.

$- 4$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 4$	$32$	$- 128$	$448$
	$1$	$- 8$	$32$	$- 112$	$432$

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.

$1 x^{3} + - 8 x^{2} + (32) x - 112 + \frac{432}{x + 4}$

Step 6.12

Simplify the quotient polynomial.

$x^{3} - 8 x^{2} + 32 x - 112 + \frac{432}{x + 4}$

Step 7

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 8

Apply synthetic division on $\frac{x^{4} - 4 x^{3} + 16 x - 16}{x - 8}$ when $x = 8$ .

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Step 8.1

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

$8$	$1$	$- 4$	$0$	$16$	$- 16$

Step 8.2

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$8$	$1$	$- 4$	$0$	$16$	$- 16$

	$1$

Step 8.3

Multiply the newest entry in the result $(1)$ by the divisor $(8)$ and place the result of $(8)$ under the next term in the dividend $(- 4)$ .

$8$	$1$	$- 4$	$0$	$16$	$- 16$
		$8$
	$1$

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.

$8$	$1$	$- 4$	$0$	$16$	$- 16$
		$8$
	$1$	$4$

Step 8.5

Multiply the newest entry in the result $(4)$ by the divisor $(8)$ and place the result of $(32)$ under the next term in the dividend $(0)$ .

$8$	$1$	$- 4$	$0$	$16$	$- 16$
		$8$	$32$
	$1$	$4$

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.

$8$	$1$	$- 4$	$0$	$16$	$- 16$
		$8$	$32$
	$1$	$4$	$32$

Step 8.7

Multiply the newest entry in the result $(32)$ by the divisor $(8)$ and place the result of $(256)$ under the next term in the dividend $(16)$ .

$8$	$1$	$- 4$	$0$	$16$	$- 16$
		$8$	$32$	$256$
	$1$	$4$	$32$

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.

$8$	$1$	$- 4$	$0$	$16$	$- 16$
		$8$	$32$	$256$
	$1$	$4$	$32$	$272$

Step 8.9

Multiply the newest entry in the result $(272)$ by the divisor $(8)$ and place the result of $(2176)$ under the next term in the dividend $(- 16)$ .

$8$	$1$	$- 4$	$0$	$16$	$- 16$
		$8$	$32$	$256$	$2176$
	$1$	$4$	$32$	$272$

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.

$8$	$1$	$- 4$	$0$	$16$	$- 16$
		$8$	$32$	$256$	$2176$
	$1$	$4$	$32$	$272$	$2160$

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.

$1 x^{3} + 4 x^{2} + (32) x + 272 + \frac{2160}{x - 8}$

Step 8.12

Simplify the quotient polynomial.

$x^{3} + 4 x^{2} + 32 x + 272 + \frac{2160}{x - 8}$

Step 9

Since $8 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $8$ is an upper bound for the real roots of the function.

Upper Bound: $8$

Step 10

Apply synthetic division on $\frac{x^{4} - 4 x^{3} + 16 x - 16}{x + 8}$ when $x = - 8$ .

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Step 10.1

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

$- 8$	$1$	$- 4$	$0$	$16$	$- 16$

Step 10.2

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$- 8$	$1$	$- 4$	$0$	$16$	$- 16$

	$1$

Step 10.3

Multiply the newest entry in the result $(1)$ by the divisor $(- 8)$ and place the result of $(- 8)$ under the next term in the dividend $(- 4)$ .

$- 8$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 8$
	$1$

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.

$- 8$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 8$
	$1$	$- 12$

Step 10.5

Multiply the newest entry in the result $(- 12)$ by the divisor $(- 8)$ and place the result of $(96)$ under the next term in the dividend $(0)$ .

$- 8$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 8$	$96$
	$1$	$- 12$

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.

$- 8$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 8$	$96$
	$1$	$- 12$	$96$

Step 10.7

Multiply the newest entry in the result $(96)$ by the divisor $(- 8)$ and place the result of $(- 768)$ under the next term in the dividend $(16)$ .

$- 8$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 8$	$96$	$- 768$
	$1$	$- 12$	$96$

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.

$- 8$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 8$	$96$	$- 768$
	$1$	$- 12$	$96$	$- 752$

Step 10.9

Multiply the newest entry in the result $(- 752)$ by the divisor $(- 8)$ and place the result of $(6016)$ under the next term in the dividend $(- 16)$ .

$- 8$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 8$	$96$	$- 768$	$6016$
	$1$	$- 12$	$96$	$- 752$

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.

$- 8$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 8$	$96$	$- 768$	$6016$
	$1$	$- 12$	$96$	$- 752$	$6000$

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.

$1 x^{3} + - 12 x^{2} + (96) x - 752 + \frac{6000}{x + 8}$

Step 10.12

Simplify the quotient polynomial.

$x^{3} - 12 x^{2} + 96 x - 752 + \frac{6000}{x + 8}$

Step 11

Since $- 8 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 8$ is a lower bound for the real roots of the function.

Lower Bound: $- 8$

Step 12

Apply synthetic division on $\frac{x^{4} - 4 x^{3} + 16 x - 16}{x - 16}$ when $x = 16$ .

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Step 12.1

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

$16$	$1$	$- 4$	$0$	$16$	$- 16$

Step 12.2

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$16$	$1$	$- 4$	$0$	$16$	$- 16$

	$1$

Step 12.3

Multiply the newest entry in the result $(1)$ by the divisor $(16)$ and place the result of $(16)$ under the next term in the dividend $(- 4)$ .

$16$	$1$	$- 4$	$0$	$16$	$- 16$
		$16$
	$1$

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.

$16$	$1$	$- 4$	$0$	$16$	$- 16$
		$16$
	$1$	$12$

Step 12.5

Multiply the newest entry in the result $(12)$ by the divisor $(16)$ and place the result of $(192)$ under the next term in the dividend $(0)$ .

$16$	$1$	$- 4$	$0$	$16$	$- 16$
		$16$	$192$
	$1$	$12$

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.

$16$	$1$	$- 4$	$0$	$16$	$- 16$
		$16$	$192$
	$1$	$12$	$192$

Step 12.7

Multiply the newest entry in the result $(192)$ by the divisor $(16)$ and place the result of $(3072)$ under the next term in the dividend $(16)$ .

$16$	$1$	$- 4$	$0$	$16$	$- 16$
		$16$	$192$	$3072$
	$1$	$12$	$192$

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.

$16$	$1$	$- 4$	$0$	$16$	$- 16$
		$16$	$192$	$3072$
	$1$	$12$	$192$	$3088$

Step 12.9

Multiply the newest entry in the result $(3088)$ by the divisor $(16)$ and place the result of $(49408)$ under the next term in the dividend $(- 16)$ .

$16$	$1$	$- 4$	$0$	$16$	$- 16$
		$16$	$192$	$3072$	$49408$
	$1$	$12$	$192$	$3088$

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.

$16$	$1$	$- 4$	$0$	$16$	$- 16$
		$16$	$192$	$3072$	$49408$
	$1$	$12$	$192$	$3088$	$49392$

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.

$1 x^{3} + 12 x^{2} + (192) x + 3088 + \frac{49392}{x - 16}$

Step 12.12

Simplify the quotient polynomial.

$x^{3} + 12 x^{2} + 192 x + 3088 + \frac{49392}{x - 16}$

Step 13

Since $16 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $16$ is an upper bound for the real roots of the function.

Upper Bound: $16$

Step 14

Apply synthetic division on $\frac{x^{4} - 4 x^{3} + 16 x - 16}{x + 16}$ when $x = - 16$ .

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Step 14.1

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

$- 16$	$1$	$- 4$	$0$	$16$	$- 16$

Step 14.2

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$- 16$	$1$	$- 4$	$0$	$16$	$- 16$

	$1$

Step 14.3

Multiply the newest entry in the result $(1)$ by the divisor $(- 16)$ and place the result of $(- 16)$ under the next term in the dividend $(- 4)$ .

$- 16$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 16$
	$1$

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.

$- 16$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 16$
	$1$	$- 20$

Step 14.5

Multiply the newest entry in the result $(- 20)$ by the divisor $(- 16)$ and place the result of $(320)$ under the next term in the dividend $(0)$ .

$- 16$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 16$	$320$
	$1$	$- 20$

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.

$- 16$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 16$	$320$
	$1$	$- 20$	$320$

Step 14.7

Multiply the newest entry in the result $(320)$ by the divisor $(- 16)$ and place the result of $(- 5120)$ under the next term in the dividend $(16)$ .

$- 16$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 16$	$320$	$- 5120$
	$1$	$- 20$	$320$

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.

$- 16$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 16$	$320$	$- 5120$
	$1$	$- 20$	$320$	$- 5104$

Step 14.9

Multiply the newest entry in the result $(- 5104)$ by the divisor $(- 16)$ and place the result of $(81664)$ under the next term in the dividend $(- 16)$ .

$- 16$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 16$	$320$	$- 5120$	$81664$
	$1$	$- 20$	$320$	$- 5104$

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.

$- 16$	$1$	$- 4$	$0$	$16$	$- 16$
		$- 16$	$320$	$- 5120$	$81664$
	$1$	$- 20$	$320$	$- 5104$	$81648$

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.

$1 x^{3} + - 20 x^{2} + (320) x - 5104 + \frac{81648}{x + 16}$

Step 14.12

Simplify the quotient polynomial.

$x^{3} - 20 x^{2} + 320 x - 5104 + \frac{81648}{x + 16}$

Step 15

Since $- 16 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 16$ is a lower bound for the real roots of the function.

Lower Bound: $- 16$

Step 16

Determine the upper and lower bounds.

Upper Bounds: $4, 8, 16$

Lower Bounds: $- 2, - 4, - 8, - 16$

Step 17