Algebra Examples

$x^{4} - 48 x^{2} - 49$

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.

$p = \pm 1, \pm 7, \pm 49$

$q = \pm 1$

Step 1.2

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

$\pm 1, \pm 7, \pm 49$

Step 2

Apply synthetic division on $\frac{x^{4} - 48 x^{2} - 49}{x - 7}$ when $x = 7$ .

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Step 2.1

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

$7$	$1$	$0$	$- 48$	$0$	$- 49$

Step 2.2

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

$7$	$1$	$0$	$- 48$	$0$	$- 49$

	$1$

Step 2.3

Multiply the newest entry in the result $(1)$ by the divisor $(7)$ and place the result of $(7)$ under the next term in the dividend $(0)$ .

$7$	$1$	$0$	$- 48$	$0$	$- 49$
		$7$
	$1$

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.

$7$	$1$	$0$	$- 48$	$0$	$- 49$
		$7$
	$1$	$7$

Step 2.5

Multiply the newest entry in the result $(7)$ by the divisor $(7)$ and place the result of $(49)$ under the next term in the dividend $(- 48)$ .

$7$	$1$	$0$	$- 48$	$0$	$- 49$
		$7$	$49$
	$1$	$7$

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.

$7$	$1$	$0$	$- 48$	$0$	$- 49$
		$7$	$49$
	$1$	$7$	$1$

Step 2.7

Multiply the newest entry in the result $(1)$ by the divisor $(7)$ and place the result of $(7)$ under the next term in the dividend $(0)$ .

$7$	$1$	$0$	$- 48$	$0$	$- 49$
		$7$	$49$	$7$
	$1$	$7$	$1$

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.

$7$	$1$	$0$	$- 48$	$0$	$- 49$
		$7$	$49$	$7$
	$1$	$7$	$1$	$7$

Step 2.9

Multiply the newest entry in the result $(7)$ by the divisor $(7)$ and place the result of $(49)$ under the next term in the dividend $(- 49)$ .

$7$	$1$	$0$	$- 48$	$0$	$- 49$
		$7$	$49$	$7$	$49$
	$1$	$7$	$1$	$7$

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.

$7$	$1$	$0$	$- 48$	$0$	$- 49$
		$7$	$49$	$7$	$49$
	$1$	$7$	$1$	$7$	$0$

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.

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

Step 2.12

Simplify the quotient polynomial.

$x^{3} + 7 x^{2} + x + 7$

Step 3

Since $7 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $7$ is an upper bound for the real roots of the function.

Upper Bound: $7$

Step 4

Apply synthetic division on $\frac{x^{4} - 48 x^{2} - 49}{x + 7}$ when $x = - 7$ .

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Step 4.1

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

$- 7$	$1$	$0$	$- 48$	$0$	$- 49$

Step 4.2

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

$- 7$	$1$	$0$	$- 48$	$0$	$- 49$

	$1$

Step 4.3

Multiply the newest entry in the result $(1)$ by the divisor $(- 7)$ and place the result of $(- 7)$ under the next term in the dividend $(0)$ .

$- 7$	$1$	$0$	$- 48$	$0$	$- 49$
		$- 7$
	$1$

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.

$- 7$	$1$	$0$	$- 48$	$0$	$- 49$
		$- 7$
	$1$	$- 7$

Step 4.5

Multiply the newest entry in the result $(- 7)$ by the divisor $(- 7)$ and place the result of $(49)$ under the next term in the dividend $(- 48)$ .

$- 7$	$1$	$0$	$- 48$	$0$	$- 49$
		$- 7$	$49$
	$1$	$- 7$

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.

$- 7$	$1$	$0$	$- 48$	$0$	$- 49$
		$- 7$	$49$
	$1$	$- 7$	$1$

Step 4.7

Multiply the newest entry in the result $(1)$ by the divisor $(- 7)$ and place the result of $(- 7)$ under the next term in the dividend $(0)$ .

$- 7$	$1$	$0$	$- 48$	$0$	$- 49$
		$- 7$	$49$	$- 7$
	$1$	$- 7$	$1$

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.

$- 7$	$1$	$0$	$- 48$	$0$	$- 49$
		$- 7$	$49$	$- 7$
	$1$	$- 7$	$1$	$- 7$

Step 4.9

Multiply the newest entry in the result $(- 7)$ by the divisor $(- 7)$ and place the result of $(49)$ under the next term in the dividend $(- 49)$ .

$- 7$	$1$	$0$	$- 48$	$0$	$- 49$
		$- 7$	$49$	$- 7$	$49$
	$1$	$- 7$	$1$	$- 7$

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.

$- 7$	$1$	$0$	$- 48$	$0$	$- 49$
		$- 7$	$49$	$- 7$	$49$
	$1$	$- 7$	$1$	$- 7$	$0$

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.

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

Step 4.12

Simplify the quotient polynomial.

$x^{3} - 7 x^{2} + x - 7$

Step 5

Since $- 7 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 7$ is a lower bound for the real roots of the function.

Lower Bound: $- 7$

Step 6

Apply synthetic division on $\frac{x^{4} - 48 x^{2} - 49}{x - 49}$ when $x = 49$ .

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Step 6.1

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

$49$	$1$	$0$	$- 48$	$0$	$- 49$

Step 6.2

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

$49$	$1$	$0$	$- 48$	$0$	$- 49$

	$1$

Step 6.3

Multiply the newest entry in the result $(1)$ by the divisor $(49)$ and place the result of $(49)$ under the next term in the dividend $(0)$ .

$49$	$1$	$0$	$- 48$	$0$	$- 49$
		$49$
	$1$

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.

$49$	$1$	$0$	$- 48$	$0$	$- 49$
		$49$
	$1$	$49$

Step 6.5

Multiply the newest entry in the result $(49)$ by the divisor $(49)$ and place the result of $(2401)$ under the next term in the dividend $(- 48)$ .

$49$	$1$	$0$	$- 48$	$0$	$- 49$
		$49$	$2401$
	$1$	$49$

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.

$49$	$1$	$0$	$- 48$	$0$	$- 49$
		$49$	$2401$
	$1$	$49$	$2353$

Step 6.7

Multiply the newest entry in the result $(2353)$ by the divisor $(49)$ and place the result of $(115297)$ under the next term in the dividend $(0)$ .

$49$	$1$	$0$	$- 48$	$0$	$- 49$
		$49$	$2401$	$115297$
	$1$	$49$	$2353$

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.

$49$	$1$	$0$	$- 48$	$0$	$- 49$
		$49$	$2401$	$115297$
	$1$	$49$	$2353$	$115297$

Step 6.9

Multiply the newest entry in the result $(115297)$ by the divisor $(49)$ and place the result of $(5649553)$ under the next term in the dividend $(- 49)$ .

$49$	$1$	$0$	$- 48$	$0$	$- 49$
		$49$	$2401$	$115297$	$5649553$
	$1$	$49$	$2353$	$115297$

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.

$49$	$1$	$0$	$- 48$	$0$	$- 49$
		$49$	$2401$	$115297$	$5649553$
	$1$	$49$	$2353$	$115297$	$5649504$

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.

$1 x^{3} + 49 x^{2} + (2353) x + 115297 + \frac{5649504}{x - 49}$

Step 6.12

Simplify the quotient polynomial.

$x^{3} + 49 x^{2} + 2353 x + 115297 + \frac{5649504}{x - 49}$

Step 7

Since $49 > 0$ and all of the signs in the bottom row of the synthetic division are positive, $49$ is an upper bound for the real roots of the function.

Upper Bound: $49$

Step 8

Apply synthetic division on $\frac{x^{4} - 48 x^{2} - 49}{x + 49}$ when $x = - 49$ .

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Step 8.1

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

$- 49$	$1$	$0$	$- 48$	$0$	$- 49$

Step 8.2

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

$- 49$	$1$	$0$	$- 48$	$0$	$- 49$

	$1$

Step 8.3

Multiply the newest entry in the result $(1)$ by the divisor $(- 49)$ and place the result of $(- 49)$ under the next term in the dividend $(0)$ .

$- 49$	$1$	$0$	$- 48$	$0$	$- 49$
		$- 49$
	$1$

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.

$- 49$	$1$	$0$	$- 48$	$0$	$- 49$
		$- 49$
	$1$	$- 49$

Step 8.5

Multiply the newest entry in the result $(- 49)$ by the divisor $(- 49)$ and place the result of $(2401)$ under the next term in the dividend $(- 48)$ .

$- 49$	$1$	$0$	$- 48$	$0$	$- 49$
		$- 49$	$2401$
	$1$	$- 49$

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.

$- 49$	$1$	$0$	$- 48$	$0$	$- 49$
		$- 49$	$2401$
	$1$	$- 49$	$2353$

Step 8.7

Multiply the newest entry in the result $(2353)$ by the divisor $(- 49)$ and place the result of $(- 115297)$ under the next term in the dividend $(0)$ .

$- 49$	$1$	$0$	$- 48$	$0$	$- 49$
		$- 49$	$2401$	$- 115297$
	$1$	$- 49$	$2353$

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.

$- 49$	$1$	$0$	$- 48$	$0$	$- 49$
		$- 49$	$2401$	$- 115297$
	$1$	$- 49$	$2353$	$- 115297$

Step 8.9

Multiply the newest entry in the result $(- 115297)$ by the divisor $(- 49)$ and place the result of $(5649553)$ under the next term in the dividend $(- 49)$ .

$- 49$	$1$	$0$	$- 48$	$0$	$- 49$
		$- 49$	$2401$	$- 115297$	$5649553$
	$1$	$- 49$	$2353$	$- 115297$

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.

$- 49$	$1$	$0$	$- 48$	$0$	$- 49$
		$- 49$	$2401$	$- 115297$	$5649553$
	$1$	$- 49$	$2353$	$- 115297$	$5649504$

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.

$1 x^{3} + - 49 x^{2} + (2353) x - 115297 + \frac{5649504}{x + 49}$

Step 8.12

Simplify the quotient polynomial.

$x^{3} - 49 x^{2} + 2353 x - 115297 + \frac{5649504}{x + 49}$

Step 9

Since $- 49 < 0$ and the signs in the bottom row of the synthetic division alternate sign, $- 49$ is a lower bound for the real roots of the function.

Lower Bound: $- 49$

Step 10

Determine the upper and lower bounds.

Upper Bounds: $7, 49$

Lower Bounds: $- 7, - 49$

Step 11