Finite Math Examples

$2 x^{10} - 6 x^{9} + 10 x^{9} - 126 x^{8}$

Step 1

Add $- 6 x^{9}$ and $10 x^{9}$ .

$y = 2 x^{10} + 4 x^{9} - 126 x^{8}$

Step 2

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.

$p = \pm 0$

$q = \pm 1, \pm 2$

Step 3

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

$0$

Step 4

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is $0$ , which means it is a root.

$2 {(0)}^{10} + 4 {(0)}^{9} - 126 {(0)}^{8}$

Step 5

Simplify the expression. In this case, the expression is equal to $0$ so $x = 0$ is a root of the polynomial.

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Step 5.1

Simplify each term.

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Step 5.1.1

Raising $0$ to any positive power yields $0$ .

$2 \cdot 0 + 4 {(0)}^{9} - 126 {(0)}^{8}$

Step 5.1.2

Multiply $2$ by $0$ .

$0 + 4 {(0)}^{9} - 126 {(0)}^{8}$

Step 5.1.3

Raising $0$ to any positive power yields $0$ .

$0 + 4 \cdot 0 - 126 {(0)}^{8}$

Step 5.1.4

Multiply $4$ by $0$ .

$0 + 0 - 126 {(0)}^{8}$

Step 5.1.5

Raising $0$ to any positive power yields $0$ .

$0 + 0 - 126 \cdot 0$

Step 5.1.6

Multiply $- 126$ by $0$ .

$0 + 0 + 0$

$0 + 0 + 0$

Step 5.2

Simplify by adding numbers.

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Step 5.2.1

Add $0$ and $0$ .

$0 + 0$

Step 5.2.2

Add $0$ and $0$ .

$0$

$0$

$0$

Step 6

Since $0$ is a known root, divide the polynomial by $x + 0$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 x^{10} + 4 x^{9} - 126 x^{8}}{x + 0}$

Step 7

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

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Step 7.1

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

$0$	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

Step 7.2

The first number in the dividend $(2)$ is put into the first position of the result area (below the horizontal line).

$0$	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

	$2$

Step 7.3

Multiply the newest entry in the result $(2)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(4)$ .

$0$	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
		$0$
	$2$

Step 7.4

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

$0$	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
		$0$
	$2$	$4$

Step 7.5

Multiply the newest entry in the result $(4)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(- 126)$ .

$0$	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
		$0$	$0$
	$2$	$4$

Step 7.6

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

$0$	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
		$0$	$0$
	$2$	$4$	$- 126$

Step 7.7

Multiply the newest entry in the result $(- 126)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(0)$ .

$0$	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
		$0$	$0$	$0$
	$2$	$4$	$- 126$

Step 7.8

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

$0$	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
		$0$	$0$	$0$
	$2$	$4$	$- 126$	$0$

Step 7.9

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

$0$	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
		$0$	$0$	$0$	$0$
	$2$	$4$	$- 126$	$0$

Step 7.10

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

$0$	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
		$0$	$0$	$0$	$0$
	$2$	$4$	$- 126$	$0$	$0$

Step 7.11

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

$0$	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
		$0$	$0$	$0$	$0$	$0$
	$2$	$4$	$- 126$	$0$	$0$

Step 7.12

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

$0$	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
		$0$	$0$	$0$	$0$	$0$
	$2$	$4$	$- 126$	$0$	$0$	$0$

Step 7.13

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

$0$	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
		$0$	$0$	$0$	$0$	$0$	$0$
	$2$	$4$	$- 126$	$0$	$0$	$0$

Step 7.14

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

$0$	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
		$0$	$0$	$0$	$0$	$0$	$0$
	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$

Step 7.15

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

$0$	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
		$0$	$0$	$0$	$0$	$0$	$0$	$0$
	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$

Step 7.16

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

$0$	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
		$0$	$0$	$0$	$0$	$0$	$0$	$0$
	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$

Step 7.17

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

$0$	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
		$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$

Step 7.18

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

$0$	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
		$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$	$0$

Step 7.19

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

$0$	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
		$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$	$0$

Step 7.20

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

$0$	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
		$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

Step 7.21

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

$0$	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
		$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

Step 7.22

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

$0$	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
		$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
	$2$	$4$	$- 126$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

Step 7.23

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

$2 x^{9} + 4 x^{8} - 126 x^{7} + 0 x^{6} + 0 x^{5} + 0 x^{4} + 0 x^{3} + 0 x^{2} + (0) x + 0$

Step 7.24

Simplify the quotient polynomial.

$2 x^{9} + 4 x^{8} - 126 x^{7}$

$2 x^{9} + 4 x^{8} - 126 x^{7}$

Step 8

Factor $2 x^{7}$ out of $2 x^{9} + 4 x^{8} - 126 x^{7}$ .

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Step 8.1

Factor $2 x^{7}$ out of $2 x^{9}$ .

$(x + 0) (2 x^{7} (x^{2}) + 4 x^{8} - 126 x^{7})$

Step 8.2

Factor $2 x^{7}$ out of $4 x^{8}$ .

$(x + 0) (2 x^{7} (x^{2}) + 2 x^{7} (2 x) - 126 x^{7})$

Step 8.3

Factor $2 x^{7}$ out of $- 126 x^{7}$ .

$(x + 0) (2 x^{7} (x^{2}) + 2 x^{7} (2 x) + 2 x^{7} (- 63))$

Step 8.4

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

$(x + 0) (2 x^{7} (x^{2} + 2 x) + 2 x^{7} (- 63))$

Step 8.5

Factor $2 x^{7}$ out of $2 x^{7} (x^{2} + 2 x) + 2 x^{7} (- 63)$ .

$(x + 0) (2 x^{7} (x^{2} + 2 x - 63))$

$(x + 0) (2 x^{7} (x^{2} + 2 x - 63))$

Step 9

Factor.

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Step 9.1

Factor $x^{2} + 2 x - 63$ using the AC method.

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Step 9.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 63$ and whose sum is $2$ .

$- 7, 9$

Step 9.1.2

Write the factored form using these integers.

$(x + 0) (2 x^{7} ((x - 7) (x + 9)))$

$(x + 0) (2 x^{7} ((x - 7) (x + 9)))$

Step 9.2

Remove unnecessary parentheses.

$(x + 0) (2 x^{7} (x - 7) (x + 9))$

$(x + 0) (2 x^{7} (x - 7) (x + 9))$

Step 10

Add $- 6 x^{9}$ and $10 x^{9}$ .

$2 x^{10} + 4 x^{9} - 126 x^{8} = 0$

Step 11

Factor the left side of the equation.

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Step 11.1

Factor $2 x^{8}$ out of $2 x^{10} + 4 x^{9} - 126 x^{8}$ .

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Step 11.1.1

Factor $2 x^{8}$ out of $2 x^{10}$ .

$2 x^{8} (x^{2}) + 4 x^{9} - 126 x^{8} = 0$

Step 11.1.2

Factor $2 x^{8}$ out of $4 x^{9}$ .

$2 x^{8} (x^{2}) + 2 x^{8} (2 x) - 126 x^{8} = 0$

Step 11.1.3

Factor $2 x^{8}$ out of $- 126 x^{8}$ .

$2 x^{8} (x^{2}) + 2 x^{8} (2 x) + 2 x^{8} (- 63) = 0$

Step 11.1.4

Factor $2 x^{8}$ out of $2 x^{8} (x^{2}) + 2 x^{8} (2 x)$ .

$2 x^{8} (x^{2} + 2 x) + 2 x^{8} (- 63) = 0$

Step 11.1.5

Factor $2 x^{8}$ out of $2 x^{8} (x^{2} + 2 x) + 2 x^{8} (- 63)$ .

$2 x^{8} (x^{2} + 2 x - 63) = 0$

$2 x^{8} (x^{2} + 2 x - 63) = 0$

Step 11.2

Factor.

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Step 11.2.1

Factor $x^{2} + 2 x - 63$ using the AC method.

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Step 11.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 63$ and whose sum is $2$ .

$- 7, 9$

Step 11.2.1.2

Write the factored form using these integers.

$2 x^{8} ((x - 7) (x + 9)) = 0$

$2 x^{8} ((x - 7) (x + 9)) = 0$

Step 11.2.2

Remove unnecessary parentheses.

$2 x^{8} (x - 7) (x + 9) = 0$

$2 x^{8} (x - 7) (x + 9) = 0$

$2 x^{8} (x - 7) (x + 9) = 0$

Step 12

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{8} = 0$

$x - 7 = 0$

$x + 9 = 0$

Step 13

Set $x^{8}$ equal to $0$ and solve for $x$ .

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Step 13.1

Set $x^{8}$ equal to $0$ .

$x^{8} = 0$

Step 13.2

Solve $x^{8} = 0$ for $x$ .

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Step 13.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[8]{0}$

Step 13.2.2

Simplify $\pm \sqrt[8]{0}$ .

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Step 13.2.2.1

Rewrite $0$ as $0^{8}$ .

$x = \pm \sqrt[8]{0^{8}}$

Step 13.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 13.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

$x = 0$

$x = 0$

$x = 0$

Step 14

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 14.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 14.2

Add $7$ to both sides of the equation.

$x = 7$

$x = 7$

Step 15

Set $x + 9$ equal to $0$ and solve for $x$ .

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Step 15.1

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

Step 15.2

Subtract $9$ from both sides of the equation.

$x = - 9$

$x = - 9$

Step 16

The final solution is all the values that make $2 x^{8} (x - 7) (x + 9) = 0$ true.

$x = 0, 7, - 9$

Step 17

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$