Finite Math Examples

$65 x^{6} + 60 x^{3} - 15 x$

Step 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.

$p = \pm 0$

$q = \pm 1, \pm 5, \pm 13, \pm 65$

Step 2

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

$0$

Step 3

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.

$65 {(0)}^{6} + 60 {(0)}^{3} - 15 \cdot 0$

Step 4

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 4.1

Simplify each term.

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Step 4.1.1

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

$65 \cdot 0 + 60 {(0)}^{3} - 15 \cdot 0$

Step 4.1.2

Multiply $65$ by $0$ .

$0 + 60 {(0)}^{3} - 15 \cdot 0$

Step 4.1.3

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

$0 + 60 \cdot 0 - 15 \cdot 0$

Step 4.1.4

Multiply $60$ by $0$ .

$0 + 0 - 15 \cdot 0$

Step 4.1.5

Multiply $- 15$ by $0$ .

$0 + 0 + 0$

$0 + 0 + 0$

Step 4.2

Simplify by adding numbers.

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Step 4.2.1

Add $0$ and $0$ .

$0 + 0$

Step 4.2.2

Add $0$ and $0$ .

$0$

$0$

$0$

Step 5

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.

$\frac{65 x^{6} + 60 x^{3} - 15 x}{x + 0}$

Step 6

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

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Step 6.1

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

$0$	$65$	$0$	$0$	$60$	$0$	$- 15$	$0$

Step 6.2

The first number in the dividend $(65)$ is put into the first position of the result area (below the horizontal line).

$0$	$65$	$0$	$0$	$60$	$0$	$- 15$	$0$

	$65$

Step 6.3

Multiply the newest entry in the result $(65)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(0)$ .

$0$	$65$	$0$	$0$	$60$	$0$	$- 15$	$0$
		$0$
	$65$

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.

$0$	$65$	$0$	$0$	$60$	$0$	$- 15$	$0$
		$0$
	$65$	$0$

Step 6.5

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 $(0)$ .

$0$	$65$	$0$	$0$	$60$	$0$	$- 15$	$0$
		$0$	$0$
	$65$	$0$

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.

$0$	$65$	$0$	$0$	$60$	$0$	$- 15$	$0$
		$0$	$0$
	$65$	$0$	$0$

Step 6.7

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 $(60)$ .

$0$	$65$	$0$	$0$	$60$	$0$	$- 15$	$0$
		$0$	$0$	$0$
	$65$	$0$	$0$

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.

$0$	$65$	$0$	$0$	$60$	$0$	$- 15$	$0$
		$0$	$0$	$0$
	$65$	$0$	$0$	$60$

Step 6.9

Multiply the newest entry in the result $(60)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(0)$ .

$0$	$65$	$0$	$0$	$60$	$0$	$- 15$	$0$
		$0$	$0$	$0$	$0$
	$65$	$0$	$0$	$60$

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.

$0$	$65$	$0$	$0$	$60$	$0$	$- 15$	$0$
		$0$	$0$	$0$	$0$
	$65$	$0$	$0$	$60$	$0$

Step 6.11

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 $(- 15)$ .

$0$	$65$	$0$	$0$	$60$	$0$	$- 15$	$0$
		$0$	$0$	$0$	$0$	$0$
	$65$	$0$	$0$	$60$	$0$

Step 6.12

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

$0$	$65$	$0$	$0$	$60$	$0$	$- 15$	$0$
		$0$	$0$	$0$	$0$	$0$
	$65$	$0$	$0$	$60$	$0$	$- 15$

Step 6.13

Multiply the newest entry in the result $(- 15)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(0)$ .

$0$	$65$	$0$	$0$	$60$	$0$	$- 15$	$0$
		$0$	$0$	$0$	$0$	$0$	$0$
	$65$	$0$	$0$	$60$	$0$	$- 15$

Step 6.14

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

$0$	$65$	$0$	$0$	$60$	$0$	$- 15$	$0$
		$0$	$0$	$0$	$0$	$0$	$0$
	$65$	$0$	$0$	$60$	$0$	$- 15$	$0$

Step 6.15

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

$65 x^{5} + 0 x^{4} + 0 x^{3} + 60 x^{2} + (0) x - 15$

Step 6.16

Simplify the quotient polynomial.

$65 x^{5} + 60 x^{2} - 15$

$65 x^{5} + 60 x^{2} - 15$

Step 7

Factor $5$ out of $65 x^{5} + 60 x^{2} - 15$ .

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Step 7.1

Factor $5$ out of $65 x^{5}$ .

$(x + 0) (5 (13 x^{5}) + 60 x^{2} - 15)$

Step 7.2

Factor $5$ out of $60 x^{2}$ .

$(x + 0) (5 (13 x^{5}) + 5 (12 x^{2}) - 15)$

Step 7.3

Factor $5$ out of $- 15$ .

$(x + 0) (5 (13 x^{5}) + 5 (12 x^{2}) + 5 (- 3))$

Step 7.4

Factor $5$ out of $5 (13 x^{5}) + 5 (12 x^{2})$ .

$(x + 0) (5 (13 x^{5} + 12 x^{2}) + 5 (- 3))$

Step 7.5

Factor $5$ out of $5 (13 x^{5} + 12 x^{2}) + 5 (- 3)$ .

$(x + 0) (5 (13 x^{5} + 12 x^{2} - 3))$

$(x + 0) (5 (13 x^{5} + 12 x^{2} - 3))$

Step 8

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 0.84246268, - 0.55336178, 0, 0.4735107$

Step 9

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