Finite Math Examples

$5 x^{2} - 3 x - 2$

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.

$p = \pm 1, \pm 2$

$q = \pm 1, \pm 5$

Step 2

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

$\pm 1, \pm \frac{1}{5}, \pm 2, \pm \frac{2}{5}$

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.

$5 {(1)}^{2} - 3 \cdot 1 - 2$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = 1$ is a root of the polynomial.

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Step 4.1

Simplify each term.

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Step 4.1.1

One to any power is one.

$5 \cdot 1 - 3 \cdot 1 - 2$

Step 4.1.2

Multiply $5$ by $1$ .

$5 - 3 \cdot 1 - 2$

Step 4.1.3

Multiply $- 3$ by $1$ .

$5 - 3 - 2$

Step 4.2

Simplify by subtracting numbers.

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Step 4.2.1

Subtract $3$ from $5$ .

$2 - 2$

Step 4.2.2

Subtract $2$ from $2$ .

$0$

Step 5

Since $1$ is a known root, divide the polynomial by $x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{5 x^{2} - 3 x - 2}{x - 1}$

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.

$1$	$5$	$- 3$	$- 2$

Step 6.2

The first number in the dividend $(5)$ is put into the first position of the result area (below the horizontal line).

$1$	$5$	$- 3$	$- 2$

	$5$

Step 6.3

Multiply the newest entry in the result $(5)$ by the divisor $(1)$ and place the result of $(5)$ under the next term in the dividend $(- 3)$ .

$1$	$5$	$- 3$	$- 2$
		$5$
	$5$

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.

$1$	$5$	$- 3$	$- 2$
		$5$
	$5$	$2$

Step 6.5

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

$1$	$5$	$- 3$	$- 2$
		$5$	$2$
	$5$	$2$

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.

$1$	$5$	$- 3$	$- 2$
		$5$	$2$
	$5$	$2$	$0$

Step 6.7

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

$(5) x + 2$

Step 6.8

Simplify the quotient polynomial.

$5 x + 2$

Step 7

Solve the equation to find any remaining roots.

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Step 7.1

Subtract $2$ from both sides of the equation.

$5 x = - 2$

Step 7.2

Divide each term in $5 x = - 2$ by $5$ and simplify.

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Step 7.2.1

Divide each term in $5 x = - 2$ by $5$ .

$\frac{5 x}{5} = \frac{- 2}{5}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $5$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 2}{5}$

Step 7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2}{5}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Move the negative in front of the fraction.

$x = - \frac{2}{5}$

Step 8

The polynomial can be written as a set of linear factors.

$(x - 1) (x + \frac{2}{5})$

Step 9

These are the roots (zeros) of the polynomial $5 x^{2} - 3 x - 2$ .

$x = 1, - \frac{2}{5}$

Step 10