Finite Math Examples

$3 x^{2} - 6 x$

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.

$p = \pm 0$

$q = \pm 1, \pm 3$

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.

$3 {(0)}^{2} - 6 \cdot 0$

Step 4

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

Tap for more steps...

Step 4.1

Simplify each term.

Tap for more steps...

Step 4.1.1

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

$3 \cdot 0 - 6 \cdot 0$

Step 4.1.2

Multiply $3$ by $0$ .

$0 - 6 \cdot 0$

Step 4.1.3

Multiply $- 6$ by $0$ .

$0 + 0$

Step 4.2

Add $0$ and $0$ .

$0$

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.

$\frac{3 x^{2} - 6 x}{x + 0}$

Step 6

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

Tap for more steps...

Step 6.1

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

$0$	$3$	$- 6$	$0$

Step 6.2

The first number in the dividend $(3)$ is put into the first position of the result area (below the horizontal line).

$0$	$3$	$- 6$	$0$

	$3$

Step 6.3

Multiply the newest entry in the result $(3)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(- 6)$ .

$0$	$3$	$- 6$	$0$
		$0$
	$3$

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.

$0$	$3$	$- 6$	$0$
		$0$
	$3$	$- 6$

Step 6.5

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

$0$	$3$	$- 6$	$0$
		$0$	$0$
	$3$	$- 6$

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.

$0$	$3$	$- 6$	$0$
		$0$	$0$
	$3$	$- 6$	$0$

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.

$(3) x - 6$

Step 6.8

Simplify the quotient polynomial.

$3 x - 6$

Step 7

Factor $3$ out of $3 x - 6$ .

Tap for more steps...

Step 7.1

Factor $3$ out of $3 x$ .

$(x + 0) (3 (x) - 6)$

Step 7.2

Factor $3$ out of $- 6$ .

$(x + 0) (3 x + 3 \cdot - 2)$

Step 7.3

Factor $3$ out of $3 x + 3 \cdot - 2$ .

$(x + 0) (3 (x - 2))$

Step 8

Factor $3 x$ out of $3 x^{2} - 6 x$ .

Tap for more steps...

Step 8.1

Factor $3 x$ out of $3 x^{2}$ .

$3 x (x) - 6 x = 0$

Step 8.2

Factor $3 x$ out of $- 6 x$ .

$3 x (x) + 3 x (- 2) = 0$

Step 8.3

Factor $3 x$ out of $3 x (x) + 3 x (- 2)$ .

$3 x (x - 2) = 0$

Step 9

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

$x = 0$

$x - 2 = 0$

Step 10

Set $x$ equal to $0$ .

$x = 0$

Step 11

Set $x - 2$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 11.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 11.2

Add $2$ to both sides of the equation.

$x = 2$

Step 12

The final solution is all the values that make $3 x (x - 2) = 0$ true.

$x = 0, 2$

Step 13