Finite Math Examples

$\frac{1}{3} \cdot (6 m + 15) + 3$

Step 1

Simplify $\frac{1}{3} \cdot (6 m + 15) + 3$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$y = \frac{1}{3} (6 m) + \frac{1}{3} \cdot 15 + 3$

Step 1.1.2

Cancel the common factor of $3$ .

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Step 1.1.2.1

Factor $3$ out of $6 m$ .

$y = \frac{1}{3} (3 (2 m)) + \frac{1}{3} \cdot 15 + 3$

Step 1.1.2.2

Cancel the common factor.

$y = \frac{1}{3} (3 (2 m)) + \frac{1}{3} \cdot 15 + 3$

Step 1.1.2.3

Rewrite the expression.

$y = 2 m + \frac{1}{3} \cdot 15 + 3$

Step 1.1.3

Cancel the common factor of $3$ .

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Step 1.1.3.1

Factor $3$ out of $15$ .

$y = 2 m + \frac{1}{3} \cdot (3 (5)) + 3$

Step 1.1.3.2

Cancel the common factor.

$y = 2 m + \frac{1}{3} \cdot (3 \cdot 5) + 3$

Step 1.1.3.3

Rewrite the expression.

$y = 2 m + 5 + 3$

Step 1.2

Add $5$ and $3$ .

$y = 2 m + 8$

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.

$p = \pm 1, \pm 2, \pm 4, \pm 8$

$q = \pm 1, \pm 2$

Step 3

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

$\pm 1, \pm \frac{1}{2}, \pm 2, \pm 4, \pm 8$

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.

$2 (- 4) + 8$

Step 5

Simplify the expression. In this case, the expression is equal to $0$ so $m = - 4$ is a root of the polynomial.

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Step 5.1

Multiply $2$ by $- 4$ .

$- 8 + 8$

Step 5.2

Add $- 8$ and $8$ .

$0$

Step 6

Since $- 4$ is a known root, divide the polynomial by $m + 4$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 m + 8}{m + 4}$

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.

$- 4$	$2$	$8$

Step 7.2

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

$- 4$	$2$	$8$

	$2$

Step 7.3

Multiply the newest entry in the result $(2)$ by the divisor $(- 4)$ and place the result of $(- 8)$ under the next term in the dividend $(8)$ .

$- 4$	$2$	$8$
		$- 8$
	$2$

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.

$- 4$	$2$	$8$
		$- 8$
	$2$	$0$

Step 7.5

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

$2$

Step 8

Since $2 \neq 0$ , there are no solutions.

No solution

Step 9

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

$m + 4$

Step 10

These are the roots (zeros) of the polynomial $2 m + 8$ .

$m = - 4$

Step 11

Simplify $\frac{1}{3} \cdot (6 m + 15) + 3$ .

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Step 11.1

Simplify each term.

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Step 11.1.1

Apply the distributive property.

$\frac{1}{3} (6 m) + \frac{1}{3} \cdot 15 + 3 = 0$

Step 11.1.2

Cancel the common factor of $3$ .

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Step 11.1.2.1

Factor $3$ out of $6 m$ .

$\frac{1}{3} (3 (2 m)) + \frac{1}{3} \cdot 15 + 3 = 0$

Step 11.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 (2 m)) + \frac{1}{3} \cdot 15 + 3 = 0$

Step 11.1.2.3

Rewrite the expression.

$2 m + \frac{1}{3} \cdot 15 + 3 = 0$

Step 11.1.3

Cancel the common factor of $3$ .

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Step 11.1.3.1

Factor $3$ out of $15$ .

$2 m + \frac{1}{3} \cdot (3 (5)) + 3 = 0$

Step 11.1.3.2

Cancel the common factor.

$2 m + \frac{1}{3} \cdot (3 \cdot 5) + 3 = 0$

Step 11.1.3.3

Rewrite the expression.

$2 m + 5 + 3 = 0$

Step 11.2

Add $5$ and $3$ .

$2 m + 8 = 0$

Step 12

Subtract $8$ from both sides of the equation.

$2 m = - 8$

Step 13

Divide each term in $2 m = - 8$ by $2$ and simplify.

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Step 13.1

Divide each term in $2 m = - 8$ by $2$ .

$\frac{2 m}{2} = \frac{- 8}{2}$

Step 13.2

Simplify the left side.

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Step 13.2.1

Cancel the common factor of $2$ .

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Step 13.2.1.1

Cancel the common factor.

$\frac{2 m}{2} = \frac{- 8}{2}$

Step 13.2.1.2

Divide $m$ by $1$ .

$m = \frac{- 8}{2}$

Step 13.3

Simplify the right side.

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Step 13.3.1

Divide $- 8$ by $2$ .

$m = - 4$

Step 14