Finite Math Examples

$3 b^{3} - 5 b^{2} + 2 b$

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.

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

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $b = 0$ 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 $0$ .

$3 \cdot 0 - 5 {(0)}^{2} + 2 (0)$

Step 4.1.2

Multiply $3$ by $0$ .

$0 - 5 {(0)}^{2} + 2 (0)$

Step 4.1.3

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

$0 - 5 \cdot 0 + 2 (0)$

Step 4.1.4

Multiply $- 5$ by $0$ .

$0 + 0 + 2 (0)$

Step 4.1.5

Multiply $2$ by $0$ .

$0 + 0 + 0$

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 $0$ .

$0$

Step 5

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

$\frac{3 b^{3} - 5 b^{2} + 2 b}{b + 0}$

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.

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

Step 6.2

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

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

	$3$

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 $(- 5)$ .

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

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.

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

Step 6.5

Multiply the newest entry in the result $(- 5)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(2)$ .

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

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.

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

Step 6.7

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 $(0)$ .

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

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.

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

Step 6.9

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

$3 b^{2} + - 5 b + 2$

Step 6.10

Simplify the quotient polynomial.

$3 b^{2} - 5 b + 2$

Step 7

Factor by grouping.

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Step 7.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 2 = 6$ and whose sum is $b = - 5$ .

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Step 7.1.1

Factor $- 5$ out of $- 5 b$ .

$(b + 0) + 3 b^{2} - 5 b + 2$

Step 7.1.2

Rewrite $- 5$ as $- 2$ plus $- 3$

$(b + 0) + 3 b^{2} + (- 2 - 3) b + 2$

Step 7.1.3

Apply the distributive property.

$(b + 0) + 3 b^{2} - 2 b - 3 b + 2$

Step 7.2

Factor out the greatest common factor from each group.

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Step 7.2.1

Group the first two terms and the last two terms.

$(b + 0) (3 b^{2} - 2 b) - 3 b + 2$

Step 7.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 7.3

Factor the polynomial by factoring out the greatest common factor, $3 b - 2$ .

$(b + 0) + (3 b - 2) (b - 1)$

$(b + 0) (3 b - 2) (b - 1)$

Step 8

Factor the left side of the equation.

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Step 8.1

Factor $b$ out of $3 b^{3} - 5 b^{2} + 2 b$ .

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Step 8.1.1

Factor $b$ out of $3 b^{3}$ .

$b (3 b^{2}) - 5 b^{2} + 2 b = 0$

Step 8.1.2

Factor $b$ out of $- 5 b^{2}$ .

$b (3 b^{2}) + b (- 5 b) + 2 b = 0$

Step 8.1.3

Factor $b$ out of $2 b$ .

$b (3 b^{2}) + b (- 5 b) + b \cdot 2 = 0$

Step 8.1.4

Factor $b$ out of $b (3 b^{2}) + b (- 5 b)$ .

$b (3 b^{2} - 5 b) + b \cdot 2 = 0$

Step 8.1.5

Factor $b$ out of $b (3 b^{2} - 5 b) + b \cdot 2$ .

$b (3 b^{2} - 5 b + 2) = 0$

Step 8.2

Factor.

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Step 8.2.1

Factor by grouping.

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Step 8.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 2 = 6$ and whose sum is $b = - 5$ .

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Step 8.2.1.1.1

Factor $- 5$ out of $- 5 b$ .

$b (3 b^{2} - 5 b + 2) = 0$

Step 8.2.1.1.2

Rewrite $- 5$ as $- 2$ plus $- 3$

$b (3 b^{2} + (- 2 - 3) b + 2) = 0$

Step 8.2.1.1.3

Apply the distributive property.

$b (3 b^{2} - 2 b - 3 b + 2) = 0$

Step 8.2.1.2

Factor out the greatest common factor from each group.

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Step 8.2.1.2.1

Group the first two terms and the last two terms.

$b ((3 b^{2} - 2 b) - 3 b + 2) = 0$

Step 8.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$b (b (3 b - 2) - (3 b - 2)) = 0$

Step 8.2.1.3

Factor the polynomial by factoring out the greatest common factor, $3 b - 2$ .

$b ((3 b - 2) (b - 1)) = 0$

Step 8.2.2

Remove unnecessary parentheses.

$b (3 b - 2) (b - 1) = 0$

Step 9

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

$b = 0$

$3 b - 2 = 0$

$b - 1 = 0$

Step 10

Set $b$ equal to $0$ .

$b = 0$

Step 11

Set $3 b - 2$ equal to $0$ and solve for $b$ .

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Step 11.1

Set $3 b - 2$ equal to $0$ .

$3 b - 2 = 0$

Step 11.2

Solve $3 b - 2 = 0$ for $b$ .

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Step 11.2.1

Add $2$ to both sides of the equation.

$3 b = 2$

Step 11.2.2

Divide each term in $3 b = 2$ by $3$ and simplify.

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Step 11.2.2.1

Divide each term in $3 b = 2$ by $3$ .

$\frac{3 b}{3} = \frac{2}{3}$

Step 11.2.2.2

Simplify the left side.

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Step 11.2.2.2.1

Cancel the common factor of $3$ .

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Step 11.2.2.2.1.1

Cancel the common factor.

$\frac{3 b}{3} = \frac{2}{3}$

Step 11.2.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{2}{3}$

Step 12

Set $b - 1$ equal to $0$ and solve for $b$ .

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Step 12.1

Set $b - 1$ equal to $0$ .

$b - 1 = 0$

Step 12.2

Add $1$ to both sides of the equation.

$b = 1$

Step 13

The final solution is all the values that make $b (3 b - 2) (b - 1) = 0$ true.

$b = 0, \frac{2}{3}, 1$

Step 14