Basic Math Examples

$\frac{b + 9}{1} + \frac{4}{b^{2} + b - 6}$

Step 1

Simplify each term.

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Step 1.1

Divide $b + 9$ by $1$ .

$b + 9 + \frac{4}{b^{2} + b - 6}$

Step 1.2

Factor $b^{2} + b - 6$ using the AC method.

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Step 1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 1.2.2

Write the factored form using these integers.

$b + 9 + \frac{4}{(b - 2) (b + 3)}$

Step 2

To write $b$ as a fraction with a common denominator, multiply by $\frac{(b - 2) (b + 3)}{(b - 2) (b + 3)}$ .

$b \cdot \frac{(b - 2) (b + 3)}{(b - 2) (b + 3)} + \frac{4}{(b - 2) (b + 3)} + 9$

Step 3

Simplify terms.

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Step 3.1

Combine $b$ and $\frac{(b - 2) (b + 3)}{(b - 2) (b + 3)}$ .

$\frac{b ((b - 2) (b + 3))}{(b - 2) (b + 3)} + \frac{4}{(b - 2) (b + 3)} + 9$

Step 3.2

Combine the numerators over the common denominator.

$\frac{b ((b - 2) (b + 3)) + 4}{(b - 2) (b + 3)} + 9$

Step 4

Simplify the numerator.

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Step 4.1

Expand $(b - 2) (b + 3)$ using the FOIL Method.

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Step 4.1.1

Apply the distributive property.

$\frac{b (b (b + 3) - 2 (b + 3)) + 4}{(b - 2) (b + 3)} + 9$

Step 4.1.2

Apply the distributive property.

$\frac{b (b \cdot b + b \cdot 3 - 2 (b + 3)) + 4}{(b - 2) (b + 3)} + 9$

Step 4.1.3

Apply the distributive property.

$\frac{b (b \cdot b + b \cdot 3 - 2 b - 2 \cdot 3) + 4}{(b - 2) (b + 3)} + 9$

Step 4.2

Simplify and combine like terms.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $b$ by $b$ .

$\frac{b (b^{2} + b \cdot 3 - 2 b - 2 \cdot 3) + 4}{(b - 2) (b + 3)} + 9$

Step 4.2.1.2

Move $3$ to the left of $b$ .

$\frac{b (b^{2} + 3 \cdot b - 2 b - 2 \cdot 3) + 4}{(b - 2) (b + 3)} + 9$

Step 4.2.1.3

Multiply $- 2$ by $3$ .

$\frac{b (b^{2} + 3 b - 2 b - 6) + 4}{(b - 2) (b + 3)} + 9$

Step 4.2.2

Subtract $2 b$ from $3 b$ .

$\frac{b (b^{2} + b - 6) + 4}{(b - 2) (b + 3)} + 9$

Step 4.3

Apply the distributive property.

$\frac{b \cdot b^{2} + b \cdot b + b \cdot - 6 + 4}{(b - 2) (b + 3)} + 9$

Step 4.4

Simplify.

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Step 4.4.1

Multiply $b$ by $b^{2}$ by adding the exponents.

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Step 4.4.1.1

Multiply $b$ by $b^{2}$ .

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Step 4.4.1.1.1

Raise $b$ to the power of $1$ .

$\frac{b^{1} b^{2} + b \cdot b + b \cdot - 6 + 4}{(b - 2) (b + 3)} + 9$

Step 4.4.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{b^{1 + 2} + b \cdot b + b \cdot - 6 + 4}{(b - 2) (b + 3)} + 9$

Step 4.4.1.2

Add $1$ and $2$ .

$\frac{b^{3} + b \cdot b + b \cdot - 6 + 4}{(b - 2) (b + 3)} + 9$

Step 4.4.2

Multiply $b$ by $b$ .

$\frac{b^{3} + b^{2} + b \cdot - 6 + 4}{(b - 2) (b + 3)} + 9$

Step 4.4.3

Move $- 6$ to the left of $b$ .

$\frac{b^{3} + b^{2} - 6 \cdot b + 4}{(b - 2) (b + 3)} + 9$

Step 4.5

Factor $b^{3} + b^{2} - 6 b + 4$ using the rational roots test.

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Step 4.5.1

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.

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

$q = \pm 1$

Step 4.5.2

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

$\pm 1, \pm 4, \pm 2$

Step 4.5.3

Substitute $1$ and simplify the expression. In this case, the expression is equal to $0$ so $1$ is a root of the polynomial.

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Step 4.5.3.1

Substitute $1$ into the polynomial.

$1^{3} + 1^{2} - 6 \cdot 1 + 4$

Step 4.5.3.2

Raise $1$ to the power of $3$ .

$1 + 1^{2} - 6 \cdot 1 + 4$

Step 4.5.3.3

Raise $1$ to the power of $2$ .

$1 + 1 - 6 \cdot 1 + 4$

Step 4.5.3.4

Add $1$ and $1$ .

$2 - 6 \cdot 1 + 4$

Step 4.5.3.5

Multiply $- 6$ by $1$ .

$2 - 6 + 4$

Step 4.5.3.6

Subtract $6$ from $2$ .

$- 4 + 4$

Step 4.5.3.7

Add $- 4$ and $4$ .

$0$

Step 4.5.4

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

$\frac{b^{3} + b^{2} - 6 b + 4}{b - 1}$

Step 4.5.5

Divide $b^{3} + b^{2} - 6 b + 4$ by $b - 1$ .

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Step 4.5.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$b$

$1$

$b^{3}$

$b^{2}$

$6 b$

$4$

Step 4.5.5.2

Divide the highest order term in the dividend $b^{3}$ by the highest order term in divisor $b$ .

					$b^{2}$
	$b$	-	$1$		$b^{3}$	+	$b^{2}$	-	$6 b$	+	$4$

Step 4.5.5.3

Multiply the new quotient term by the divisor.

				$b^{2}$
$b$	-	$1$		$b^{3}$	+	$b^{2}$	-	$6 b$	+	$4$
			+	$b^{3}$	-	$b^{2}$

Step 4.5.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $b^{3} - b^{2}$

				$b^{2}$
$b$	-	$1$		$b^{3}$	+	$b^{2}$	-	$6 b$	+	$4$
			-	$b^{3}$	+	$b^{2}$

Step 4.5.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$b^{2}$
$b$	-	$1$		$b^{3}$	+	$b^{2}$	-	$6 b$	+	$4$
			-	$b^{3}$	+	$b^{2}$
					+	$2 b^{2}$

Step 4.5.5.6

Pull the next terms from the original dividend down into the current dividend.

				$b^{2}$
$b$	-	$1$		$b^{3}$	+	$b^{2}$	-	$6 b$	+	$4$
			-	$b^{3}$	+	$b^{2}$
					+	$2 b^{2}$	-	$6 b$

Step 4.5.5.7

Divide the highest order term in the dividend $2 b^{2}$ by the highest order term in divisor $b$ .

				$b^{2}$	+	$2 b$
$b$	-	$1$		$b^{3}$	+	$b^{2}$	-	$6 b$	+	$4$
			-	$b^{3}$	+	$b^{2}$
					+	$2 b^{2}$	-	$6 b$

Step 4.5.5.8

Multiply the new quotient term by the divisor.

				$b^{2}$	+	$2 b$
$b$	-	$1$		$b^{3}$	+	$b^{2}$	-	$6 b$	+	$4$
			-	$b^{3}$	+	$b^{2}$
					+	$2 b^{2}$	-	$6 b$
					+	$2 b^{2}$	-	$2 b$

Step 4.5.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $2 b^{2} - 2 b$

				$b^{2}$	+	$2 b$
$b$	-	$1$		$b^{3}$	+	$b^{2}$	-	$6 b$	+	$4$
			-	$b^{3}$	+	$b^{2}$
					+	$2 b^{2}$	-	$6 b$
					-	$2 b^{2}$	+	$2 b$

Step 4.5.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$b^{2}$	+	$2 b$
$b$	-	$1$		$b^{3}$	+	$b^{2}$	-	$6 b$	+	$4$
			-	$b^{3}$	+	$b^{2}$
					+	$2 b^{2}$	-	$6 b$
					-	$2 b^{2}$	+	$2 b$
							-	$4 b$

Step 4.5.5.11

Pull the next terms from the original dividend down into the current dividend.

				$b^{2}$	+	$2 b$
$b$	-	$1$		$b^{3}$	+	$b^{2}$	-	$6 b$	+	$4$
			-	$b^{3}$	+	$b^{2}$
					+	$2 b^{2}$	-	$6 b$
					-	$2 b^{2}$	+	$2 b$
							-	$4 b$	+	$4$

Step 4.5.5.12

Divide the highest order term in the dividend $- 4 b$ by the highest order term in divisor $b$ .

				$b^{2}$	+	$2 b$	-	$4$
$b$	-	$1$		$b^{3}$	+	$b^{2}$	-	$6 b$	+	$4$
			-	$b^{3}$	+	$b^{2}$
					+	$2 b^{2}$	-	$6 b$
					-	$2 b^{2}$	+	$2 b$
							-	$4 b$	+	$4$

Step 4.5.5.13

Multiply the new quotient term by the divisor.

				$b^{2}$	+	$2 b$	-	$4$
$b$	-	$1$		$b^{3}$	+	$b^{2}$	-	$6 b$	+	$4$
			-	$b^{3}$	+	$b^{2}$
					+	$2 b^{2}$	-	$6 b$
					-	$2 b^{2}$	+	$2 b$
							-	$4 b$	+	$4$
							-	$4 b$	+	$4$

Step 4.5.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 b + 4$

				$b^{2}$	+	$2 b$	-	$4$
$b$	-	$1$		$b^{3}$	+	$b^{2}$	-	$6 b$	+	$4$
			-	$b^{3}$	+	$b^{2}$
					+	$2 b^{2}$	-	$6 b$
					-	$2 b^{2}$	+	$2 b$
							-	$4 b$	+	$4$
							+	$4 b$	-	$4$

Step 4.5.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$b^{2}$	+	$2 b$	-	$4$
$b$	-	$1$		$b^{3}$	+	$b^{2}$	-	$6 b$	+	$4$
			-	$b^{3}$	+	$b^{2}$
					+	$2 b^{2}$	-	$6 b$
					-	$2 b^{2}$	+	$2 b$
							-	$4 b$	+	$4$
							+	$4 b$	-	$4$
										$0$

Step 4.5.5.16

Since the remander is $0$ , the final answer is the quotient.

$b^{2} + 2 b - 4$

Step 4.5.6

Write $b^{3} + b^{2} - 6 b + 4$ as a set of factors.

$\frac{(b - 1) (b^{2} + 2 b - 4)}{(b - 2) (b + 3)} + 9$

Step 5

To write $9$ as a fraction with a common denominator, multiply by $\frac{(b - 2) (b + 3)}{(b - 2) (b + 3)}$ .

$\frac{(b - 1) (b^{2} + 2 b - 4)}{(b - 2) (b + 3)} + 9 \cdot \frac{(b - 2) (b + 3)}{(b - 2) (b + 3)}$

Step 6

Simplify terms.

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Step 6.1

Combine $9$ and $\frac{(b - 2) (b + 3)}{(b - 2) (b + 3)}$ .

$\frac{(b - 1) (b^{2} + 2 b - 4)}{(b - 2) (b + 3)} + \frac{9 ((b - 2) (b + 3))}{(b - 2) (b + 3)}$

Step 6.2

Combine the numerators over the common denominator.

$\frac{(b - 1) (b^{2} + 2 b - 4) + 9 ((b - 2) (b + 3))}{(b - 2) (b + 3)}$

Step 7

Simplify the numerator.

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Step 7.1

Expand $(b - 1) (b^{2} + 2 b - 4)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{b \cdot b^{2} + b (2 b) + b \cdot - 4 - 1 b^{2} - 1 (2 b) - 1 \cdot - 4 + 9 (b - 2) (b + 3)}{(b - 2) (b + 3)}$

Step 7.2

Simplify each term.

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Step 7.2.1

Multiply $b$ by $b^{2}$ by adding the exponents.

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Step 7.2.1.1

Multiply $b$ by $b^{2}$ .

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Step 7.2.1.1.1

Raise $b$ to the power of $1$ .

$\frac{b^{1} b^{2} + b (2 b) + b \cdot - 4 - 1 b^{2} - 1 (2 b) - 1 \cdot - 4 + 9 (b - 2) (b + 3)}{(b - 2) (b + 3)}$

Step 7.2.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{b^{1 + 2} + b (2 b) + b \cdot - 4 - 1 b^{2} - 1 (2 b) - 1 \cdot - 4 + 9 (b - 2) (b + 3)}{(b - 2) (b + 3)}$

Step 7.2.1.2

Add $1$ and $2$ .

$\frac{b^{3} + b (2 b) + b \cdot - 4 - 1 b^{2} - 1 (2 b) - 1 \cdot - 4 + 9 (b - 2) (b + 3)}{(b - 2) (b + 3)}$

Step 7.2.2

Rewrite using the commutative property of multiplication.

$\frac{b^{3} + 2 b \cdot b + b \cdot - 4 - 1 b^{2} - 1 (2 b) - 1 \cdot - 4 + 9 (b - 2) (b + 3)}{(b - 2) (b + 3)}$

Step 7.2.3

Multiply $b$ by $b$ by adding the exponents.

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Step 7.2.3.1

Move $b$ .

$\frac{b^{3} + 2 (b \cdot b) + b \cdot - 4 - 1 b^{2} - 1 (2 b) - 1 \cdot - 4 + 9 (b - 2) (b + 3)}{(b - 2) (b + 3)}$

Step 7.2.3.2

Multiply $b$ by $b$ .

$\frac{b^{3} + 2 b^{2} + b \cdot - 4 - 1 b^{2} - 1 (2 b) - 1 \cdot - 4 + 9 (b - 2) (b + 3)}{(b - 2) (b + 3)}$

Step 7.2.4

Move $- 4$ to the left of $b$ .

$\frac{b^{3} + 2 b^{2} - 4 \cdot b - 1 b^{2} - 1 (2 b) - 1 \cdot - 4 + 9 (b - 2) (b + 3)}{(b - 2) (b + 3)}$

Step 7.2.5

Rewrite $- 1 b^{2}$ as $- b^{2}$ .

$\frac{b^{3} + 2 b^{2} - 4 b - b^{2} - 1 (2 b) - 1 \cdot - 4 + 9 (b - 2) (b + 3)}{(b - 2) (b + 3)}$

Step 7.2.6

Multiply $2$ by $- 1$ .

$\frac{b^{3} + 2 b^{2} - 4 b - b^{2} - 2 b - 1 \cdot - 4 + 9 (b - 2) (b + 3)}{(b - 2) (b + 3)}$

Step 7.2.7

Multiply $- 1$ by $- 4$ .

$\frac{b^{3} + 2 b^{2} - 4 b - b^{2} - 2 b + 4 + 9 (b - 2) (b + 3)}{(b - 2) (b + 3)}$

Step 7.3

Subtract $b^{2}$ from $2 b^{2}$ .

$\frac{b^{3} + b^{2} - 4 b - 2 b + 4 + 9 (b - 2) (b + 3)}{(b - 2) (b + 3)}$

Step 7.4

Subtract $2 b$ from $- 4 b$ .

$\frac{b^{3} + b^{2} - 6 b + 4 + 9 (b - 2) (b + 3)}{(b - 2) (b + 3)}$

Step 7.5

Apply the distributive property.

$\frac{b^{3} + b^{2} - 6 b + 4 + (9 b + 9 \cdot - 2) (b + 3)}{(b - 2) (b + 3)}$

Step 7.6

Multiply $9$ by $- 2$ .

$\frac{b^{3} + b^{2} - 6 b + 4 + (9 b - 18) (b + 3)}{(b - 2) (b + 3)}$

Step 7.7

Expand $(9 b - 18) (b + 3)$ using the FOIL Method.

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Step 7.7.1

Apply the distributive property.

$\frac{b^{3} + b^{2} - 6 b + 4 + 9 b (b + 3) - 18 (b + 3)}{(b - 2) (b + 3)}$

Step 7.7.2

Apply the distributive property.

$\frac{b^{3} + b^{2} - 6 b + 4 + 9 b \cdot b + 9 b \cdot 3 - 18 (b + 3)}{(b - 2) (b + 3)}$

Step 7.7.3

Apply the distributive property.

$\frac{b^{3} + b^{2} - 6 b + 4 + 9 b \cdot b + 9 b \cdot 3 - 18 b - 18 \cdot 3}{(b - 2) (b + 3)}$

Step 7.8

Simplify and combine like terms.

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Step 7.8.1

Simplify each term.

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Step 7.8.1.1

Multiply $b$ by $b$ by adding the exponents.

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Step 7.8.1.1.1

Move $b$ .

$\frac{b^{3} + b^{2} - 6 b + 4 + 9 (b \cdot b) + 9 b \cdot 3 - 18 b - 18 \cdot 3}{(b - 2) (b + 3)}$

Step 7.8.1.1.2

Multiply $b$ by $b$ .

$\frac{b^{3} + b^{2} - 6 b + 4 + 9 b^{2} + 9 b \cdot 3 - 18 b - 18 \cdot 3}{(b - 2) (b + 3)}$

Step 7.8.1.2

Multiply $3$ by $9$ .

$\frac{b^{3} + b^{2} - 6 b + 4 + 9 b^{2} + 27 b - 18 b - 18 \cdot 3}{(b - 2) (b + 3)}$

Step 7.8.1.3

Multiply $- 18$ by $3$ .

$\frac{b^{3} + b^{2} - 6 b + 4 + 9 b^{2} + 27 b - 18 b - 54}{(b - 2) (b + 3)}$

Step 7.8.2

Subtract $18 b$ from $27 b$ .

$\frac{b^{3} + b^{2} - 6 b + 4 + 9 b^{2} + 9 b - 54}{(b - 2) (b + 3)}$

Step 7.9

Add $b^{2}$ and $9 b^{2}$ .

$\frac{b^{3} + 10 b^{2} - 6 b + 4 + 9 b - 54}{(b - 2) (b + 3)}$

Step 7.10

Add $- 6 b$ and $9 b$ .

$\frac{b^{3} + 10 b^{2} + 3 b + 4 - 54}{(b - 2) (b + 3)}$

Step 7.11

Subtract $54$ from $4$ .

$\frac{b^{3} + 10 b^{2} + 3 b - 50}{(b - 2) (b + 3)}$