Algebra Examples

$(b^{3} - 5 b^{2} + 8) (- 5 b + 4)$

Step 1

Expand $(b^{3} - 5 b^{2} + 8) (- 5 b + 4)$ by multiplying each term in the first expression by each term in the second expression.

$b^{3} (- 5 b) + b^{3} \cdot 4 - 5 b^{2} (- 5 b) - 5 b^{2} \cdot 4 + 8 (- 5 b) + 8 \cdot 4$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite using the commutative property of multiplication.

$- 5 b^{3} b + b^{3} \cdot 4 - 5 b^{2} (- 5 b) - 5 b^{2} \cdot 4 + 8 (- 5 b) + 8 \cdot 4$

Step 2.1.2

Multiply $b^{3}$ by $b$ by adding the exponents.

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Step 2.1.2.1

Move $b$ .

$- 5 (b \cdot b^{3}) + b^{3} \cdot 4 - 5 b^{2} (- 5 b) - 5 b^{2} \cdot 4 + 8 (- 5 b) + 8 \cdot 4$

Step 2.1.2.2

Multiply $b$ by $b^{3}$ .

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Step 2.1.2.2.1

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

$- 5 (b^{1} b^{3}) + b^{3} \cdot 4 - 5 b^{2} (- 5 b) - 5 b^{2} \cdot 4 + 8 (- 5 b) + 8 \cdot 4$

Step 2.1.2.2.2

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

$- 5 b^{1 + 3} + b^{3} \cdot 4 - 5 b^{2} (- 5 b) - 5 b^{2} \cdot 4 + 8 (- 5 b) + 8 \cdot 4$

Step 2.1.2.3

Add $1$ and $3$ .

$- 5 b^{4} + b^{3} \cdot 4 - 5 b^{2} (- 5 b) - 5 b^{2} \cdot 4 + 8 (- 5 b) + 8 \cdot 4$

Step 2.1.3

Move $4$ to the left of $b^{3}$ .

$- 5 b^{4} + 4 \cdot b^{3} - 5 b^{2} (- 5 b) - 5 b^{2} \cdot 4 + 8 (- 5 b) + 8 \cdot 4$

Step 2.1.4

Rewrite using the commutative property of multiplication.

$- 5 b^{4} + 4 b^{3} - 5 \cdot - 5 b^{2} b - 5 b^{2} \cdot 4 + 8 (- 5 b) + 8 \cdot 4$

Step 2.1.5

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

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Step 2.1.5.1

Move $b$ .

$- 5 b^{4} + 4 b^{3} - 5 \cdot - 5 (b \cdot b^{2}) - 5 b^{2} \cdot 4 + 8 (- 5 b) + 8 \cdot 4$

Step 2.1.5.2

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

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Step 2.1.5.2.1

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

$- 5 b^{4} + 4 b^{3} - 5 \cdot - 5 (b^{1} b^{2}) - 5 b^{2} \cdot 4 + 8 (- 5 b) + 8 \cdot 4$

Step 2.1.5.2.2

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

$- 5 b^{4} + 4 b^{3} - 5 \cdot - 5 b^{1 + 2} - 5 b^{2} \cdot 4 + 8 (- 5 b) + 8 \cdot 4$

Step 2.1.5.3

Add $1$ and $2$ .

$- 5 b^{4} + 4 b^{3} - 5 \cdot - 5 b^{3} - 5 b^{2} \cdot 4 + 8 (- 5 b) + 8 \cdot 4$

Step 2.1.6

Multiply $- 5$ by $- 5$ .

$- 5 b^{4} + 4 b^{3} + 25 b^{3} - 5 b^{2} \cdot 4 + 8 (- 5 b) + 8 \cdot 4$

Step 2.1.7

Multiply $4$ by $- 5$ .

$- 5 b^{4} + 4 b^{3} + 25 b^{3} - 20 b^{2} + 8 (- 5 b) + 8 \cdot 4$

Step 2.1.8

Multiply $- 5$ by $8$ .

$- 5 b^{4} + 4 b^{3} + 25 b^{3} - 20 b^{2} - 40 b + 8 \cdot 4$

Step 2.1.9

Multiply $8$ by $4$ .

$- 5 b^{4} + 4 b^{3} + 25 b^{3} - 20 b^{2} - 40 b + 32$

Step 2.2

Add $4 b^{3}$ and $25 b^{3}$ .

$- 5 b^{4} + 29 b^{3} - 20 b^{2} - 40 b + 32$