Algebra Examples

${(a + (- b + c))}^{3}$

Step 1

Use the binomial expansion theorem to find each term. The binomial theorem states ${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

$\sum_{k = 0}^{3} \frac{3!}{(3 - k)! k!} \cdot {(a)}^{3 - k} \cdot {(- b + c)}^{k}$

Step 2

Expand the summation.

$\frac{3!}{(3 - 0)! 0!} {(a)}^{3 - 0} \cdot {(- b + c)}^{0} + \frac{3!}{(3 - 1)! 1!} {(a)}^{3 - 1} \cdot {(- b + c)}^{1} + \frac{3!}{(3 - 2)! 2!} {(a)}^{3 - 2} \cdot {(- b + c)}^{2} + \frac{3!}{(3 - 3)! 3!} {(a)}^{3 - 3} \cdot {(- b + c)}^{3}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(a)}^{3} \cdot {(- b + c)}^{0} + 3 \cdot {(a)}^{2} \cdot {(- b + c)}^{1} + 3 \cdot {(a)}^{1} \cdot {(- b + c)}^{2} + 1 \cdot {(a)}^{0} \cdot {(- b + c)}^{3}$

Step 4

Simplify each term.

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Step 4.1

Multiply ${(a)}^{3}$ by $1$ .

${(a)}^{3} \cdot {(- b + c)}^{0} + 3 \cdot {(a)}^{2} \cdot {(- b + c)}^{1} + 3 \cdot {(a)}^{1} \cdot {(- b + c)}^{2} + 1 \cdot {(a)}^{0} \cdot {(- b + c)}^{3}$

Step 4.2

Anything raised to $0$ is $1$ .

$a^{3} \cdot 1 + 3 \cdot {(a)}^{2} \cdot {(- b + c)}^{1} + 3 \cdot {(a)}^{1} \cdot {(- b + c)}^{2} + 1 \cdot {(a)}^{0} \cdot {(- b + c)}^{3}$

Step 4.3

Multiply $a^{3}$ by $1$ .

$a^{3} + 3 \cdot {(a)}^{2} \cdot {(- b + c)}^{1} + 3 \cdot {(a)}^{1} \cdot {(- b + c)}^{2} + 1 \cdot {(a)}^{0} \cdot {(- b + c)}^{3}$

Step 4.4

Simplify.

$a^{3} + 3 a^{2} \cdot (- b + c) + 3 \cdot {(a)}^{1} \cdot {(- b + c)}^{2} + 1 \cdot {(a)}^{0} \cdot {(- b + c)}^{3}$

Step 4.5

Apply the distributive property.

$a^{3} + 3 a^{2} (- b) + 3 a^{2} c + 3 \cdot {(a)}^{1} \cdot {(- b + c)}^{2} + 1 \cdot {(a)}^{0} \cdot {(- b + c)}^{3}$

Step 4.6

Rewrite using the commutative property of multiplication.

$a^{3} + 3 \cdot - 1 a^{2} b + 3 a^{2} c + 3 \cdot {(a)}^{1} \cdot {(- b + c)}^{2} + 1 \cdot {(a)}^{0} \cdot {(- b + c)}^{3}$

Step 4.7

Multiply $3$ by $- 1$ .

$a^{3} - 3 a^{2} b + 3 a^{2} c + 3 \cdot {(a)}^{1} \cdot {(- b + c)}^{2} + 1 \cdot {(a)}^{0} \cdot {(- b + c)}^{3}$

Step 4.8

Simplify.

$a^{3} - 3 a^{2} b + 3 a^{2} c + 3 \cdot a \cdot {(- b + c)}^{2} + 1 \cdot {(a)}^{0} \cdot {(- b + c)}^{3}$

Step 4.9

Rewrite ${(- b + c)}^{2}$ as $(- b + c) (- b + c)$ .

$a^{3} - 3 a^{2} b + 3 a^{2} c + 3 a \cdot ((- b + c) (- b + c)) + 1 \cdot {(a)}^{0} \cdot {(- b + c)}^{3}$

Step 4.10

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

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Step 4.10.1

Apply the distributive property.

$a^{3} - 3 a^{2} b + 3 a^{2} c + 3 a \cdot (- b (- b + c) + c (- b + c)) + 1 \cdot {(a)}^{0} \cdot {(- b + c)}^{3}$

Step 4.10.2

Apply the distributive property.

$a^{3} - 3 a^{2} b + 3 a^{2} c + 3 a \cdot (- b (- b) - b c + c (- b + c)) + 1 \cdot {(a)}^{0} \cdot {(- b + c)}^{3}$

Step 4.10.3

Apply the distributive property.

$a^{3} - 3 a^{2} b + 3 a^{2} c + 3 a \cdot (- b (- b) - b c + c (- b) + c \cdot c) + 1 \cdot {(a)}^{0} \cdot {(- b + c)}^{3}$

Step 4.11

Simplify and combine like terms.

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Step 4.11.1

Simplify each term.

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Step 4.11.1.1

Rewrite using the commutative property of multiplication.

$a^{3} - 3 a^{2} b + 3 a^{2} c + 3 a \cdot (- 1 \cdot - 1 b \cdot b - b c + c (- b) + c \cdot c) + 1 \cdot {(a)}^{0} \cdot {(- b + c)}^{3}$

Step 4.11.1.2

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

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Step 4.11.1.2.1

Move $b$ .

$a^{3} - 3 a^{2} b + 3 a^{2} c + 3 a \cdot (- 1 \cdot - 1 (b \cdot b) - b c + c (- b) + c \cdot c) + 1 \cdot {(a)}^{0} \cdot {(- b + c)}^{3}$

Step 4.11.1.2.2

Multiply $b$ by $b$ .

$a^{3} - 3 a^{2} b + 3 a^{2} c + 3 a \cdot (- 1 \cdot - 1 b^{2} - b c + c (- b) + c \cdot c) + 1 \cdot {(a)}^{0} \cdot {(- b + c)}^{3}$

Step 4.11.1.3

Multiply $- 1$ by $- 1$ .

$a^{3} - 3 a^{2} b + 3 a^{2} c + 3 a \cdot (1 b^{2} - b c + c (- b) + c \cdot c) + 1 \cdot {(a)}^{0} \cdot {(- b + c)}^{3}$

Step 4.11.1.4

Multiply $b^{2}$ by $1$ .

$a^{3} - 3 a^{2} b + 3 a^{2} c + 3 a \cdot (b^{2} - b c + c (- b) + c \cdot c) + 1 \cdot {(a)}^{0} \cdot {(- b + c)}^{3}$

Step 4.11.1.5

Rewrite using the commutative property of multiplication.

$a^{3} - 3 a^{2} b + 3 a^{2} c + 3 a \cdot (b^{2} - b c - c b + c \cdot c) + 1 \cdot {(a)}^{0} \cdot {(- b + c)}^{3}$

Step 4.11.1.6

Multiply $c$ by $c$ .

$a^{3} - 3 a^{2} b + 3 a^{2} c + 3 a \cdot (b^{2} - b c - c b + c^{2}) + 1 \cdot {(a)}^{0} \cdot {(- b + c)}^{3}$

Step 4.11.2

Subtract $c b$ from $- b c$ .

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Step 4.11.2.1

Move $c$ .

$a^{3} - 3 a^{2} b + 3 a^{2} c + 3 a \cdot (b^{2} - b c - b c + c^{2}) + 1 \cdot {(a)}^{0} \cdot {(- b + c)}^{3}$

Step 4.11.2.2

Subtract $b c$ from $- b c$ .

$a^{3} - 3 a^{2} b + 3 a^{2} c + 3 a \cdot (b^{2} - 2 b c + c^{2}) + 1 \cdot {(a)}^{0} \cdot {(- b + c)}^{3}$

Step 4.12

Apply the distributive property.

$a^{3} - 3 a^{2} b + 3 a^{2} c + 3 a b^{2} + 3 a (- 2 b c) + 3 a c^{2} + 1 \cdot {(a)}^{0} \cdot {(- b + c)}^{3}$

Step 4.13

Multiply $- 2$ by $3$ .

$a^{3} - 3 a^{2} b + 3 a^{2} c + 3 a b^{2} - 6 a (b c) + 3 a c^{2} + 1 \cdot {(a)}^{0} \cdot {(- b + c)}^{3}$

Step 4.14

Multiply ${(a)}^{0}$ by $1$ .

$a^{3} - 3 a^{2} b + 3 a^{2} c + 3 a b^{2} - 6 a b c + 3 a c^{2} + {(a)}^{0} \cdot {(- b + c)}^{3}$

Step 4.15

Anything raised to $0$ is $1$ .

$a^{3} - 3 a^{2} b + 3 a^{2} c + 3 a b^{2} - 6 a b c + 3 a c^{2} + 1 \cdot {(- b + c)}^{3}$

Step 4.16

Multiply ${(- b + c)}^{3}$ by $1$ .

$a^{3} - 3 a^{2} b + 3 a^{2} c + 3 a b^{2} - 6 a b c + 3 a c^{2} + {(- b + c)}^{3}$

Step 4.17

Use the Binomial Theorem.

$a^{3} - 3 a^{2} b + 3 a^{2} c + 3 a b^{2} - 6 a b c + 3 a c^{2} + {(- b)}^{3} + 3 {(- b)}^{2} c + 3 (- b) c^{2} + c^{3}$

Step 4.18

Simplify each term.

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Step 4.18.1

Apply the product rule to $- b$ .

$a^{3} - 3 a^{2} b + 3 a^{2} c + 3 a b^{2} - 6 a b c + 3 a c^{2} + {(- 1)}^{3} b^{3} + 3 {(- b)}^{2} c + 3 (- b) c^{2} + c^{3}$

Step 4.18.2

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

$a^{3} - 3 a^{2} b + 3 a^{2} c + 3 a b^{2} - 6 a b c + 3 a c^{2} - b^{3} + 3 {(- b)}^{2} c + 3 (- b) c^{2} + c^{3}$

Step 4.18.3

Apply the product rule to $- b$ .

$a^{3} - 3 a^{2} b + 3 a^{2} c + 3 a b^{2} - 6 a b c + 3 a c^{2} - b^{3} + 3 ({(- 1)}^{2} b^{2}) c + 3 (- b) c^{2} + c^{3}$

Step 4.18.4

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

$a^{3} - 3 a^{2} b + 3 a^{2} c + 3 a b^{2} - 6 a b c + 3 a c^{2} - b^{3} + 3 (1 b^{2}) c + 3 (- b) c^{2} + c^{3}$

Step 4.18.5

Multiply $b^{2}$ by $1$ .

$a^{3} - 3 a^{2} b + 3 a^{2} c + 3 a b^{2} - 6 a b c + 3 a c^{2} - b^{3} + 3 b^{2} c + 3 (- b) c^{2} + c^{3}$

Step 4.18.6

Multiply $- 1$ by $3$ .

$a^{3} - 3 a^{2} b + 3 a^{2} c + 3 a b^{2} - 6 a b c + 3 a c^{2} - b^{3} + 3 b^{2} c - 3 b c^{2} + c^{3}$