Algebra Examples

${(m - n)}^{6} + {(m - n)}^{2}$

Step 1

Factor ${(m - n)}^{2}$ out of ${(m - n)}^{6} + {(m - n)}^{2}$ .

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Step 1.1

Factor ${(m - n)}^{2}$ out of ${(m - n)}^{6}$ .

${(m - n)}^{2} {(m - n)}^{4} + {(m - n)}^{2}$

Step 1.2

Multiply by $1$ .

${(m - n)}^{2} {(m - n)}^{4} + {(m - n)}^{2} \cdot 1$

Step 1.3

Factor ${(m - n)}^{2}$ out of ${(m - n)}^{2} {(m - n)}^{4} + {(m - n)}^{2} \cdot 1$ .

${(m - n)}^{2} ({(m - n)}^{4} + 1)$

Step 2

Use the Binomial Theorem.

${(m - n)}^{2} (m^{4} + 4 m^{3} (- n) + 6 m^{2} {(- n)}^{2} + 4 m {(- n)}^{3} + {(- n)}^{4} + 1)$

Step 3

Simplify each term.

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Step 3.1

Rewrite using the commutative property of multiplication.

${(m - n)}^{2} (m^{4} + 4 \cdot - 1 m^{3} n + 6 m^{2} {(- n)}^{2} + 4 m {(- n)}^{3} + {(- n)}^{4} + 1)$

Step 3.2

Multiply $4$ by $- 1$ .

${(m - n)}^{2} (m^{4} - 4 m^{3} n + 6 m^{2} {(- n)}^{2} + 4 m {(- n)}^{3} + {(- n)}^{4} + 1)$

Step 3.3

Apply the product rule to $- n$ .

${(m - n)}^{2} (m^{4} - 4 m^{3} n + 6 m^{2} ({(- 1)}^{2} n^{2}) + 4 m {(- n)}^{3} + {(- n)}^{4} + 1)$

Step 3.4

Rewrite using the commutative property of multiplication.

${(m - n)}^{2} (m^{4} - 4 m^{3} n + 6 \cdot {(- 1)}^{2} m^{2} n^{2} + 4 m {(- n)}^{3} + {(- n)}^{4} + 1)$

Step 3.5

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

${(m - n)}^{2} (m^{4} - 4 m^{3} n + 6 \cdot 1 m^{2} n^{2} + 4 m {(- n)}^{3} + {(- n)}^{4} + 1)$

Step 3.6

Multiply $6$ by $1$ .

${(m - n)}^{2} (m^{4} - 4 m^{3} n + 6 m^{2} n^{2} + 4 m {(- n)}^{3} + {(- n)}^{4} + 1)$

Step 3.7

Apply the product rule to $- n$ .

${(m - n)}^{2} (m^{4} - 4 m^{3} n + 6 m^{2} n^{2} + 4 m ({(- 1)}^{3} n^{3}) + {(- n)}^{4} + 1)$

Step 3.8

Rewrite using the commutative property of multiplication.

${(m - n)}^{2} (m^{4} - 4 m^{3} n + 6 m^{2} n^{2} + 4 \cdot {(- 1)}^{3} m n^{3} + {(- n)}^{4} + 1)$

Step 3.9

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

${(m - n)}^{2} (m^{4} - 4 m^{3} n + 6 m^{2} n^{2} + 4 \cdot - 1 m n^{3} + {(- n)}^{4} + 1)$

Step 3.10

Multiply $4$ by $- 1$ .

${(m - n)}^{2} (m^{4} - 4 m^{3} n + 6 m^{2} n^{2} - 4 m n^{3} + {(- n)}^{4} + 1)$

Step 3.11

Apply the product rule to $- n$ .

${(m - n)}^{2} (m^{4} - 4 m^{3} n + 6 m^{2} n^{2} - 4 m n^{3} + {(- 1)}^{4} n^{4} + 1)$

Step 3.12

Raise $- 1$ to the power of $4$ .

${(m - n)}^{2} (m^{4} - 4 m^{3} n + 6 m^{2} n^{2} - 4 m n^{3} + 1 n^{4} + 1)$

Step 3.13

Multiply $n^{4}$ by $1$ .

${(m - n)}^{2} (m^{4} - 4 m^{3} n + 6 m^{2} n^{2} - 4 m n^{3} + n^{4} + 1)$