Simplify (x^3-2x^2+x-2)(x^2-2x+2)

$(x^{3} - 2 x^{2} + x - 2) (x^{2} - 2 x + 2)$

Step 1

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

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

Step 2

Simplify terms.

Simplify each term.

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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

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

Add $3$ and $2$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

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

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

$x^{5} - 2 (x^{1} x^{3}) + x^{3} \cdot 2 - 2 x^{2} x^{2} - 2 x^{2} (- 2 x) - 2 x^{2} \cdot 2 + x \cdot x^{2} + x (- 2 x) + x \cdot 2 - 2 x^{2} - 2 (- 2 x) - 2 \cdot 2$

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

$x^{5} - 2 x^{1 + 3} + x^{3} \cdot 2 - 2 x^{2} x^{2} - 2 x^{2} (- 2 x) - 2 x^{2} \cdot 2 + x \cdot x^{2} + x (- 2 x) + x \cdot 2 - 2 x^{2} - 2 (- 2 x) - 2 \cdot 2$

Add $1$ and $3$ .

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

Move $2$ to the left of $x^{3}$ .

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

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

Move $x^{2}$ .

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

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

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

Add $2$ and $2$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Multiply $- 2$ by $- 2$ .

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

Multiply $2$ by $- 2$ .

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

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

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

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

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

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

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

Add $1$ and $2$ .

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

Rewrite using the commutative property of multiplication.

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

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

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

Multiply $x$ by $x$ .

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

Move $2$ to the left of $x$ .

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

Multiply $- 2$ by $- 2$ .

$x^{5} - 2 x^{4} + 2 x^{3} - 2 x^{4} + 4 x^{3} - 4 x^{2} + x^{3} - 2 x^{2} + 2 x - 2 x^{2} + 4 x - 2 \cdot 2$

Multiply $- 2$ by $2$ .

$x^{5} - 2 x^{4} + 2 x^{3} - 2 x^{4} + 4 x^{3} - 4 x^{2} + x^{3} - 2 x^{2} + 2 x - 2 x^{2} + 4 x - 4$

Simplify by adding terms.

Subtract $2 x^{4}$ from $- 2 x^{4}$ .

$x^{5} - 4 x^{4} + 2 x^{3} + 4 x^{3} - 4 x^{2} + x^{3} - 2 x^{2} + 2 x - 2 x^{2} + 4 x - 4$

Add $2 x^{3}$ and $4 x^{3}$ .

$x^{5} - 4 x^{4} + 6 x^{3} - 4 x^{2} + x^{3} - 2 x^{2} + 2 x - 2 x^{2} + 4 x - 4$

Add $6 x^{3}$ and $x^{3}$ .

$x^{5} - 4 x^{4} + 7 x^{3} - 4 x^{2} - 2 x^{2} + 2 x - 2 x^{2} + 4 x - 4$

Subtract $2 x^{2}$ from $- 4 x^{2}$ .

$x^{5} - 4 x^{4} + 7 x^{3} - 6 x^{2} + 2 x - 2 x^{2} + 4 x - 4$

Subtract $2 x^{2}$ from $- 6 x^{2}$ .

$x^{5} - 4 x^{4} + 7 x^{3} - 8 x^{2} + 2 x + 4 x - 4$

Add $2 x$ and $4 x$ .

$x^{5} - 4 x^{4} + 7 x^{3} - 8 x^{2} + 6 x - 4$

Simplify (x^3-2x^2+x-2)(x^2-2x+2)

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