Find the Product (3x^3-9x+7)(x^2-2x+1)

$(3 x^{3} - 9 x + 7) (x^{2} - 2 x + 1)$

Expand $(3 x^{3} - 9 x + 7) (x^{2} - 2 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$3 x^{3} x^{2} + 3 x^{3} (- 2 x) + 3 x^{3} \cdot 1 - 9 x \cdot x^{2} - 9 x (- 2 x) - 9 x \cdot 1 + 7 x^{2} + 7 (- 2 x) + 7 \cdot 1$

Simplify terms.

Simplify each term.

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

Move $x^{2}$ .

$3 (x^{2} x^{3}) + 3 x^{3} (- 2 x) + 3 x^{3} \cdot 1 - 9 x \cdot x^{2} - 9 x (- 2 x) - 9 x \cdot 1 + 7 x^{2} + 7 (- 2 x) + 7 \cdot 1$

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

$3 x^{2 + 3} + 3 x^{3} (- 2 x) + 3 x^{3} \cdot 1 - 9 x \cdot x^{2} - 9 x (- 2 x) - 9 x \cdot 1 + 7 x^{2} + 7 (- 2 x) + 7 \cdot 1$

Add $2$ and $3$ .

$3 x^{5} + 3 x^{3} (- 2 x) + 3 x^{3} \cdot 1 - 9 x \cdot x^{2} - 9 x (- 2 x) - 9 x \cdot 1 + 7 x^{2} + 7 (- 2 x) + 7 \cdot 1$

Rewrite using the commutative property of multiplication.

$3 x^{5} + 3 \cdot - 2 x^{3} x + 3 x^{3} \cdot 1 - 9 x \cdot x^{2} - 9 x (- 2 x) - 9 x \cdot 1 + 7 x^{2} + 7 (- 2 x) + 7 \cdot 1$

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

Move $x$ .

$3 x^{5} + 3 \cdot - 2 (x \cdot x^{3}) + 3 x^{3} \cdot 1 - 9 x \cdot x^{2} - 9 x (- 2 x) - 9 x \cdot 1 + 7 x^{2} + 7 (- 2 x) + 7 \cdot 1$

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

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

$3 x^{5} + 3 \cdot - 2 (x^{1} x^{3}) + 3 x^{3} \cdot 1 - 9 x \cdot x^{2} - 9 x (- 2 x) - 9 x \cdot 1 + 7 x^{2} + 7 (- 2 x) + 7 \cdot 1$

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

$3 x^{5} + 3 \cdot - 2 x^{1 + 3} + 3 x^{3} \cdot 1 - 9 x \cdot x^{2} - 9 x (- 2 x) - 9 x \cdot 1 + 7 x^{2} + 7 (- 2 x) + 7 \cdot 1$

Add $1$ and $3$ .

$3 x^{5} + 3 \cdot - 2 x^{4} + 3 x^{3} \cdot 1 - 9 x \cdot x^{2} - 9 x (- 2 x) - 9 x \cdot 1 + 7 x^{2} + 7 (- 2 x) + 7 \cdot 1$

Multiply $3$ by $- 2$ .

$3 x^{5} - 6 x^{4} + 3 x^{3} \cdot 1 - 9 x \cdot x^{2} - 9 x (- 2 x) - 9 x \cdot 1 + 7 x^{2} + 7 (- 2 x) + 7 \cdot 1$

Multiply $3$ by $1$ .

$3 x^{5} - 6 x^{4} + 3 x^{3} - 9 x \cdot x^{2} - 9 x (- 2 x) - 9 x \cdot 1 + 7 x^{2} + 7 (- 2 x) + 7 \cdot 1$

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

Move $x^{2}$ .

$3 x^{5} - 6 x^{4} + 3 x^{3} - 9 (x^{2} x) - 9 x (- 2 x) - 9 x \cdot 1 + 7 x^{2} + 7 (- 2 x) + 7 \cdot 1$

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

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

$3 x^{5} - 6 x^{4} + 3 x^{3} - 9 (x^{2} x^{1}) - 9 x (- 2 x) - 9 x \cdot 1 + 7 x^{2} + 7 (- 2 x) + 7 \cdot 1$

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

$3 x^{5} - 6 x^{4} + 3 x^{3} - 9 x^{2 + 1} - 9 x (- 2 x) - 9 x \cdot 1 + 7 x^{2} + 7 (- 2 x) + 7 \cdot 1$

Add $2$ and $1$ .

$3 x^{5} - 6 x^{4} + 3 x^{3} - 9 x^{3} - 9 x (- 2 x) - 9 x \cdot 1 + 7 x^{2} + 7 (- 2 x) + 7 \cdot 1$

Rewrite using the commutative property of multiplication.

$3 x^{5} - 6 x^{4} + 3 x^{3} - 9 x^{3} - 9 \cdot - 2 x \cdot x - 9 x \cdot 1 + 7 x^{2} + 7 (- 2 x) + 7 \cdot 1$

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

Move $x$ .

$3 x^{5} - 6 x^{4} + 3 x^{3} - 9 x^{3} - 9 \cdot - 2 (x \cdot x) - 9 x \cdot 1 + 7 x^{2} + 7 (- 2 x) + 7 \cdot 1$

Multiply $x$ by $x$ .

$3 x^{5} - 6 x^{4} + 3 x^{3} - 9 x^{3} - 9 \cdot - 2 x^{2} - 9 x \cdot 1 + 7 x^{2} + 7 (- 2 x) + 7 \cdot 1$

Multiply $- 9$ by $- 2$ .

$3 x^{5} - 6 x^{4} + 3 x^{3} - 9 x^{3} + 18 x^{2} - 9 x \cdot 1 + 7 x^{2} + 7 (- 2 x) + 7 \cdot 1$

Multiply $- 9$ by $1$ .

$3 x^{5} - 6 x^{4} + 3 x^{3} - 9 x^{3} + 18 x^{2} - 9 x + 7 x^{2} + 7 (- 2 x) + 7 \cdot 1$

Multiply $- 2$ by $7$ .

$3 x^{5} - 6 x^{4} + 3 x^{3} - 9 x^{3} + 18 x^{2} - 9 x + 7 x^{2} - 14 x + 7 \cdot 1$

Multiply $7$ by $1$ .

$3 x^{5} - 6 x^{4} + 3 x^{3} - 9 x^{3} + 18 x^{2} - 9 x + 7 x^{2} - 14 x + 7$

Simplify by adding terms.

Subtract $9 x^{3}$ from $3 x^{3}$ .

$3 x^{5} - 6 x^{4} - 6 x^{3} + 18 x^{2} - 9 x + 7 x^{2} - 14 x + 7$

Add $18 x^{2}$ and $7 x^{2}$ .

$3 x^{5} - 6 x^{4} - 6 x^{3} + 25 x^{2} - 9 x - 14 x + 7$

Subtract $14 x$ from $- 9 x$ .

$3 x^{5} - 6 x^{4} - 6 x^{3} + 25 x^{2} - 23 x + 7$

Find the Product (3x^3-9x+7)(x^2-2x+1)

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