Find the Degree f(x)=(x+4)^2(x-1)^5(x-2)

$f (x) = {(x + 4)}^{2} {(x - 1)}^{5} (x - 2)$

Step 1

Simplify and reorder the polynomial.

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

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

Expand $(x + 4) (x + 4)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $4$ to the left of $x$ .

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

Multiply $4$ by $4$ .

$(x^{2} + 4 x + 4 x + 16) {(x - 1)}^{5} (x - 2)$

Add $4 x$ and $4 x$ .

$(x^{2} + 8 x + 16) {(x - 1)}^{5} (x - 2)$

Use the Binomial Theorem.

$(x^{2} + 8 x + 16) (x^{5} + 5 x^{4} \cdot - 1 + 10 x^{3} {(- 1)}^{2} + 10 x^{2} {(- 1)}^{3} + 5 x {(- 1)}^{4} + {(- 1)}^{5}) (x - 2)$

Simplify each term.

Multiply $- 1$ by $5$ .

$(x^{2} + 8 x + 16) (x^{5} - 5 x^{4} + 10 x^{3} {(- 1)}^{2} + 10 x^{2} {(- 1)}^{3} + 5 x {(- 1)}^{4} + {(- 1)}^{5}) (x - 2)$

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

$(x^{2} + 8 x + 16) (x^{5} - 5 x^{4} + 10 x^{3} \cdot 1 + 10 x^{2} {(- 1)}^{3} + 5 x {(- 1)}^{4} + {(- 1)}^{5}) (x - 2)$

Multiply $10$ by $1$ .

$(x^{2} + 8 x + 16) (x^{5} - 5 x^{4} + 10 x^{3} + 10 x^{2} {(- 1)}^{3} + 5 x {(- 1)}^{4} + {(- 1)}^{5}) (x - 2)$

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

$(x^{2} + 8 x + 16) (x^{5} - 5 x^{4} + 10 x^{3} + 10 x^{2} \cdot - 1 + 5 x {(- 1)}^{4} + {(- 1)}^{5}) (x - 2)$

Multiply $- 1$ by $10$ .

$(x^{2} + 8 x + 16) (x^{5} - 5 x^{4} + 10 x^{3} - 10 x^{2} + 5 x {(- 1)}^{4} + {(- 1)}^{5}) (x - 2)$

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

$(x^{2} + 8 x + 16) (x^{5} - 5 x^{4} + 10 x^{3} - 10 x^{2} + 5 x \cdot 1 + {(- 1)}^{5}) (x - 2)$

Multiply $5$ by $1$ .

$(x^{2} + 8 x + 16) (x^{5} - 5 x^{4} + 10 x^{3} - 10 x^{2} + 5 x + {(- 1)}^{5}) (x - 2)$

Raise $- 1$ to the power of $5$ .

$(x^{2} + 8 x + 16) (x^{5} - 5 x^{4} + 10 x^{3} - 10 x^{2} + 5 x - 1) (x - 2)$

Expand $(x^{2} + 8 x + 16) (x^{5} - 5 x^{4} + 10 x^{3} - 10 x^{2} + 5 x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$(x^{2} x^{5} + x^{2} (- 5 x^{4}) + x^{2} (10 x^{3}) + x^{2} (- 10 x^{2}) + x^{2} (5 x) + x^{2} \cdot - 1 + 8 x \cdot x^{5} + 8 x (- 5 x^{4}) + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Simplify terms.

Simplify each term.

Multiply $x^{2}$ by $x^{5}$ by adding the exponents.

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

$(x^{2 + 5} + x^{2} (- 5 x^{4}) + x^{2} (10 x^{3}) + x^{2} (- 10 x^{2}) + x^{2} (5 x) + x^{2} \cdot - 1 + 8 x \cdot x^{5} + 8 x (- 5 x^{4}) + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Add $2$ and $5$ .

$(x^{7} + x^{2} (- 5 x^{4}) + x^{2} (10 x^{3}) + x^{2} (- 10 x^{2}) + x^{2} (5 x) + x^{2} \cdot - 1 + 8 x \cdot x^{5} + 8 x (- 5 x^{4}) + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Rewrite using the commutative property of multiplication.

$(x^{7} - 5 x^{2} x^{4} + x^{2} (10 x^{3}) + x^{2} (- 10 x^{2}) + x^{2} (5 x) + x^{2} \cdot - 1 + 8 x \cdot x^{5} + 8 x (- 5 x^{4}) + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Multiply $x^{2}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$(x^{7} - 5 (x^{4} x^{2}) + x^{2} (10 x^{3}) + x^{2} (- 10 x^{2}) + x^{2} (5 x) + x^{2} \cdot - 1 + 8 x \cdot x^{5} + 8 x (- 5 x^{4}) + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

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

$(x^{7} - 5 x^{4 + 2} + x^{2} (10 x^{3}) + x^{2} (- 10 x^{2}) + x^{2} (5 x) + x^{2} \cdot - 1 + 8 x \cdot x^{5} + 8 x (- 5 x^{4}) + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Add $4$ and $2$ .

$(x^{7} - 5 x^{6} + x^{2} (10 x^{3}) + x^{2} (- 10 x^{2}) + x^{2} (5 x) + x^{2} \cdot - 1 + 8 x \cdot x^{5} + 8 x (- 5 x^{4}) + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Rewrite using the commutative property of multiplication.

$(x^{7} - 5 x^{6} + 10 x^{2} x^{3} + x^{2} (- 10 x^{2}) + x^{2} (5 x) + x^{2} \cdot - 1 + 8 x \cdot x^{5} + 8 x (- 5 x^{4}) + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

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

Move $x^{3}$ .

$(x^{7} - 5 x^{6} + 10 (x^{3} x^{2}) + x^{2} (- 10 x^{2}) + x^{2} (5 x) + x^{2} \cdot - 1 + 8 x \cdot x^{5} + 8 x (- 5 x^{4}) + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

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

$(x^{7} - 5 x^{6} + 10 x^{3 + 2} + x^{2} (- 10 x^{2}) + x^{2} (5 x) + x^{2} \cdot - 1 + 8 x \cdot x^{5} + 8 x (- 5 x^{4}) + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Add $3$ and $2$ .

$(x^{7} - 5 x^{6} + 10 x^{5} + x^{2} (- 10 x^{2}) + x^{2} (5 x) + x^{2} \cdot - 1 + 8 x \cdot x^{5} + 8 x (- 5 x^{4}) + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Rewrite using the commutative property of multiplication.

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{2} x^{2} + x^{2} (5 x) + x^{2} \cdot - 1 + 8 x \cdot x^{5} + 8 x (- 5 x^{4}) + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

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

Move $x^{2}$ .

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 (x^{2} x^{2}) + x^{2} (5 x) + x^{2} \cdot - 1 + 8 x \cdot x^{5} + 8 x (- 5 x^{4}) + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

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

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{2 + 2} + x^{2} (5 x) + x^{2} \cdot - 1 + 8 x \cdot x^{5} + 8 x (- 5 x^{4}) + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Add $2$ and $2$ .

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + x^{2} (5 x) + x^{2} \cdot - 1 + 8 x \cdot x^{5} + 8 x (- 5 x^{4}) + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Rewrite using the commutative property of multiplication.

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{2} x + x^{2} \cdot - 1 + 8 x \cdot x^{5} + 8 x (- 5 x^{4}) + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

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

Move $x$ .

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 (x \cdot x^{2}) + x^{2} \cdot - 1 + 8 x \cdot x^{5} + 8 x (- 5 x^{4}) + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

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

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

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 (x^{1} x^{2}) + x^{2} \cdot - 1 + 8 x \cdot x^{5} + 8 x (- 5 x^{4}) + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

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

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{1 + 2} + x^{2} \cdot - 1 + 8 x \cdot x^{5} + 8 x (- 5 x^{4}) + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Add $1$ and $2$ .

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} + x^{2} \cdot - 1 + 8 x \cdot x^{5} + 8 x (- 5 x^{4}) + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Move $- 1$ to the left of $x^{2}$ .

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - 1 \cdot x^{2} + 8 x \cdot x^{5} + 8 x (- 5 x^{4}) + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x \cdot x^{5} + 8 x (- 5 x^{4}) + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Multiply $x$ by $x^{5}$ by adding the exponents.

Move $x^{5}$ .

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 (x^{5} x) + 8 x (- 5 x^{4}) + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Multiply $x^{5}$ by $x$ .

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

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 (x^{5} x^{1}) + 8 x (- 5 x^{4}) + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

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

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{5 + 1} + 8 x (- 5 x^{4}) + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Add $5$ and $1$ .

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} + 8 x (- 5 x^{4}) + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Rewrite using the commutative property of multiplication.

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} + 8 \cdot - 5 x \cdot x^{4} + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Multiply $x$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} + 8 \cdot - 5 (x^{4} x) + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Multiply $x^{4}$ by $x$ .

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

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} + 8 \cdot - 5 (x^{4} x^{1}) + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

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

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} + 8 \cdot - 5 x^{4 + 1} + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Add $4$ and $1$ .

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} + 8 \cdot - 5 x^{5} + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Multiply $8$ by $- 5$ .

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} - 40 x^{5} + 8 x (10 x^{3}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Rewrite using the commutative property of multiplication.

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} - 40 x^{5} + 8 \cdot 10 x \cdot x^{3} + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

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

Move $x^{3}$ .

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} - 40 x^{5} + 8 \cdot 10 (x^{3} x) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

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

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

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} - 40 x^{5} + 8 \cdot 10 (x^{3} x^{1}) + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

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

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} - 40 x^{5} + 8 \cdot 10 x^{3 + 1} + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Add $3$ and $1$ .

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} - 40 x^{5} + 8 \cdot 10 x^{4} + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Multiply $8$ by $10$ .

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} - 40 x^{5} + 80 x^{4} + 8 x (- 10 x^{2}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Rewrite using the commutative property of multiplication.

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} - 40 x^{5} + 80 x^{4} + 8 \cdot - 10 x \cdot x^{2} + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

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

Move $x^{2}$ .

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} - 40 x^{5} + 80 x^{4} + 8 \cdot - 10 (x^{2} x) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

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

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

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} - 40 x^{5} + 80 x^{4} + 8 \cdot - 10 (x^{2} x^{1}) + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

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

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} - 40 x^{5} + 80 x^{4} + 8 \cdot - 10 x^{2 + 1} + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Add $2$ and $1$ .

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} - 40 x^{5} + 80 x^{4} + 8 \cdot - 10 x^{3} + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Multiply $8$ by $- 10$ .

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} - 40 x^{5} + 80 x^{4} - 80 x^{3} + 8 x (5 x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Rewrite using the commutative property of multiplication.

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} - 40 x^{5} + 80 x^{4} - 80 x^{3} + 8 \cdot 5 x \cdot x + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

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

Move $x$ .

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} - 40 x^{5} + 80 x^{4} - 80 x^{3} + 8 \cdot 5 (x \cdot x) + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Multiply $x$ by $x$ .

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} - 40 x^{5} + 80 x^{4} - 80 x^{3} + 8 \cdot 5 x^{2} + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Multiply $8$ by $5$ .

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} - 40 x^{5} + 80 x^{4} - 80 x^{3} + 40 x^{2} + 8 x \cdot - 1 + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Multiply $- 1$ by $8$ .

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} - 40 x^{5} + 80 x^{4} - 80 x^{3} + 40 x^{2} - 8 x + 16 x^{5} + 16 (- 5 x^{4}) + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Multiply $- 5$ by $16$ .

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} - 40 x^{5} + 80 x^{4} - 80 x^{3} + 40 x^{2} - 8 x + 16 x^{5} - 80 x^{4} + 16 (10 x^{3}) + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Multiply $10$ by $16$ .

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} - 40 x^{5} + 80 x^{4} - 80 x^{3} + 40 x^{2} - 8 x + 16 x^{5} - 80 x^{4} + 160 x^{3} + 16 (- 10 x^{2}) + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Multiply $- 10$ by $16$ .

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} - 40 x^{5} + 80 x^{4} - 80 x^{3} + 40 x^{2} - 8 x + 16 x^{5} - 80 x^{4} + 160 x^{3} - 160 x^{2} + 16 (5 x) + 16 \cdot - 1) (x - 2)$

Multiply $5$ by $16$ .

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} - 40 x^{5} + 80 x^{4} - 80 x^{3} + 40 x^{2} - 8 x + 16 x^{5} - 80 x^{4} + 160 x^{3} - 160 x^{2} + 80 x + 16 \cdot - 1) (x - 2)$

Multiply $16$ by $- 1$ .

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} - 40 x^{5} + 80 x^{4} - 80 x^{3} + 40 x^{2} - 8 x + 16 x^{5} - 80 x^{4} + 160 x^{3} - 160 x^{2} + 80 x - 16) (x - 2)$

Simplify by adding terms.

Combine the opposite terms in $x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} - 40 x^{5} + 80 x^{4} - 80 x^{3} + 40 x^{2} - 8 x + 16 x^{5} - 80 x^{4} + 160 x^{3} - 160 x^{2} + 80 x - 16$ .

Subtract $80 x^{4}$ from $80 x^{4}$ .

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} - 40 x^{5} - 80 x^{3} + 40 x^{2} - 8 x + 16 x^{5} + 0 + 160 x^{3} - 160 x^{2} + 80 x - 16) (x - 2)$

Add $x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} - 40 x^{5} - 80 x^{3} + 40 x^{2} - 8 x + 16 x^{5}$ and $0$ .

$(x^{7} - 5 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} + 8 x^{6} - 40 x^{5} - 80 x^{3} + 40 x^{2} - 8 x + 16 x^{5} + 160 x^{3} - 160 x^{2} + 80 x - 16) (x - 2)$

Add $- 5 x^{6}$ and $8 x^{6}$ .

$(x^{7} + 3 x^{6} + 10 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} - 40 x^{5} - 80 x^{3} + 40 x^{2} - 8 x + 16 x^{5} + 160 x^{3} - 160 x^{2} + 80 x - 16) (x - 2)$

Subtract $40 x^{5}$ from $10 x^{5}$ .

$(x^{7} + 3 x^{6} - 30 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} - 80 x^{3} + 40 x^{2} - 8 x + 16 x^{5} + 160 x^{3} - 160 x^{2} + 80 x - 16) (x - 2)$

Add $- 30 x^{5}$ and $16 x^{5}$ .

$(x^{7} + 3 x^{6} - 14 x^{5} - 10 x^{4} + 5 x^{3} - x^{2} - 80 x^{3} + 40 x^{2} - 8 x + 160 x^{3} - 160 x^{2} + 80 x - 16) (x - 2)$

Subtract $80 x^{3}$ from $5 x^{3}$ .

$(x^{7} + 3 x^{6} - 14 x^{5} - 10 x^{4} - 75 x^{3} - x^{2} + 40 x^{2} - 8 x + 160 x^{3} - 160 x^{2} + 80 x - 16) (x - 2)$

Add $- 75 x^{3}$ and $160 x^{3}$ .

$(x^{7} + 3 x^{6} - 14 x^{5} - 10 x^{4} + 85 x^{3} - x^{2} + 40 x^{2} - 8 x - 160 x^{2} + 80 x - 16) (x - 2)$

Add $- x^{2}$ and $40 x^{2}$ .

$(x^{7} + 3 x^{6} - 14 x^{5} - 10 x^{4} + 85 x^{3} + 39 x^{2} - 8 x - 160 x^{2} + 80 x - 16) (x - 2)$

Subtract $160 x^{2}$ from $39 x^{2}$ .

$(x^{7} + 3 x^{6} - 14 x^{5} - 10 x^{4} + 85 x^{3} - 121 x^{2} - 8 x + 80 x - 16) (x - 2)$

Add $- 8 x$ and $80 x$ .

$(x^{7} + 3 x^{6} - 14 x^{5} - 10 x^{4} + 85 x^{3} - 121 x^{2} + 72 x - 16) (x - 2)$

Expand $(x^{7} + 3 x^{6} - 14 x^{5} - 10 x^{4} + 85 x^{3} - 121 x^{2} + 72 x - 16) (x - 2)$ by multiplying each term in the first expression by each term in the second expression.

$x^{7} x + x^{7} \cdot - 2 + 3 x^{6} x + 3 x^{6} \cdot - 2 - 14 x^{5} x - 14 x^{5} \cdot - 2 - 10 x^{4} x - 10 x^{4} \cdot - 2 + 85 x^{3} x + 85 x^{3} \cdot - 2 - 121 x^{2} x - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

Simplify terms.

Simplify each term.

Multiply $x^{7}$ by $x$ by adding the exponents.

Multiply $x^{7}$ by $x$ .

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

$x^{7} x^{1} + x^{7} \cdot - 2 + 3 x^{6} x + 3 x^{6} \cdot - 2 - 14 x^{5} x - 14 x^{5} \cdot - 2 - 10 x^{4} x - 10 x^{4} \cdot - 2 + 85 x^{3} x + 85 x^{3} \cdot - 2 - 121 x^{2} x - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

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

$x^{7 + 1} + x^{7} \cdot - 2 + 3 x^{6} x + 3 x^{6} \cdot - 2 - 14 x^{5} x - 14 x^{5} \cdot - 2 - 10 x^{4} x - 10 x^{4} \cdot - 2 + 85 x^{3} x + 85 x^{3} \cdot - 2 - 121 x^{2} x - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

Add $7$ and $1$ .

$x^{8} + x^{7} \cdot - 2 + 3 x^{6} x + 3 x^{6} \cdot - 2 - 14 x^{5} x - 14 x^{5} \cdot - 2 - 10 x^{4} x - 10 x^{4} \cdot - 2 + 85 x^{3} x + 85 x^{3} \cdot - 2 - 121 x^{2} x - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

Move $- 2$ to the left of $x^{7}$ .

$x^{8} - 2 \cdot x^{7} + 3 x^{6} x + 3 x^{6} \cdot - 2 - 14 x^{5} x - 14 x^{5} \cdot - 2 - 10 x^{4} x - 10 x^{4} \cdot - 2 + 85 x^{3} x + 85 x^{3} \cdot - 2 - 121 x^{2} x - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

Multiply $x^{6}$ by $x$ by adding the exponents.

Move $x$ .

$x^{8} - 2 x^{7} + 3 (x \cdot x^{6}) + 3 x^{6} \cdot - 2 - 14 x^{5} x - 14 x^{5} \cdot - 2 - 10 x^{4} x - 10 x^{4} \cdot - 2 + 85 x^{3} x + 85 x^{3} \cdot - 2 - 121 x^{2} x - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

Multiply $x$ by $x^{6}$ .

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

$x^{8} - 2 x^{7} + 3 (x^{1} x^{6}) + 3 x^{6} \cdot - 2 - 14 x^{5} x - 14 x^{5} \cdot - 2 - 10 x^{4} x - 10 x^{4} \cdot - 2 + 85 x^{3} x + 85 x^{3} \cdot - 2 - 121 x^{2} x - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

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

$x^{8} - 2 x^{7} + 3 x^{1 + 6} + 3 x^{6} \cdot - 2 - 14 x^{5} x - 14 x^{5} \cdot - 2 - 10 x^{4} x - 10 x^{4} \cdot - 2 + 85 x^{3} x + 85 x^{3} \cdot - 2 - 121 x^{2} x - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

Add $1$ and $6$ .

$x^{8} - 2 x^{7} + 3 x^{7} + 3 x^{6} \cdot - 2 - 14 x^{5} x - 14 x^{5} \cdot - 2 - 10 x^{4} x - 10 x^{4} \cdot - 2 + 85 x^{3} x + 85 x^{3} \cdot - 2 - 121 x^{2} x - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

Multiply $- 2$ by $3$ .

$x^{8} - 2 x^{7} + 3 x^{7} - 6 x^{6} - 14 x^{5} x - 14 x^{5} \cdot - 2 - 10 x^{4} x - 10 x^{4} \cdot - 2 + 85 x^{3} x + 85 x^{3} \cdot - 2 - 121 x^{2} x - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

Multiply $x^{5}$ by $x$ by adding the exponents.

Move $x$ .

$x^{8} - 2 x^{7} + 3 x^{7} - 6 x^{6} - 14 (x \cdot x^{5}) - 14 x^{5} \cdot - 2 - 10 x^{4} x - 10 x^{4} \cdot - 2 + 85 x^{3} x + 85 x^{3} \cdot - 2 - 121 x^{2} x - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

Multiply $x$ by $x^{5}$ .

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

$x^{8} - 2 x^{7} + 3 x^{7} - 6 x^{6} - 14 (x^{1} x^{5}) - 14 x^{5} \cdot - 2 - 10 x^{4} x - 10 x^{4} \cdot - 2 + 85 x^{3} x + 85 x^{3} \cdot - 2 - 121 x^{2} x - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

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

$x^{8} - 2 x^{7} + 3 x^{7} - 6 x^{6} - 14 x^{1 + 5} - 14 x^{5} \cdot - 2 - 10 x^{4} x - 10 x^{4} \cdot - 2 + 85 x^{3} x + 85 x^{3} \cdot - 2 - 121 x^{2} x - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

Add $1$ and $5$ .

$x^{8} - 2 x^{7} + 3 x^{7} - 6 x^{6} - 14 x^{6} - 14 x^{5} \cdot - 2 - 10 x^{4} x - 10 x^{4} \cdot - 2 + 85 x^{3} x + 85 x^{3} \cdot - 2 - 121 x^{2} x - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

Multiply $- 2$ by $- 14$ .

$x^{8} - 2 x^{7} + 3 x^{7} - 6 x^{6} - 14 x^{6} + 28 x^{5} - 10 x^{4} x - 10 x^{4} \cdot - 2 + 85 x^{3} x + 85 x^{3} \cdot - 2 - 121 x^{2} x - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

Multiply $x^{4}$ by $x$ by adding the exponents.

Move $x$ .

$x^{8} - 2 x^{7} + 3 x^{7} - 6 x^{6} - 14 x^{6} + 28 x^{5} - 10 (x \cdot x^{4}) - 10 x^{4} \cdot - 2 + 85 x^{3} x + 85 x^{3} \cdot - 2 - 121 x^{2} x - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

Multiply $x$ by $x^{4}$ .

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

$x^{8} - 2 x^{7} + 3 x^{7} - 6 x^{6} - 14 x^{6} + 28 x^{5} - 10 (x^{1} x^{4}) - 10 x^{4} \cdot - 2 + 85 x^{3} x + 85 x^{3} \cdot - 2 - 121 x^{2} x - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

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

$x^{8} - 2 x^{7} + 3 x^{7} - 6 x^{6} - 14 x^{6} + 28 x^{5} - 10 x^{1 + 4} - 10 x^{4} \cdot - 2 + 85 x^{3} x + 85 x^{3} \cdot - 2 - 121 x^{2} x - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

Add $1$ and $4$ .

$x^{8} - 2 x^{7} + 3 x^{7} - 6 x^{6} - 14 x^{6} + 28 x^{5} - 10 x^{5} - 10 x^{4} \cdot - 2 + 85 x^{3} x + 85 x^{3} \cdot - 2 - 121 x^{2} x - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

Multiply $- 2$ by $- 10$ .

$x^{8} - 2 x^{7} + 3 x^{7} - 6 x^{6} - 14 x^{6} + 28 x^{5} - 10 x^{5} + 20 x^{4} + 85 x^{3} x + 85 x^{3} \cdot - 2 - 121 x^{2} x - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

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

Move $x$ .

$x^{8} - 2 x^{7} + 3 x^{7} - 6 x^{6} - 14 x^{6} + 28 x^{5} - 10 x^{5} + 20 x^{4} + 85 (x \cdot x^{3}) + 85 x^{3} \cdot - 2 - 121 x^{2} x - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

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

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

$x^{8} - 2 x^{7} + 3 x^{7} - 6 x^{6} - 14 x^{6} + 28 x^{5} - 10 x^{5} + 20 x^{4} + 85 (x^{1} x^{3}) + 85 x^{3} \cdot - 2 - 121 x^{2} x - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

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

$x^{8} - 2 x^{7} + 3 x^{7} - 6 x^{6} - 14 x^{6} + 28 x^{5} - 10 x^{5} + 20 x^{4} + 85 x^{1 + 3} + 85 x^{3} \cdot - 2 - 121 x^{2} x - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

Add $1$ and $3$ .

$x^{8} - 2 x^{7} + 3 x^{7} - 6 x^{6} - 14 x^{6} + 28 x^{5} - 10 x^{5} + 20 x^{4} + 85 x^{4} + 85 x^{3} \cdot - 2 - 121 x^{2} x - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

Multiply $- 2$ by $85$ .

$x^{8} - 2 x^{7} + 3 x^{7} - 6 x^{6} - 14 x^{6} + 28 x^{5} - 10 x^{5} + 20 x^{4} + 85 x^{4} - 170 x^{3} - 121 x^{2} x - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

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

Move $x$ .

$x^{8} - 2 x^{7} + 3 x^{7} - 6 x^{6} - 14 x^{6} + 28 x^{5} - 10 x^{5} + 20 x^{4} + 85 x^{4} - 170 x^{3} - 121 (x \cdot x^{2}) - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

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

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

$x^{8} - 2 x^{7} + 3 x^{7} - 6 x^{6} - 14 x^{6} + 28 x^{5} - 10 x^{5} + 20 x^{4} + 85 x^{4} - 170 x^{3} - 121 (x^{1} x^{2}) - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

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

$x^{8} - 2 x^{7} + 3 x^{7} - 6 x^{6} - 14 x^{6} + 28 x^{5} - 10 x^{5} + 20 x^{4} + 85 x^{4} - 170 x^{3} - 121 x^{1 + 2} - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

Add $1$ and $2$ .

$x^{8} - 2 x^{7} + 3 x^{7} - 6 x^{6} - 14 x^{6} + 28 x^{5} - 10 x^{5} + 20 x^{4} + 85 x^{4} - 170 x^{3} - 121 x^{3} - 121 x^{2} \cdot - 2 + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

Multiply $- 2$ by $- 121$ .

$x^{8} - 2 x^{7} + 3 x^{7} - 6 x^{6} - 14 x^{6} + 28 x^{5} - 10 x^{5} + 20 x^{4} + 85 x^{4} - 170 x^{3} - 121 x^{3} + 242 x^{2} + 72 x \cdot x + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

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

Move $x$ .

$x^{8} - 2 x^{7} + 3 x^{7} - 6 x^{6} - 14 x^{6} + 28 x^{5} - 10 x^{5} + 20 x^{4} + 85 x^{4} - 170 x^{3} - 121 x^{3} + 242 x^{2} + 72 (x \cdot x) + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

Multiply $x$ by $x$ .

$x^{8} - 2 x^{7} + 3 x^{7} - 6 x^{6} - 14 x^{6} + 28 x^{5} - 10 x^{5} + 20 x^{4} + 85 x^{4} - 170 x^{3} - 121 x^{3} + 242 x^{2} + 72 x^{2} + 72 x \cdot - 2 - 16 x - 16 \cdot - 2$

Multiply $- 2$ by $72$ .

$x^{8} - 2 x^{7} + 3 x^{7} - 6 x^{6} - 14 x^{6} + 28 x^{5} - 10 x^{5} + 20 x^{4} + 85 x^{4} - 170 x^{3} - 121 x^{3} + 242 x^{2} + 72 x^{2} - 144 x - 16 x - 16 \cdot - 2$

Multiply $- 16$ by $- 2$ .

$x^{8} - 2 x^{7} + 3 x^{7} - 6 x^{6} - 14 x^{6} + 28 x^{5} - 10 x^{5} + 20 x^{4} + 85 x^{4} - 170 x^{3} - 121 x^{3} + 242 x^{2} + 72 x^{2} - 144 x - 16 x + 32$

Simplify by adding terms.

Add $- 2 x^{7}$ and $3 x^{7}$ .

$x^{8} + x^{7} - 6 x^{6} - 14 x^{6} + 28 x^{5} - 10 x^{5} + 20 x^{4} + 85 x^{4} - 170 x^{3} - 121 x^{3} + 242 x^{2} + 72 x^{2} - 144 x - 16 x + 32$

Subtract $14 x^{6}$ from $- 6 x^{6}$ .

$x^{8} + x^{7} - 20 x^{6} + 28 x^{5} - 10 x^{5} + 20 x^{4} + 85 x^{4} - 170 x^{3} - 121 x^{3} + 242 x^{2} + 72 x^{2} - 144 x - 16 x + 32$

Subtract $10 x^{5}$ from $28 x^{5}$ .

$x^{8} + x^{7} - 20 x^{6} + 18 x^{5} + 20 x^{4} + 85 x^{4} - 170 x^{3} - 121 x^{3} + 242 x^{2} + 72 x^{2} - 144 x - 16 x + 32$

Add $20 x^{4}$ and $85 x^{4}$ .

$x^{8} + x^{7} - 20 x^{6} + 18 x^{5} + 105 x^{4} - 170 x^{3} - 121 x^{3} + 242 x^{2} + 72 x^{2} - 144 x - 16 x + 32$

Subtract $121 x^{3}$ from $- 170 x^{3}$ .

$x^{8} + x^{7} - 20 x^{6} + 18 x^{5} + 105 x^{4} - 291 x^{3} + 242 x^{2} + 72 x^{2} - 144 x - 16 x + 32$

Add $242 x^{2}$ and $72 x^{2}$ .

$x^{8} + x^{7} - 20 x^{6} + 18 x^{5} + 105 x^{4} - 291 x^{3} + 314 x^{2} - 144 x - 16 x + 32$

Subtract $16 x$ from $- 144 x$ .

$x^{8} + x^{7} - 20 x^{6} + 18 x^{5} + 105 x^{4} - 291 x^{3} + 314 x^{2} - 160 x + 32$

Step 2

The largest exponent is the degree of the polynomial.

$8$

Find the Degree f(x)=(x+4)^2(x-1)^5(x-2)

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