Find the Degree f(x)=(x+4)^2(x-2)^3(x-1)

$f (x) = {(x + 4)}^{2} {(x - 2)}^{3} (x - 1)$

Step 1

Simplify and reorder the polynomial.

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

Multiply $4$ by $4$ .

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

Add $4 x$ and $4 x$ .

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

Use the Binomial Theorem.

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

Simplify each term.

Multiply $- 2$ by $3$ .

$(x^{2} + 8 x + 16) (x^{3} - 6 x^{2} + 3 x {(- 2)}^{2} + {(- 2)}^{3}) (x - 1)$

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

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

Multiply $4$ by $3$ .

$(x^{2} + 8 x + 16) (x^{3} - 6 x^{2} + 12 x + {(- 2)}^{3}) (x - 1)$

Raise $- 2$ to the power of $3$ .

$(x^{2} + 8 x + 16) (x^{3} - 6 x^{2} + 12 x - 8) (x - 1)$

Expand $(x^{2} + 8 x + 16) (x^{3} - 6 x^{2} + 12 x - 8)$ by multiplying each term in the first expression by each term in the second expression.

$(x^{2} x^{3} + x^{2} (- 6 x^{2}) + x^{2} (12 x) + x^{2} \cdot - 8 + 8 x \cdot x^{3} + 8 x (- 6 x^{2}) + 8 x (12 x) + 8 x \cdot - 8 + 16 x^{3} + 16 (- 6 x^{2}) + 16 (12 x) + 16 \cdot - 8) (x - 1)$

Simplify terms.

Simplify each term.

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

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

$(x^{2 + 3} + x^{2} (- 6 x^{2}) + x^{2} (12 x) + x^{2} \cdot - 8 + 8 x \cdot x^{3} + 8 x (- 6 x^{2}) + 8 x (12 x) + 8 x \cdot - 8 + 16 x^{3} + 16 (- 6 x^{2}) + 16 (12 x) + 16 \cdot - 8) (x - 1)$

Add $2$ and $3$ .

$(x^{5} + x^{2} (- 6 x^{2}) + x^{2} (12 x) + x^{2} \cdot - 8 + 8 x \cdot x^{3} + 8 x (- 6 x^{2}) + 8 x (12 x) + 8 x \cdot - 8 + 16 x^{3} + 16 (- 6 x^{2}) + 16 (12 x) + 16 \cdot - 8) (x - 1)$

Rewrite using the commutative property of multiplication.

$(x^{5} - 6 x^{2} x^{2} + x^{2} (12 x) + x^{2} \cdot - 8 + 8 x \cdot x^{3} + 8 x (- 6 x^{2}) + 8 x (12 x) + 8 x \cdot - 8 + 16 x^{3} + 16 (- 6 x^{2}) + 16 (12 x) + 16 \cdot - 8) (x - 1)$

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

Move $x^{2}$ .

$(x^{5} - 6 (x^{2} x^{2}) + x^{2} (12 x) + x^{2} \cdot - 8 + 8 x \cdot x^{3} + 8 x (- 6 x^{2}) + 8 x (12 x) + 8 x \cdot - 8 + 16 x^{3} + 16 (- 6 x^{2}) + 16 (12 x) + 16 \cdot - 8) (x - 1)$

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

$(x^{5} - 6 x^{2 + 2} + x^{2} (12 x) + x^{2} \cdot - 8 + 8 x \cdot x^{3} + 8 x (- 6 x^{2}) + 8 x (12 x) + 8 x \cdot - 8 + 16 x^{3} + 16 (- 6 x^{2}) + 16 (12 x) + 16 \cdot - 8) (x - 1)$

Add $2$ and $2$ .

$(x^{5} - 6 x^{4} + x^{2} (12 x) + x^{2} \cdot - 8 + 8 x \cdot x^{3} + 8 x (- 6 x^{2}) + 8 x (12 x) + 8 x \cdot - 8 + 16 x^{3} + 16 (- 6 x^{2}) + 16 (12 x) + 16 \cdot - 8) (x - 1)$

Rewrite using the commutative property of multiplication.

$(x^{5} - 6 x^{4} + 12 x^{2} x + x^{2} \cdot - 8 + 8 x \cdot x^{3} + 8 x (- 6 x^{2}) + 8 x (12 x) + 8 x \cdot - 8 + 16 x^{3} + 16 (- 6 x^{2}) + 16 (12 x) + 16 \cdot - 8) (x - 1)$

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

Move $x$ .

$(x^{5} - 6 x^{4} + 12 (x \cdot x^{2}) + x^{2} \cdot - 8 + 8 x \cdot x^{3} + 8 x (- 6 x^{2}) + 8 x (12 x) + 8 x \cdot - 8 + 16 x^{3} + 16 (- 6 x^{2}) + 16 (12 x) + 16 \cdot - 8) (x - 1)$

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

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

$(x^{5} - 6 x^{4} + 12 (x^{1} x^{2}) + x^{2} \cdot - 8 + 8 x \cdot x^{3} + 8 x (- 6 x^{2}) + 8 x (12 x) + 8 x \cdot - 8 + 16 x^{3} + 16 (- 6 x^{2}) + 16 (12 x) + 16 \cdot - 8) (x - 1)$

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

$(x^{5} - 6 x^{4} + 12 x^{1 + 2} + x^{2} \cdot - 8 + 8 x \cdot x^{3} + 8 x (- 6 x^{2}) + 8 x (12 x) + 8 x \cdot - 8 + 16 x^{3} + 16 (- 6 x^{2}) + 16 (12 x) + 16 \cdot - 8) (x - 1)$

Add $1$ and $2$ .

$(x^{5} - 6 x^{4} + 12 x^{3} + x^{2} \cdot - 8 + 8 x \cdot x^{3} + 8 x (- 6 x^{2}) + 8 x (12 x) + 8 x \cdot - 8 + 16 x^{3} + 16 (- 6 x^{2}) + 16 (12 x) + 16 \cdot - 8) (x - 1)$

Move $- 8$ to the left of $x^{2}$ .

$(x^{5} - 6 x^{4} + 12 x^{3} - 8 \cdot x^{2} + 8 x \cdot x^{3} + 8 x (- 6 x^{2}) + 8 x (12 x) + 8 x \cdot - 8 + 16 x^{3} + 16 (- 6 x^{2}) + 16 (12 x) + 16 \cdot - 8) (x - 1)$

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

Move $x^{3}$ .

$(x^{5} - 6 x^{4} + 12 x^{3} - 8 x^{2} + 8 (x^{3} x) + 8 x (- 6 x^{2}) + 8 x (12 x) + 8 x \cdot - 8 + 16 x^{3} + 16 (- 6 x^{2}) + 16 (12 x) + 16 \cdot - 8) (x - 1)$

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

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

$(x^{5} - 6 x^{4} + 12 x^{3} - 8 x^{2} + 8 (x^{3} x^{1}) + 8 x (- 6 x^{2}) + 8 x (12 x) + 8 x \cdot - 8 + 16 x^{3} + 16 (- 6 x^{2}) + 16 (12 x) + 16 \cdot - 8) (x - 1)$

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

$(x^{5} - 6 x^{4} + 12 x^{3} - 8 x^{2} + 8 x^{3 + 1} + 8 x (- 6 x^{2}) + 8 x (12 x) + 8 x \cdot - 8 + 16 x^{3} + 16 (- 6 x^{2}) + 16 (12 x) + 16 \cdot - 8) (x - 1)$

Add $3$ and $1$ .

$(x^{5} - 6 x^{4} + 12 x^{3} - 8 x^{2} + 8 x^{4} + 8 x (- 6 x^{2}) + 8 x (12 x) + 8 x \cdot - 8 + 16 x^{3} + 16 (- 6 x^{2}) + 16 (12 x) + 16 \cdot - 8) (x - 1)$

Rewrite using the commutative property of multiplication.

$(x^{5} - 6 x^{4} + 12 x^{3} - 8 x^{2} + 8 x^{4} + 8 \cdot - 6 x \cdot x^{2} + 8 x (12 x) + 8 x \cdot - 8 + 16 x^{3} + 16 (- 6 x^{2}) + 16 (12 x) + 16 \cdot - 8) (x - 1)$

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

Move $x^{2}$ .

$(x^{5} - 6 x^{4} + 12 x^{3} - 8 x^{2} + 8 x^{4} + 8 \cdot - 6 (x^{2} x) + 8 x (12 x) + 8 x \cdot - 8 + 16 x^{3} + 16 (- 6 x^{2}) + 16 (12 x) + 16 \cdot - 8) (x - 1)$

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

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

$(x^{5} - 6 x^{4} + 12 x^{3} - 8 x^{2} + 8 x^{4} + 8 \cdot - 6 (x^{2} x^{1}) + 8 x (12 x) + 8 x \cdot - 8 + 16 x^{3} + 16 (- 6 x^{2}) + 16 (12 x) + 16 \cdot - 8) (x - 1)$

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

$(x^{5} - 6 x^{4} + 12 x^{3} - 8 x^{2} + 8 x^{4} + 8 \cdot - 6 x^{2 + 1} + 8 x (12 x) + 8 x \cdot - 8 + 16 x^{3} + 16 (- 6 x^{2}) + 16 (12 x) + 16 \cdot - 8) (x - 1)$

Add $2$ and $1$ .

$(x^{5} - 6 x^{4} + 12 x^{3} - 8 x^{2} + 8 x^{4} + 8 \cdot - 6 x^{3} + 8 x (12 x) + 8 x \cdot - 8 + 16 x^{3} + 16 (- 6 x^{2}) + 16 (12 x) + 16 \cdot - 8) (x - 1)$

Multiply $8$ by $- 6$ .

$(x^{5} - 6 x^{4} + 12 x^{3} - 8 x^{2} + 8 x^{4} - 48 x^{3} + 8 x (12 x) + 8 x \cdot - 8 + 16 x^{3} + 16 (- 6 x^{2}) + 16 (12 x) + 16 \cdot - 8) (x - 1)$

Rewrite using the commutative property of multiplication.

$(x^{5} - 6 x^{4} + 12 x^{3} - 8 x^{2} + 8 x^{4} - 48 x^{3} + 8 \cdot 12 x \cdot x + 8 x \cdot - 8 + 16 x^{3} + 16 (- 6 x^{2}) + 16 (12 x) + 16 \cdot - 8) (x - 1)$

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

Move $x$ .

$(x^{5} - 6 x^{4} + 12 x^{3} - 8 x^{2} + 8 x^{4} - 48 x^{3} + 8 \cdot 12 (x \cdot x) + 8 x \cdot - 8 + 16 x^{3} + 16 (- 6 x^{2}) + 16 (12 x) + 16 \cdot - 8) (x - 1)$

Multiply $x$ by $x$ .

$(x^{5} - 6 x^{4} + 12 x^{3} - 8 x^{2} + 8 x^{4} - 48 x^{3} + 8 \cdot 12 x^{2} + 8 x \cdot - 8 + 16 x^{3} + 16 (- 6 x^{2}) + 16 (12 x) + 16 \cdot - 8) (x - 1)$

Multiply $8$ by $12$ .

$(x^{5} - 6 x^{4} + 12 x^{3} - 8 x^{2} + 8 x^{4} - 48 x^{3} + 96 x^{2} + 8 x \cdot - 8 + 16 x^{3} + 16 (- 6 x^{2}) + 16 (12 x) + 16 \cdot - 8) (x - 1)$

Multiply $- 8$ by $8$ .

$(x^{5} - 6 x^{4} + 12 x^{3} - 8 x^{2} + 8 x^{4} - 48 x^{3} + 96 x^{2} - 64 x + 16 x^{3} + 16 (- 6 x^{2}) + 16 (12 x) + 16 \cdot - 8) (x - 1)$

Multiply $- 6$ by $16$ .

$(x^{5} - 6 x^{4} + 12 x^{3} - 8 x^{2} + 8 x^{4} - 48 x^{3} + 96 x^{2} - 64 x + 16 x^{3} - 96 x^{2} + 16 (12 x) + 16 \cdot - 8) (x - 1)$

Multiply $12$ by $16$ .

$(x^{5} - 6 x^{4} + 12 x^{3} - 8 x^{2} + 8 x^{4} - 48 x^{3} + 96 x^{2} - 64 x + 16 x^{3} - 96 x^{2} + 192 x + 16 \cdot - 8) (x - 1)$

Multiply $16$ by $- 8$ .

$(x^{5} - 6 x^{4} + 12 x^{3} - 8 x^{2} + 8 x^{4} - 48 x^{3} + 96 x^{2} - 64 x + 16 x^{3} - 96 x^{2} + 192 x - 128) (x - 1)$

Simplify by adding terms.

Combine the opposite terms in $x^{5} - 6 x^{4} + 12 x^{3} - 8 x^{2} + 8 x^{4} - 48 x^{3} + 96 x^{2} - 64 x + 16 x^{3} - 96 x^{2} + 192 x - 128$ .

Subtract $96 x^{2}$ from $96 x^{2}$ .

$(x^{5} - 6 x^{4} + 12 x^{3} - 8 x^{2} + 8 x^{4} - 48 x^{3} - 64 x + 16 x^{3} + 0 + 192 x - 128) (x - 1)$

Add $x^{5} - 6 x^{4} + 12 x^{3} - 8 x^{2} + 8 x^{4} - 48 x^{3} - 64 x + 16 x^{3}$ and $0$ .

$(x^{5} - 6 x^{4} + 12 x^{3} - 8 x^{2} + 8 x^{4} - 48 x^{3} - 64 x + 16 x^{3} + 192 x - 128) (x - 1)$

Add $- 6 x^{4}$ and $8 x^{4}$ .

$(x^{5} + 2 x^{4} + 12 x^{3} - 8 x^{2} - 48 x^{3} - 64 x + 16 x^{3} + 192 x - 128) (x - 1)$

Subtract $48 x^{3}$ from $12 x^{3}$ .

$(x^{5} + 2 x^{4} - 36 x^{3} - 8 x^{2} - 64 x + 16 x^{3} + 192 x - 128) (x - 1)$

Add $- 36 x^{3}$ and $16 x^{3}$ .

$(x^{5} + 2 x^{4} - 20 x^{3} - 8 x^{2} - 64 x + 192 x - 128) (x - 1)$

Add $- 64 x$ and $192 x$ .

$(x^{5} + 2 x^{4} - 20 x^{3} - 8 x^{2} + 128 x - 128) (x - 1)$

Expand $(x^{5} + 2 x^{4} - 20 x^{3} - 8 x^{2} + 128 x - 128) (x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$x^{5} x + x^{5} \cdot - 1 + 2 x^{4} x + 2 x^{4} \cdot - 1 - 20 x^{3} x - 20 x^{3} \cdot - 1 - 8 x^{2} x - 8 x^{2} \cdot - 1 + 128 x \cdot x + 128 x \cdot - 1 - 128 x - 128 \cdot - 1$

Simplify terms.

Simplify each term.

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

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

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

$x^{5} x^{1} + x^{5} \cdot - 1 + 2 x^{4} x + 2 x^{4} \cdot - 1 - 20 x^{3} x - 20 x^{3} \cdot - 1 - 8 x^{2} x - 8 x^{2} \cdot - 1 + 128 x \cdot x + 128 x \cdot - 1 - 128 x - 128 \cdot - 1$

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

$x^{5 + 1} + x^{5} \cdot - 1 + 2 x^{4} x + 2 x^{4} \cdot - 1 - 20 x^{3} x - 20 x^{3} \cdot - 1 - 8 x^{2} x - 8 x^{2} \cdot - 1 + 128 x \cdot x + 128 x \cdot - 1 - 128 x - 128 \cdot - 1$

Add $5$ and $1$ .

$x^{6} + x^{5} \cdot - 1 + 2 x^{4} x + 2 x^{4} \cdot - 1 - 20 x^{3} x - 20 x^{3} \cdot - 1 - 8 x^{2} x - 8 x^{2} \cdot - 1 + 128 x \cdot x + 128 x \cdot - 1 - 128 x - 128 \cdot - 1$

Move $- 1$ to the left of $x^{5}$ .

$x^{6} - 1 \cdot x^{5} + 2 x^{4} x + 2 x^{4} \cdot - 1 - 20 x^{3} x - 20 x^{3} \cdot - 1 - 8 x^{2} x - 8 x^{2} \cdot - 1 + 128 x \cdot x + 128 x \cdot - 1 - 128 x - 128 \cdot - 1$

Rewrite $- 1 x^{5}$ as $- x^{5}$ .

$x^{6} - x^{5} + 2 x^{4} x + 2 x^{4} \cdot - 1 - 20 x^{3} x - 20 x^{3} \cdot - 1 - 8 x^{2} x - 8 x^{2} \cdot - 1 + 128 x \cdot x + 128 x \cdot - 1 - 128 x - 128 \cdot - 1$

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

Move $x$ .

$x^{6} - x^{5} + 2 (x \cdot x^{4}) + 2 x^{4} \cdot - 1 - 20 x^{3} x - 20 x^{3} \cdot - 1 - 8 x^{2} x - 8 x^{2} \cdot - 1 + 128 x \cdot x + 128 x \cdot - 1 - 128 x - 128 \cdot - 1$

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

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

$x^{6} - x^{5} + 2 (x^{1} x^{4}) + 2 x^{4} \cdot - 1 - 20 x^{3} x - 20 x^{3} \cdot - 1 - 8 x^{2} x - 8 x^{2} \cdot - 1 + 128 x \cdot x + 128 x \cdot - 1 - 128 x - 128 \cdot - 1$

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

$x^{6} - x^{5} + 2 x^{1 + 4} + 2 x^{4} \cdot - 1 - 20 x^{3} x - 20 x^{3} \cdot - 1 - 8 x^{2} x - 8 x^{2} \cdot - 1 + 128 x \cdot x + 128 x \cdot - 1 - 128 x - 128 \cdot - 1$

Add $1$ and $4$ .

$x^{6} - x^{5} + 2 x^{5} + 2 x^{4} \cdot - 1 - 20 x^{3} x - 20 x^{3} \cdot - 1 - 8 x^{2} x - 8 x^{2} \cdot - 1 + 128 x \cdot x + 128 x \cdot - 1 - 128 x - 128 \cdot - 1$

Multiply $- 1$ by $2$ .

$x^{6} - x^{5} + 2 x^{5} - 2 x^{4} - 20 x^{3} x - 20 x^{3} \cdot - 1 - 8 x^{2} x - 8 x^{2} \cdot - 1 + 128 x \cdot x + 128 x \cdot - 1 - 128 x - 128 \cdot - 1$

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

Move $x$ .

$x^{6} - x^{5} + 2 x^{5} - 2 x^{4} - 20 (x \cdot x^{3}) - 20 x^{3} \cdot - 1 - 8 x^{2} x - 8 x^{2} \cdot - 1 + 128 x \cdot x + 128 x \cdot - 1 - 128 x - 128 \cdot - 1$

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

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

$x^{6} - x^{5} + 2 x^{5} - 2 x^{4} - 20 (x^{1} x^{3}) - 20 x^{3} \cdot - 1 - 8 x^{2} x - 8 x^{2} \cdot - 1 + 128 x \cdot x + 128 x \cdot - 1 - 128 x - 128 \cdot - 1$

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

$x^{6} - x^{5} + 2 x^{5} - 2 x^{4} - 20 x^{1 + 3} - 20 x^{3} \cdot - 1 - 8 x^{2} x - 8 x^{2} \cdot - 1 + 128 x \cdot x + 128 x \cdot - 1 - 128 x - 128 \cdot - 1$

Add $1$ and $3$ .

$x^{6} - x^{5} + 2 x^{5} - 2 x^{4} - 20 x^{4} - 20 x^{3} \cdot - 1 - 8 x^{2} x - 8 x^{2} \cdot - 1 + 128 x \cdot x + 128 x \cdot - 1 - 128 x - 128 \cdot - 1$

Multiply $- 1$ by $- 20$ .

$x^{6} - x^{5} + 2 x^{5} - 2 x^{4} - 20 x^{4} + 20 x^{3} - 8 x^{2} x - 8 x^{2} \cdot - 1 + 128 x \cdot x + 128 x \cdot - 1 - 128 x - 128 \cdot - 1$

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

Move $x$ .

$x^{6} - x^{5} + 2 x^{5} - 2 x^{4} - 20 x^{4} + 20 x^{3} - 8 (x \cdot x^{2}) - 8 x^{2} \cdot - 1 + 128 x \cdot x + 128 x \cdot - 1 - 128 x - 128 \cdot - 1$

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

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

$x^{6} - x^{5} + 2 x^{5} - 2 x^{4} - 20 x^{4} + 20 x^{3} - 8 (x^{1} x^{2}) - 8 x^{2} \cdot - 1 + 128 x \cdot x + 128 x \cdot - 1 - 128 x - 128 \cdot - 1$

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

$x^{6} - x^{5} + 2 x^{5} - 2 x^{4} - 20 x^{4} + 20 x^{3} - 8 x^{1 + 2} - 8 x^{2} \cdot - 1 + 128 x \cdot x + 128 x \cdot - 1 - 128 x - 128 \cdot - 1$

Add $1$ and $2$ .

$x^{6} - x^{5} + 2 x^{5} - 2 x^{4} - 20 x^{4} + 20 x^{3} - 8 x^{3} - 8 x^{2} \cdot - 1 + 128 x \cdot x + 128 x \cdot - 1 - 128 x - 128 \cdot - 1$

Multiply $- 1$ by $- 8$ .

$x^{6} - x^{5} + 2 x^{5} - 2 x^{4} - 20 x^{4} + 20 x^{3} - 8 x^{3} + 8 x^{2} + 128 x \cdot x + 128 x \cdot - 1 - 128 x - 128 \cdot - 1$

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

Move $x$ .

$x^{6} - x^{5} + 2 x^{5} - 2 x^{4} - 20 x^{4} + 20 x^{3} - 8 x^{3} + 8 x^{2} + 128 (x \cdot x) + 128 x \cdot - 1 - 128 x - 128 \cdot - 1$

Multiply $x$ by $x$ .

$x^{6} - x^{5} + 2 x^{5} - 2 x^{4} - 20 x^{4} + 20 x^{3} - 8 x^{3} + 8 x^{2} + 128 x^{2} + 128 x \cdot - 1 - 128 x - 128 \cdot - 1$

Multiply $- 1$ by $128$ .

$x^{6} - x^{5} + 2 x^{5} - 2 x^{4} - 20 x^{4} + 20 x^{3} - 8 x^{3} + 8 x^{2} + 128 x^{2} - 128 x - 128 x - 128 \cdot - 1$

Multiply $- 128$ by $- 1$ .

$x^{6} - x^{5} + 2 x^{5} - 2 x^{4} - 20 x^{4} + 20 x^{3} - 8 x^{3} + 8 x^{2} + 128 x^{2} - 128 x - 128 x + 128$

Simplify by adding terms.

Add $- x^{5}$ and $2 x^{5}$ .

$x^{6} + x^{5} - 2 x^{4} - 20 x^{4} + 20 x^{3} - 8 x^{3} + 8 x^{2} + 128 x^{2} - 128 x - 128 x + 128$

Subtract $20 x^{4}$ from $- 2 x^{4}$ .

$x^{6} + x^{5} - 22 x^{4} + 20 x^{3} - 8 x^{3} + 8 x^{2} + 128 x^{2} - 128 x - 128 x + 128$

Subtract $8 x^{3}$ from $20 x^{3}$ .

$x^{6} + x^{5} - 22 x^{4} + 12 x^{3} + 8 x^{2} + 128 x^{2} - 128 x - 128 x + 128$

Add $8 x^{2}$ and $128 x^{2}$ .

$x^{6} + x^{5} - 22 x^{4} + 12 x^{3} + 136 x^{2} - 128 x - 128 x + 128$

Subtract $128 x$ from $- 128 x$ .

$x^{6} + x^{5} - 22 x^{4} + 12 x^{3} + 136 x^{2} - 256 x + 128$

Step 2

The largest exponent is the degree of the polynomial.

$6$

Find the Degree f(x)=(x+4)^2(x-2)^3(x-1)

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