Find the Degree f(x)=(x+4)(x-1)^7

$f (x) = (x + 4) {(x - 1)}^{7}$

Step 1

Simplify and reorder the polynomial.

Use the Binomial Theorem.

$(x + 4) (x^{7} + 7 x^{6} \cdot - 1 + 21 x^{5} {(- 1)}^{2} + 35 x^{4} {(- 1)}^{3} + 35 x^{3} {(- 1)}^{4} + 21 x^{2} {(- 1)}^{5} + 7 x {(- 1)}^{6} + {(- 1)}^{7})$

Simplify each term.

Multiply $- 1$ by $7$ .

$(x + 4) (x^{7} - 7 x^{6} + 21 x^{5} {(- 1)}^{2} + 35 x^{4} {(- 1)}^{3} + 35 x^{3} {(- 1)}^{4} + 21 x^{2} {(- 1)}^{5} + 7 x {(- 1)}^{6} + {(- 1)}^{7})$

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

$(x + 4) (x^{7} - 7 x^{6} + 21 x^{5} \cdot 1 + 35 x^{4} {(- 1)}^{3} + 35 x^{3} {(- 1)}^{4} + 21 x^{2} {(- 1)}^{5} + 7 x {(- 1)}^{6} + {(- 1)}^{7})$

Multiply $21$ by $1$ .

$(x + 4) (x^{7} - 7 x^{6} + 21 x^{5} + 35 x^{4} {(- 1)}^{3} + 35 x^{3} {(- 1)}^{4} + 21 x^{2} {(- 1)}^{5} + 7 x {(- 1)}^{6} + {(- 1)}^{7})$

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

$(x + 4) (x^{7} - 7 x^{6} + 21 x^{5} + 35 x^{4} \cdot - 1 + 35 x^{3} {(- 1)}^{4} + 21 x^{2} {(- 1)}^{5} + 7 x {(- 1)}^{6} + {(- 1)}^{7})$

Multiply $- 1$ by $35$ .

$(x + 4) (x^{7} - 7 x^{6} + 21 x^{5} - 35 x^{4} + 35 x^{3} {(- 1)}^{4} + 21 x^{2} {(- 1)}^{5} + 7 x {(- 1)}^{6} + {(- 1)}^{7})$

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

$(x + 4) (x^{7} - 7 x^{6} + 21 x^{5} - 35 x^{4} + 35 x^{3} \cdot 1 + 21 x^{2} {(- 1)}^{5} + 7 x {(- 1)}^{6} + {(- 1)}^{7})$

Multiply $35$ by $1$ .

$(x + 4) (x^{7} - 7 x^{6} + 21 x^{5} - 35 x^{4} + 35 x^{3} + 21 x^{2} {(- 1)}^{5} + 7 x {(- 1)}^{6} + {(- 1)}^{7})$

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

$(x + 4) (x^{7} - 7 x^{6} + 21 x^{5} - 35 x^{4} + 35 x^{3} + 21 x^{2} \cdot - 1 + 7 x {(- 1)}^{6} + {(- 1)}^{7})$

Multiply $- 1$ by $21$ .

$(x + 4) (x^{7} - 7 x^{6} + 21 x^{5} - 35 x^{4} + 35 x^{3} - 21 x^{2} + 7 x {(- 1)}^{6} + {(- 1)}^{7})$

Raise $- 1$ to the power of $6$ .

$(x + 4) (x^{7} - 7 x^{6} + 21 x^{5} - 35 x^{4} + 35 x^{3} - 21 x^{2} + 7 x \cdot 1 + {(- 1)}^{7})$

Multiply $7$ by $1$ .

$(x + 4) (x^{7} - 7 x^{6} + 21 x^{5} - 35 x^{4} + 35 x^{3} - 21 x^{2} + 7 x + {(- 1)}^{7})$

Raise $- 1$ to the power of $7$ .

$(x + 4) (x^{7} - 7 x^{6} + 21 x^{5} - 35 x^{4} + 35 x^{3} - 21 x^{2} + 7 x - 1)$

Expand $(x + 4) (x^{7} - 7 x^{6} + 21 x^{5} - 35 x^{4} + 35 x^{3} - 21 x^{2} + 7 x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$x \cdot x^{7} + x (- 7 x^{6}) + x (21 x^{5}) + x (- 35 x^{4}) + x (35 x^{3}) + x (- 21 x^{2}) + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

Simplify terms.

Simplify each term.

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

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

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

$x^{1} x^{7} + x (- 7 x^{6}) + x (21 x^{5}) + x (- 35 x^{4}) + x (35 x^{3}) + x (- 21 x^{2}) + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

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

$x^{1 + 7} + x (- 7 x^{6}) + x (21 x^{5}) + x (- 35 x^{4}) + x (35 x^{3}) + x (- 21 x^{2}) + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

Add $1$ and $7$ .

$x^{8} + x (- 7 x^{6}) + x (21 x^{5}) + x (- 35 x^{4}) + x (35 x^{3}) + x (- 21 x^{2}) + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

Rewrite using the commutative property of multiplication.

$x^{8} - 7 x \cdot x^{6} + x (21 x^{5}) + x (- 35 x^{4}) + x (35 x^{3}) + x (- 21 x^{2}) + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

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

Move $x^{6}$ .

$x^{8} - 7 (x^{6} x) + x (21 x^{5}) + x (- 35 x^{4}) + x (35 x^{3}) + x (- 21 x^{2}) + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

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

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

$x^{8} - 7 (x^{6} x^{1}) + x (21 x^{5}) + x (- 35 x^{4}) + x (35 x^{3}) + x (- 21 x^{2}) + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

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

$x^{8} - 7 x^{6 + 1} + x (21 x^{5}) + x (- 35 x^{4}) + x (35 x^{3}) + x (- 21 x^{2}) + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

Add $6$ and $1$ .

$x^{8} - 7 x^{7} + x (21 x^{5}) + x (- 35 x^{4}) + x (35 x^{3}) + x (- 21 x^{2}) + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

Rewrite using the commutative property of multiplication.

$x^{8} - 7 x^{7} + 21 x \cdot x^{5} + x (- 35 x^{4}) + x (35 x^{3}) + x (- 21 x^{2}) + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

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

Move $x^{5}$ .

$x^{8} - 7 x^{7} + 21 (x^{5} x) + x (- 35 x^{4}) + x (35 x^{3}) + x (- 21 x^{2}) + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

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

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

$x^{8} - 7 x^{7} + 21 (x^{5} x^{1}) + x (- 35 x^{4}) + x (35 x^{3}) + x (- 21 x^{2}) + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

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

$x^{8} - 7 x^{7} + 21 x^{5 + 1} + x (- 35 x^{4}) + x (35 x^{3}) + x (- 21 x^{2}) + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

Add $5$ and $1$ .

$x^{8} - 7 x^{7} + 21 x^{6} + x (- 35 x^{4}) + x (35 x^{3}) + x (- 21 x^{2}) + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

Rewrite using the commutative property of multiplication.

$x^{8} - 7 x^{7} + 21 x^{6} - 35 x \cdot x^{4} + x (35 x^{3}) + x (- 21 x^{2}) + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

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

Move $x^{4}$ .

$x^{8} - 7 x^{7} + 21 x^{6} - 35 (x^{4} x) + x (35 x^{3}) + x (- 21 x^{2}) + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

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

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

$x^{8} - 7 x^{7} + 21 x^{6} - 35 (x^{4} x^{1}) + x (35 x^{3}) + x (- 21 x^{2}) + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

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

$x^{8} - 7 x^{7} + 21 x^{6} - 35 x^{4 + 1} + x (35 x^{3}) + x (- 21 x^{2}) + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

Add $4$ and $1$ .

$x^{8} - 7 x^{7} + 21 x^{6} - 35 x^{5} + x (35 x^{3}) + x (- 21 x^{2}) + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

Rewrite using the commutative property of multiplication.

$x^{8} - 7 x^{7} + 21 x^{6} - 35 x^{5} + 35 x \cdot x^{3} + x (- 21 x^{2}) + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

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

Move $x^{3}$ .

$x^{8} - 7 x^{7} + 21 x^{6} - 35 x^{5} + 35 (x^{3} x) + x (- 21 x^{2}) + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

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

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

$x^{8} - 7 x^{7} + 21 x^{6} - 35 x^{5} + 35 (x^{3} x^{1}) + x (- 21 x^{2}) + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

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

$x^{8} - 7 x^{7} + 21 x^{6} - 35 x^{5} + 35 x^{3 + 1} + x (- 21 x^{2}) + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

Add $3$ and $1$ .

$x^{8} - 7 x^{7} + 21 x^{6} - 35 x^{5} + 35 x^{4} + x (- 21 x^{2}) + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

Rewrite using the commutative property of multiplication.

$x^{8} - 7 x^{7} + 21 x^{6} - 35 x^{5} + 35 x^{4} - 21 x \cdot x^{2} + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

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

Move $x^{2}$ .

$x^{8} - 7 x^{7} + 21 x^{6} - 35 x^{5} + 35 x^{4} - 21 (x^{2} x) + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

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

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

$x^{8} - 7 x^{7} + 21 x^{6} - 35 x^{5} + 35 x^{4} - 21 (x^{2} x^{1}) + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

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

$x^{8} - 7 x^{7} + 21 x^{6} - 35 x^{5} + 35 x^{4} - 21 x^{2 + 1} + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

Add $2$ and $1$ .

$x^{8} - 7 x^{7} + 21 x^{6} - 35 x^{5} + 35 x^{4} - 21 x^{3} + x (7 x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

Rewrite using the commutative property of multiplication.

$x^{8} - 7 x^{7} + 21 x^{6} - 35 x^{5} + 35 x^{4} - 21 x^{3} + 7 x \cdot x + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

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

Move $x$ .

$x^{8} - 7 x^{7} + 21 x^{6} - 35 x^{5} + 35 x^{4} - 21 x^{3} + 7 (x \cdot x) + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

Multiply $x$ by $x$ .

$x^{8} - 7 x^{7} + 21 x^{6} - 35 x^{5} + 35 x^{4} - 21 x^{3} + 7 x^{2} + x \cdot - 1 + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

Move $- 1$ to the left of $x$ .

$x^{8} - 7 x^{7} + 21 x^{6} - 35 x^{5} + 35 x^{4} - 21 x^{3} + 7 x^{2} - 1 \cdot x + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

Rewrite $- 1 x$ as $- x$ .

$x^{8} - 7 x^{7} + 21 x^{6} - 35 x^{5} + 35 x^{4} - 21 x^{3} + 7 x^{2} - x + 4 x^{7} + 4 (- 7 x^{6}) + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

Multiply $- 7$ by $4$ .

$x^{8} - 7 x^{7} + 21 x^{6} - 35 x^{5} + 35 x^{4} - 21 x^{3} + 7 x^{2} - x + 4 x^{7} - 28 x^{6} + 4 (21 x^{5}) + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

Multiply $21$ by $4$ .

$x^{8} - 7 x^{7} + 21 x^{6} - 35 x^{5} + 35 x^{4} - 21 x^{3} + 7 x^{2} - x + 4 x^{7} - 28 x^{6} + 84 x^{5} + 4 (- 35 x^{4}) + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

Multiply $- 35$ by $4$ .

$x^{8} - 7 x^{7} + 21 x^{6} - 35 x^{5} + 35 x^{4} - 21 x^{3} + 7 x^{2} - x + 4 x^{7} - 28 x^{6} + 84 x^{5} - 140 x^{4} + 4 (35 x^{3}) + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

Multiply $35$ by $4$ .

$x^{8} - 7 x^{7} + 21 x^{6} - 35 x^{5} + 35 x^{4} - 21 x^{3} + 7 x^{2} - x + 4 x^{7} - 28 x^{6} + 84 x^{5} - 140 x^{4} + 140 x^{3} + 4 (- 21 x^{2}) + 4 (7 x) + 4 \cdot - 1$

Multiply $- 21$ by $4$ .

$x^{8} - 7 x^{7} + 21 x^{6} - 35 x^{5} + 35 x^{4} - 21 x^{3} + 7 x^{2} - x + 4 x^{7} - 28 x^{6} + 84 x^{5} - 140 x^{4} + 140 x^{3} - 84 x^{2} + 4 (7 x) + 4 \cdot - 1$

Multiply $7$ by $4$ .

$x^{8} - 7 x^{7} + 21 x^{6} - 35 x^{5} + 35 x^{4} - 21 x^{3} + 7 x^{2} - x + 4 x^{7} - 28 x^{6} + 84 x^{5} - 140 x^{4} + 140 x^{3} - 84 x^{2} + 28 x + 4 \cdot - 1$

Multiply $4$ by $- 1$ .

$x^{8} - 7 x^{7} + 21 x^{6} - 35 x^{5} + 35 x^{4} - 21 x^{3} + 7 x^{2} - x + 4 x^{7} - 28 x^{6} + 84 x^{5} - 140 x^{4} + 140 x^{3} - 84 x^{2} + 28 x - 4$

Simplify by adding terms.

Add $- 7 x^{7}$ and $4 x^{7}$ .

$x^{8} - 3 x^{7} + 21 x^{6} - 35 x^{5} + 35 x^{4} - 21 x^{3} + 7 x^{2} - x - 28 x^{6} + 84 x^{5} - 140 x^{4} + 140 x^{3} - 84 x^{2} + 28 x - 4$

Subtract $28 x^{6}$ from $21 x^{6}$ .

$x^{8} - 3 x^{7} - 7 x^{6} - 35 x^{5} + 35 x^{4} - 21 x^{3} + 7 x^{2} - x + 84 x^{5} - 140 x^{4} + 140 x^{3} - 84 x^{2} + 28 x - 4$

Add $- 35 x^{5}$ and $84 x^{5}$ .

$x^{8} - 3 x^{7} - 7 x^{6} + 49 x^{5} + 35 x^{4} - 21 x^{3} + 7 x^{2} - x - 140 x^{4} + 140 x^{3} - 84 x^{2} + 28 x - 4$

Subtract $140 x^{4}$ from $35 x^{4}$ .

$x^{8} - 3 x^{7} - 7 x^{6} + 49 x^{5} - 105 x^{4} - 21 x^{3} + 7 x^{2} - x + 140 x^{3} - 84 x^{2} + 28 x - 4$

Add $- 21 x^{3}$ and $140 x^{3}$ .

$x^{8} - 3 x^{7} - 7 x^{6} + 49 x^{5} - 105 x^{4} + 119 x^{3} + 7 x^{2} - x - 84 x^{2} + 28 x - 4$

Subtract $84 x^{2}$ from $7 x^{2}$ .

$x^{8} - 3 x^{7} - 7 x^{6} + 49 x^{5} - 105 x^{4} + 119 x^{3} - 77 x^{2} - x + 28 x - 4$

Add $- x$ and $28 x$ .

$x^{8} - 3 x^{7} - 7 x^{6} + 49 x^{5} - 105 x^{4} + 119 x^{3} - 77 x^{2} + 27 x - 4$

Step 2

The largest exponent is the degree of the polynomial.

$8$

Find the Degree f(x)=(x+4)(x-1)^7

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