Find the Degree, Leading Term, and Leading Coefficient f(x)=(x+1)^2(x-3)^5(x-4)

$f (x) = {(x + 1)}^{2} {(x - 3)}^{5} (x - 4)$

Step 1

Simplify the polynomial, then reorder it left to right starting with the highest degree term.

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$f (x) = (x + 1) (x + 1) {(x - 3)}^{5} (x - 4)$

Expand $(x + 1) (x + 1)$ using the FOIL Method.

Apply the distributive property.

$f (x) = (x (x + 1) + 1 (x + 1)) {(x - 3)}^{5} (x - 4)$

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $x$ by $1$ .

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

Multiply $x$ by $1$ .

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

Multiply $1$ by $1$ .

$f (x) = (x^{2} + x + x + 1) {(x - 3)}^{5} (x - 4)$

Add $x$ and $x$ .

$f (x) = (x^{2} + 2 x + 1) {(x - 3)}^{5} (x - 4)$

Use the Binomial Theorem.

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

Simplify each term.

Multiply $- 3$ by $5$ .

$f (x) = (x^{2} + 2 x + 1) (x^{5} - 15 x^{4} + 10 x^{3} {(- 3)}^{2} + 10 x^{2} {(- 3)}^{3} + 5 x {(- 3)}^{4} + {(- 3)}^{5}) (x - 4)$

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

$f (x) = (x^{2} + 2 x + 1) (x^{5} - 15 x^{4} + 10 x^{3} \cdot 9 + 10 x^{2} {(- 3)}^{3} + 5 x {(- 3)}^{4} + {(- 3)}^{5}) (x - 4)$

Multiply $9$ by $10$ .

$f (x) = (x^{2} + 2 x + 1) (x^{5} - 15 x^{4} + 90 x^{3} + 10 x^{2} {(- 3)}^{3} + 5 x {(- 3)}^{4} + {(- 3)}^{5}) (x - 4)$

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

$f (x) = (x^{2} + 2 x + 1) (x^{5} - 15 x^{4} + 90 x^{3} + 10 x^{2} \cdot - 27 + 5 x {(- 3)}^{4} + {(- 3)}^{5}) (x - 4)$

Multiply $- 27$ by $10$ .

$f (x) = (x^{2} + 2 x + 1) (x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 5 x {(- 3)}^{4} + {(- 3)}^{5}) (x - 4)$

Raise $- 3$ to the power of $4$ .

$f (x) = (x^{2} + 2 x + 1) (x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 5 x \cdot 81 + {(- 3)}^{5}) (x - 4)$

Multiply $81$ by $5$ .

$f (x) = (x^{2} + 2 x + 1) (x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x + {(- 3)}^{5}) (x - 4)$

Raise $- 3$ to the power of $5$ .

$f (x) = (x^{2} + 2 x + 1) (x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 243) (x - 4)$

Expand $(x^{2} + 2 x + 1) (x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 243)$ by multiplying each term in the first expression by each term in the second expression.

$f (x) = (x^{2} x^{5} + x^{2} (- 15 x^{4}) + x^{2} (90 x^{3}) + x^{2} (- 270 x^{2}) + x^{2} (405 x) + x^{2} \cdot - 243 + 2 x \cdot x^{5} + 2 x (- 15 x^{4}) + 2 x (90 x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

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.

$f (x) = (x^{2 + 5} + x^{2} (- 15 x^{4}) + x^{2} (90 x^{3}) + x^{2} (- 270 x^{2}) + x^{2} (405 x) + x^{2} \cdot - 243 + 2 x \cdot x^{5} + 2 x (- 15 x^{4}) + 2 x (90 x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Add $2$ and $5$ .

$f (x) = (x^{7} + x^{2} (- 15 x^{4}) + x^{2} (90 x^{3}) + x^{2} (- 270 x^{2}) + x^{2} (405 x) + x^{2} \cdot - 243 + 2 x \cdot x^{5} + 2 x (- 15 x^{4}) + 2 x (90 x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Rewrite using the commutative property of multiplication.

$f (x) = (x^{7} - 15 x^{2} x^{4} + x^{2} (90 x^{3}) + x^{2} (- 270 x^{2}) + x^{2} (405 x) + x^{2} \cdot - 243 + 2 x \cdot x^{5} + 2 x (- 15 x^{4}) + 2 x (90 x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

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

Move $x^{4}$ .

$f (x) = (x^{7} - 15 (x^{4} x^{2}) + x^{2} (90 x^{3}) + x^{2} (- 270 x^{2}) + x^{2} (405 x) + x^{2} \cdot - 243 + 2 x \cdot x^{5} + 2 x (- 15 x^{4}) + 2 x (90 x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

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

$f (x) = (x^{7} - 15 x^{4 + 2} + x^{2} (90 x^{3}) + x^{2} (- 270 x^{2}) + x^{2} (405 x) + x^{2} \cdot - 243 + 2 x \cdot x^{5} + 2 x (- 15 x^{4}) + 2 x (90 x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Add $4$ and $2$ .

$f (x) = (x^{7} - 15 x^{6} + x^{2} (90 x^{3}) + x^{2} (- 270 x^{2}) + x^{2} (405 x) + x^{2} \cdot - 243 + 2 x \cdot x^{5} + 2 x (- 15 x^{4}) + 2 x (90 x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Rewrite using the commutative property of multiplication.

$f (x) = (x^{7} - 15 x^{6} + 90 x^{2} x^{3} + x^{2} (- 270 x^{2}) + x^{2} (405 x) + x^{2} \cdot - 243 + 2 x \cdot x^{5} + 2 x (- 15 x^{4}) + 2 x (90 x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

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

Move $x^{3}$ .

$f (x) = (x^{7} - 15 x^{6} + 90 (x^{3} x^{2}) + x^{2} (- 270 x^{2}) + x^{2} (405 x) + x^{2} \cdot - 243 + 2 x \cdot x^{5} + 2 x (- 15 x^{4}) + 2 x (90 x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

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

$f (x) = (x^{7} - 15 x^{6} + 90 x^{3 + 2} + x^{2} (- 270 x^{2}) + x^{2} (405 x) + x^{2} \cdot - 243 + 2 x \cdot x^{5} + 2 x (- 15 x^{4}) + 2 x (90 x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Add $3$ and $2$ .

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} + x^{2} (- 270 x^{2}) + x^{2} (405 x) + x^{2} \cdot - 243 + 2 x \cdot x^{5} + 2 x (- 15 x^{4}) + 2 x (90 x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Rewrite using the commutative property of multiplication.

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{2} x^{2} + x^{2} (405 x) + x^{2} \cdot - 243 + 2 x \cdot x^{5} + 2 x (- 15 x^{4}) + 2 x (90 x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

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

Move $x^{2}$ .

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 (x^{2} x^{2}) + x^{2} (405 x) + x^{2} \cdot - 243 + 2 x \cdot x^{5} + 2 x (- 15 x^{4}) + 2 x (90 x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

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

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{2 + 2} + x^{2} (405 x) + x^{2} \cdot - 243 + 2 x \cdot x^{5} + 2 x (- 15 x^{4}) + 2 x (90 x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Add $2$ and $2$ .

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + x^{2} (405 x) + x^{2} \cdot - 243 + 2 x \cdot x^{5} + 2 x (- 15 x^{4}) + 2 x (90 x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Rewrite using the commutative property of multiplication.

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{2} x + x^{2} \cdot - 243 + 2 x \cdot x^{5} + 2 x (- 15 x^{4}) + 2 x (90 x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

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

Move $x$ .

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 (x \cdot x^{2}) + x^{2} \cdot - 243 + 2 x \cdot x^{5} + 2 x (- 15 x^{4}) + 2 x (90 x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

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

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

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

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{1 + 2} + x^{2} \cdot - 243 + 2 x \cdot x^{5} + 2 x (- 15 x^{4}) + 2 x (90 x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Add $1$ and $2$ .

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} + x^{2} \cdot - 243 + 2 x \cdot x^{5} + 2 x (- 15 x^{4}) + 2 x (90 x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Move $- 243$ to the left of $x^{2}$ .

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 \cdot x^{2} + 2 x \cdot x^{5} + 2 x (- 15 x^{4}) + 2 x (90 x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

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

Move $x^{5}$ .

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 (x^{5} x) + 2 x (- 15 x^{4}) + 2 x (90 x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

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

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

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

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 x^{5 + 1} + 2 x (- 15 x^{4}) + 2 x (90 x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Add $5$ and $1$ .

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 x^{6} + 2 x (- 15 x^{4}) + 2 x (90 x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Rewrite using the commutative property of multiplication.

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 x^{6} + 2 \cdot (- 15 x \cdot x^{4}) + 2 x (90 x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

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

Move $x^{4}$ .

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 x^{6} + 2 \cdot (- 15 (x^{4} x)) + 2 x (90 x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

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

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

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

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 x^{6} + 2 \cdot (- 15 x^{4 + 1}) + 2 x (90 x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Add $4$ and $1$ .

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 x^{6} + 2 \cdot (- 15 x^{5}) + 2 x (90 x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Multiply $2$ by $- 15$ .

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 x^{6} - 30 x^{5} + 2 x (90 x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Rewrite using the commutative property of multiplication.

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 x^{6} - 30 x^{5} + 2 \cdot (90 x \cdot x^{3}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

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

Move $x^{3}$ .

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 x^{6} - 30 x^{5} + 2 \cdot (90 (x^{3} x)) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

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

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

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

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 x^{6} - 30 x^{5} + 2 \cdot (90 x^{3 + 1}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Add $3$ and $1$ .

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 x^{6} - 30 x^{5} + 2 \cdot (90 x^{4}) + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Multiply $2$ by $90$ .

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 x^{6} - 30 x^{5} + 180 x^{4} + 2 x (- 270 x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Rewrite using the commutative property of multiplication.

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 x^{6} - 30 x^{5} + 180 x^{4} + 2 \cdot (- 270 x \cdot x^{2}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

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

Move $x^{2}$ .

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 x^{6} - 30 x^{5} + 180 x^{4} + 2 \cdot (- 270 (x^{2} x)) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

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

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

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

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 x^{6} - 30 x^{5} + 180 x^{4} + 2 \cdot (- 270 x^{2 + 1}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Add $2$ and $1$ .

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 x^{6} - 30 x^{5} + 180 x^{4} + 2 \cdot (- 270 x^{3}) + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Multiply $2$ by $- 270$ .

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 x^{6} - 30 x^{5} + 180 x^{4} - 540 x^{3} + 2 x (405 x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Rewrite using the commutative property of multiplication.

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 x^{6} - 30 x^{5} + 180 x^{4} - 540 x^{3} + 2 \cdot (405 x \cdot x) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

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

Move $x$ .

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 x^{6} - 30 x^{5} + 180 x^{4} - 540 x^{3} + 2 \cdot (405 (x \cdot x)) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Multiply $x$ by $x$ .

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 x^{6} - 30 x^{5} + 180 x^{4} - 540 x^{3} + 2 \cdot (405 x^{2}) + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Multiply $2$ by $405$ .

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 x^{6} - 30 x^{5} + 180 x^{4} - 540 x^{3} + 810 x^{2} + 2 x \cdot - 243 + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Multiply $- 243$ by $2$ .

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 x^{6} - 30 x^{5} + 180 x^{4} - 540 x^{3} + 810 x^{2} - 486 x + 1 x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Multiply $x^{5}$ by $1$ .

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 x^{6} - 30 x^{5} + 180 x^{4} - 540 x^{3} + 810 x^{2} - 486 x + x^{5} + 1 (- 15 x^{4}) + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Multiply $- 15 x^{4}$ by $1$ .

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 x^{6} - 30 x^{5} + 180 x^{4} - 540 x^{3} + 810 x^{2} - 486 x + x^{5} - 15 x^{4} + 1 (90 x^{3}) + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Multiply $90 x^{3}$ by $1$ .

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 x^{6} - 30 x^{5} + 180 x^{4} - 540 x^{3} + 810 x^{2} - 486 x + x^{5} - 15 x^{4} + 90 x^{3} + 1 (- 270 x^{2}) + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Multiply $- 270 x^{2}$ by $1$ .

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 x^{6} - 30 x^{5} + 180 x^{4} - 540 x^{3} + 810 x^{2} - 486 x + x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 1 (405 x) + 1 \cdot - 243) (x - 4)$

Multiply $405 x$ by $1$ .

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 x^{6} - 30 x^{5} + 180 x^{4} - 540 x^{3} + 810 x^{2} - 486 x + x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x + 1 \cdot - 243) (x - 4)$

Multiply $- 243$ by $1$ .

$f (x) = (x^{7} - 15 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 2 x^{6} - 30 x^{5} + 180 x^{4} - 540 x^{3} + 810 x^{2} - 486 x + x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 243) (x - 4)$

Simplify by adding terms.

Add $- 15 x^{6}$ and $2 x^{6}$ .

$f (x) = (x^{7} - 13 x^{6} + 90 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} - 30 x^{5} + 180 x^{4} - 540 x^{3} + 810 x^{2} - 486 x + x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 243) (x - 4)$

Subtract $30 x^{5}$ from $90 x^{5}$ .

$f (x) = (x^{7} - 13 x^{6} + 60 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 180 x^{4} - 540 x^{3} + 810 x^{2} - 486 x + x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 243) (x - 4)$

Add $60 x^{5}$ and $x^{5}$ .

$f (x) = (x^{7} - 13 x^{6} + 61 x^{5} - 270 x^{4} + 405 x^{3} - 243 x^{2} + 180 x^{4} - 540 x^{3} + 810 x^{2} - 486 x - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 243) (x - 4)$

Add $- 270 x^{4}$ and $180 x^{4}$ .

$f (x) = (x^{7} - 13 x^{6} + 61 x^{5} - 90 x^{4} + 405 x^{3} - 243 x^{2} - 540 x^{3} + 810 x^{2} - 486 x - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 243) (x - 4)$

Subtract $15 x^{4}$ from $- 90 x^{4}$ .

$f (x) = (x^{7} - 13 x^{6} + 61 x^{5} - 105 x^{4} + 405 x^{3} - 243 x^{2} - 540 x^{3} + 810 x^{2} - 486 x + 90 x^{3} - 270 x^{2} + 405 x - 243) (x - 4)$

Subtract $540 x^{3}$ from $405 x^{3}$ .

$f (x) = (x^{7} - 13 x^{6} + 61 x^{5} - 105 x^{4} - 135 x^{3} - 243 x^{2} + 810 x^{2} - 486 x + 90 x^{3} - 270 x^{2} + 405 x - 243) (x - 4)$

Add $- 135 x^{3}$ and $90 x^{3}$ .

$f (x) = (x^{7} - 13 x^{6} + 61 x^{5} - 105 x^{4} - 45 x^{3} - 243 x^{2} + 810 x^{2} - 486 x - 270 x^{2} + 405 x - 243) (x - 4)$

Add $- 243 x^{2}$ and $810 x^{2}$ .

$f (x) = (x^{7} - 13 x^{6} + 61 x^{5} - 105 x^{4} - 45 x^{3} + 567 x^{2} - 486 x - 270 x^{2} + 405 x - 243) (x - 4)$

Subtract $270 x^{2}$ from $567 x^{2}$ .

$f (x) = (x^{7} - 13 x^{6} + 61 x^{5} - 105 x^{4} - 45 x^{3} + 297 x^{2} - 486 x + 405 x - 243) (x - 4)$

Add $- 486 x$ and $405 x$ .

$f (x) = (x^{7} - 13 x^{6} + 61 x^{5} - 105 x^{4} - 45 x^{3} + 297 x^{2} - 81 x - 243) (x - 4)$

Expand $(x^{7} - 13 x^{6} + 61 x^{5} - 105 x^{4} - 45 x^{3} + 297 x^{2} - 81 x - 243) (x - 4)$ by multiplying each term in the first expression by each term in the second expression.

$f (x) = x^{7} x + x^{7} \cdot - 4 - 13 x^{6} x - 13 x^{6} \cdot - 4 + 61 x^{5} x + 61 x^{5} \cdot - 4 - 105 x^{4} x - 105 x^{4} \cdot - 4 - 45 x^{3} x - 45 x^{3} \cdot - 4 + 297 x^{2} x + 297 x^{2} \cdot - 4 - 81 x \cdot x - 81 x \cdot - 4 - 243 x - 243 \cdot - 4$

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 $1$ .

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

$f (x) = x^{7 + 1} + x^{7} \cdot - 4 - 13 x^{6} x - 13 x^{6} \cdot - 4 + 61 x^{5} x + 61 x^{5} \cdot - 4 - 105 x^{4} x - 105 x^{4} \cdot - 4 - 45 x^{3} x - 45 x^{3} \cdot - 4 + 297 x^{2} x + 297 x^{2} \cdot - 4 - 81 x \cdot x - 81 x \cdot - 4 - 243 x - 243 \cdot - 4$

Add $7$ and $1$ .

$f (x) = x^{8} + x^{7} \cdot - 4 - 13 x^{6} x - 13 x^{6} \cdot - 4 + 61 x^{5} x + 61 x^{5} \cdot - 4 - 105 x^{4} x - 105 x^{4} \cdot - 4 - 45 x^{3} x - 45 x^{3} \cdot - 4 + 297 x^{2} x + 297 x^{2} \cdot - 4 - 81 x \cdot x - 81 x \cdot - 4 - 243 x - 243 \cdot - 4$

Move $- 4$ to the left of $x^{7}$ .

$f (x) = x^{8} - 4 \cdot x^{7} - 13 x^{6} x - 13 x^{6} \cdot - 4 + 61 x^{5} x + 61 x^{5} \cdot - 4 - 105 x^{4} x - 105 x^{4} \cdot - 4 - 45 x^{3} x - 45 x^{3} \cdot - 4 + 297 x^{2} x + 297 x^{2} \cdot - 4 - 81 x \cdot x - 81 x \cdot - 4 - 243 x - 243 \cdot - 4$

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

Move $x$ .

$f (x) = x^{8} - 4 x^{7} - 13 (x \cdot x^{6}) - 13 x^{6} \cdot - 4 + 61 x^{5} x + 61 x^{5} \cdot - 4 - 105 x^{4} x - 105 x^{4} \cdot - 4 - 45 x^{3} x - 45 x^{3} \cdot - 4 + 297 x^{2} x + 297 x^{2} \cdot - 4 - 81 x \cdot x - 81 x \cdot - 4 - 243 x - 243 \cdot - 4$

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

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

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

$f (x) = x^{8} - 4 x^{7} - 13 x^{1 + 6} - 13 x^{6} \cdot - 4 + 61 x^{5} x + 61 x^{5} \cdot - 4 - 105 x^{4} x - 105 x^{4} \cdot - 4 - 45 x^{3} x - 45 x^{3} \cdot - 4 + 297 x^{2} x + 297 x^{2} \cdot - 4 - 81 x \cdot x - 81 x \cdot - 4 - 243 x - 243 \cdot - 4$

Add $1$ and $6$ .

$f (x) = x^{8} - 4 x^{7} - 13 x^{7} - 13 x^{6} \cdot - 4 + 61 x^{5} x + 61 x^{5} \cdot - 4 - 105 x^{4} x - 105 x^{4} \cdot - 4 - 45 x^{3} x - 45 x^{3} \cdot - 4 + 297 x^{2} x + 297 x^{2} \cdot - 4 - 81 x \cdot x - 81 x \cdot - 4 - 243 x - 243 \cdot - 4$

Multiply $- 4$ by $- 13$ .

$f (x) = x^{8} - 4 x^{7} - 13 x^{7} + 52 x^{6} + 61 x^{5} x + 61 x^{5} \cdot - 4 - 105 x^{4} x - 105 x^{4} \cdot - 4 - 45 x^{3} x - 45 x^{3} \cdot - 4 + 297 x^{2} x + 297 x^{2} \cdot - 4 - 81 x \cdot x - 81 x \cdot - 4 - 243 x - 243 \cdot - 4$

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

Move $x$ .

$f (x) = x^{8} - 4 x^{7} - 13 x^{7} + 52 x^{6} + 61 (x \cdot x^{5}) + 61 x^{5} \cdot - 4 - 105 x^{4} x - 105 x^{4} \cdot - 4 - 45 x^{3} x - 45 x^{3} \cdot - 4 + 297 x^{2} x + 297 x^{2} \cdot - 4 - 81 x \cdot x - 81 x \cdot - 4 - 243 x - 243 \cdot - 4$

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

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

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

$f (x) = x^{8} - 4 x^{7} - 13 x^{7} + 52 x^{6} + 61 x^{1 + 5} + 61 x^{5} \cdot - 4 - 105 x^{4} x - 105 x^{4} \cdot - 4 - 45 x^{3} x - 45 x^{3} \cdot - 4 + 297 x^{2} x + 297 x^{2} \cdot - 4 - 81 x \cdot x - 81 x \cdot - 4 - 243 x - 243 \cdot - 4$

Add $1$ and $5$ .

$f (x) = x^{8} - 4 x^{7} - 13 x^{7} + 52 x^{6} + 61 x^{6} + 61 x^{5} \cdot - 4 - 105 x^{4} x - 105 x^{4} \cdot - 4 - 45 x^{3} x - 45 x^{3} \cdot - 4 + 297 x^{2} x + 297 x^{2} \cdot - 4 - 81 x \cdot x - 81 x \cdot - 4 - 243 x - 243 \cdot - 4$

Multiply $- 4$ by $61$ .

$f (x) = x^{8} - 4 x^{7} - 13 x^{7} + 52 x^{6} + 61 x^{6} - 244 x^{5} - 105 x^{4} x - 105 x^{4} \cdot - 4 - 45 x^{3} x - 45 x^{3} \cdot - 4 + 297 x^{2} x + 297 x^{2} \cdot - 4 - 81 x \cdot x - 81 x \cdot - 4 - 243 x - 243 \cdot - 4$

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

Move $x$ .

$f (x) = x^{8} - 4 x^{7} - 13 x^{7} + 52 x^{6} + 61 x^{6} - 244 x^{5} - 105 (x \cdot x^{4}) - 105 x^{4} \cdot - 4 - 45 x^{3} x - 45 x^{3} \cdot - 4 + 297 x^{2} x + 297 x^{2} \cdot - 4 - 81 x \cdot x - 81 x \cdot - 4 - 243 x - 243 \cdot - 4$

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

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

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

$f (x) = x^{8} - 4 x^{7} - 13 x^{7} + 52 x^{6} + 61 x^{6} - 244 x^{5} - 105 x^{1 + 4} - 105 x^{4} \cdot - 4 - 45 x^{3} x - 45 x^{3} \cdot - 4 + 297 x^{2} x + 297 x^{2} \cdot - 4 - 81 x \cdot x - 81 x \cdot - 4 - 243 x - 243 \cdot - 4$

Add $1$ and $4$ .

$f (x) = x^{8} - 4 x^{7} - 13 x^{7} + 52 x^{6} + 61 x^{6} - 244 x^{5} - 105 x^{5} - 105 x^{4} \cdot - 4 - 45 x^{3} x - 45 x^{3} \cdot - 4 + 297 x^{2} x + 297 x^{2} \cdot - 4 - 81 x \cdot x - 81 x \cdot - 4 - 243 x - 243 \cdot - 4$

Multiply $- 4$ by $- 105$ .

$f (x) = x^{8} - 4 x^{7} - 13 x^{7} + 52 x^{6} + 61 x^{6} - 244 x^{5} - 105 x^{5} + 420 x^{4} - 45 x^{3} x - 45 x^{3} \cdot - 4 + 297 x^{2} x + 297 x^{2} \cdot - 4 - 81 x \cdot x - 81 x \cdot - 4 - 243 x - 243 \cdot - 4$

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

Move $x$ .

$f (x) = x^{8} - 4 x^{7} - 13 x^{7} + 52 x^{6} + 61 x^{6} - 244 x^{5} - 105 x^{5} + 420 x^{4} - 45 (x \cdot x^{3}) - 45 x^{3} \cdot - 4 + 297 x^{2} x + 297 x^{2} \cdot - 4 - 81 x \cdot x - 81 x \cdot - 4 - 243 x - 243 \cdot - 4$

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

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

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

$f (x) = x^{8} - 4 x^{7} - 13 x^{7} + 52 x^{6} + 61 x^{6} - 244 x^{5} - 105 x^{5} + 420 x^{4} - 45 x^{1 + 3} - 45 x^{3} \cdot - 4 + 297 x^{2} x + 297 x^{2} \cdot - 4 - 81 x \cdot x - 81 x \cdot - 4 - 243 x - 243 \cdot - 4$

Add $1$ and $3$ .

$f (x) = x^{8} - 4 x^{7} - 13 x^{7} + 52 x^{6} + 61 x^{6} - 244 x^{5} - 105 x^{5} + 420 x^{4} - 45 x^{4} - 45 x^{3} \cdot - 4 + 297 x^{2} x + 297 x^{2} \cdot - 4 - 81 x \cdot x - 81 x \cdot - 4 - 243 x - 243 \cdot - 4$

Multiply $- 4$ by $- 45$ .

$f (x) = x^{8} - 4 x^{7} - 13 x^{7} + 52 x^{6} + 61 x^{6} - 244 x^{5} - 105 x^{5} + 420 x^{4} - 45 x^{4} + 180 x^{3} + 297 x^{2} x + 297 x^{2} \cdot - 4 - 81 x \cdot x - 81 x \cdot - 4 - 243 x - 243 \cdot - 4$

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

Move $x$ .

$f (x) = x^{8} - 4 x^{7} - 13 x^{7} + 52 x^{6} + 61 x^{6} - 244 x^{5} - 105 x^{5} + 420 x^{4} - 45 x^{4} + 180 x^{3} + 297 (x \cdot x^{2}) + 297 x^{2} \cdot - 4 - 81 x \cdot x - 81 x \cdot - 4 - 243 x - 243 \cdot - 4$

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

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

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

$f (x) = x^{8} - 4 x^{7} - 13 x^{7} + 52 x^{6} + 61 x^{6} - 244 x^{5} - 105 x^{5} + 420 x^{4} - 45 x^{4} + 180 x^{3} + 297 x^{1 + 2} + 297 x^{2} \cdot - 4 - 81 x \cdot x - 81 x \cdot - 4 - 243 x - 243 \cdot - 4$

Add $1$ and $2$ .

$f (x) = x^{8} - 4 x^{7} - 13 x^{7} + 52 x^{6} + 61 x^{6} - 244 x^{5} - 105 x^{5} + 420 x^{4} - 45 x^{4} + 180 x^{3} + 297 x^{3} + 297 x^{2} \cdot - 4 - 81 x \cdot x - 81 x \cdot - 4 - 243 x - 243 \cdot - 4$

Multiply $- 4$ by $297$ .

$f (x) = x^{8} - 4 x^{7} - 13 x^{7} + 52 x^{6} + 61 x^{6} - 244 x^{5} - 105 x^{5} + 420 x^{4} - 45 x^{4} + 180 x^{3} + 297 x^{3} - 1188 x^{2} - 81 x \cdot x - 81 x \cdot - 4 - 243 x - 243 \cdot - 4$

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

Move $x$ .

$f (x) = x^{8} - 4 x^{7} - 13 x^{7} + 52 x^{6} + 61 x^{6} - 244 x^{5} - 105 x^{5} + 420 x^{4} - 45 x^{4} + 180 x^{3} + 297 x^{3} - 1188 x^{2} - 81 (x \cdot x) - 81 x \cdot - 4 - 243 x - 243 \cdot - 4$

Multiply $x$ by $x$ .

$f (x) = x^{8} - 4 x^{7} - 13 x^{7} + 52 x^{6} + 61 x^{6} - 244 x^{5} - 105 x^{5} + 420 x^{4} - 45 x^{4} + 180 x^{3} + 297 x^{3} - 1188 x^{2} - 81 x^{2} - 81 x \cdot - 4 - 243 x - 243 \cdot - 4$

Multiply $- 4$ by $- 81$ .

$f (x) = x^{8} - 4 x^{7} - 13 x^{7} + 52 x^{6} + 61 x^{6} - 244 x^{5} - 105 x^{5} + 420 x^{4} - 45 x^{4} + 180 x^{3} + 297 x^{3} - 1188 x^{2} - 81 x^{2} + 324 x - 243 x - 243 \cdot - 4$

Multiply $- 243$ by $- 4$ .

$f (x) = x^{8} - 4 x^{7} - 13 x^{7} + 52 x^{6} + 61 x^{6} - 244 x^{5} - 105 x^{5} + 420 x^{4} - 45 x^{4} + 180 x^{3} + 297 x^{3} - 1188 x^{2} - 81 x^{2} + 324 x - 243 x + 972$

Simplify by adding terms.

Subtract $13 x^{7}$ from $- 4 x^{7}$ .

$f (x) = x^{8} - 17 x^{7} + 52 x^{6} + 61 x^{6} - 244 x^{5} - 105 x^{5} + 420 x^{4} - 45 x^{4} + 180 x^{3} + 297 x^{3} - 1188 x^{2} - 81 x^{2} + 324 x - 243 x + 972$

Add $52 x^{6}$ and $61 x^{6}$ .

$f (x) = x^{8} - 17 x^{7} + 113 x^{6} - 244 x^{5} - 105 x^{5} + 420 x^{4} - 45 x^{4} + 180 x^{3} + 297 x^{3} - 1188 x^{2} - 81 x^{2} + 324 x - 243 x + 972$

Subtract $105 x^{5}$ from $- 244 x^{5}$ .

$f (x) = x^{8} - 17 x^{7} + 113 x^{6} - 349 x^{5} + 420 x^{4} - 45 x^{4} + 180 x^{3} + 297 x^{3} - 1188 x^{2} - 81 x^{2} + 324 x - 243 x + 972$

Subtract $45 x^{4}$ from $420 x^{4}$ .

$f (x) = x^{8} - 17 x^{7} + 113 x^{6} - 349 x^{5} + 375 x^{4} + 180 x^{3} + 297 x^{3} - 1188 x^{2} - 81 x^{2} + 324 x - 243 x + 972$

Add $180 x^{3}$ and $297 x^{3}$ .

$f (x) = x^{8} - 17 x^{7} + 113 x^{6} - 349 x^{5} + 375 x^{4} + 477 x^{3} - 1188 x^{2} - 81 x^{2} + 324 x - 243 x + 972$

Subtract $81 x^{2}$ from $- 1188 x^{2}$ .

$f (x) = x^{8} - 17 x^{7} + 113 x^{6} - 349 x^{5} + 375 x^{4} + 477 x^{3} - 1269 x^{2} + 324 x - 243 x + 972$

Subtract $243 x$ from $324 x$ .

$f (x) = x^{8} - 17 x^{7} + 113 x^{6} - 349 x^{5} + 375 x^{4} + 477 x^{3} - 1269 x^{2} + 81 x + 972$

Step 2

The degree of a polynomial is the highest degree of its terms.

Identify the exponents on the variables in each term, and add them together to find the degree of each term.

$x^{8} \to 8$

$- 17 x^{7} \to 7$

$113 x^{6} \to 6$

$- 349 x^{5} \to 5$

$375 x^{4} \to 4$

$477 x^{3} \to 3$

$- 1269 x^{2} \to 2$

$81 x \to 1$

$972 \to 0$

The largest exponent is the degree of the polynomial.

$8$

Step 3

The leading term in a polynomial is the term with the highest degree.

$x^{8}$

Step 4

The leading coefficient of a polynomial is the coefficient of the leading term.

The leading term in a polynomial is the term with the highest degree.

$x^{8}$

The leading coefficient in a polynomial is the coefficient of the leading term.

$1$

Step 5

List the results.

Polynomial Degree: $8$

Leading Term: $x^{8}$

Leading Coefficient: $1$

Find the Degree, Leading Term, and Leading Coefficient f(x)=(x+1)^2(x-3)^5(x-4)

Related Questions

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