Find the Degree, Leading Term, and Leading Coefficient 5x^6(x-6)^2(x+8)(x^2-2x+1)

$5 x^{6} {(x - 6)}^{2} (x + 8) (x^{2} - 2 x + 1)$

Step 1

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

Rewrite ${(x - 6)}^{2}$ as $(x - 6) (x - 6)$ .

$5 x^{6} ((x - 6) (x - 6)) (x + 8) (x^{2} - 2 x + 1)$

Expand $(x - 6) (x - 6)$ using the FOIL Method.

Apply the distributive property.

$5 x^{6} (x (x - 6) - 6 (x - 6)) (x + 8) (x^{2} - 2 x + 1)$

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 6$ to the left of $x$ .

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

Multiply $- 6$ by $- 6$ .

$5 x^{6} (x^{2} - 6 x - 6 x + 36) (x + 8) (x^{2} - 2 x + 1)$

Subtract $6 x$ from $- 6 x$ .

$5 x^{6} (x^{2} - 12 x + 36) (x + 8) (x^{2} - 2 x + 1)$

Apply the distributive property.

$(5 x^{6} x^{2} + 5 x^{6} (- 12 x) + 5 x^{6} \cdot 36) (x + 8) (x^{2} - 2 x + 1)$

Simplify.

Multiply $x^{6}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$(5 (x^{2} x^{6}) + 5 x^{6} (- 12 x) + 5 x^{6} \cdot 36) (x + 8) (x^{2} - 2 x + 1)$

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

$(5 x^{2 + 6} + 5 x^{6} (- 12 x) + 5 x^{6} \cdot 36) (x + 8) (x^{2} - 2 x + 1)$

Add $2$ and $6$ .

$(5 x^{8} + 5 x^{6} (- 12 x) + 5 x^{6} \cdot 36) (x + 8) (x^{2} - 2 x + 1)$

Rewrite using the commutative property of multiplication.

$(5 x^{8} + 5 \cdot - 12 x^{6} x + 5 x^{6} \cdot 36) (x + 8) (x^{2} - 2 x + 1)$

Multiply $36$ by $5$ .

$(5 x^{8} + 5 \cdot - 12 x^{6} x + 180 x^{6}) (x + 8) (x^{2} - 2 x + 1)$

Simplify each term.

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

Move $x$ .

$(5 x^{8} + 5 \cdot - 12 (x \cdot x^{6}) + 180 x^{6}) (x + 8) (x^{2} - 2 x + 1)$

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

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

$(5 x^{8} + 5 \cdot - 12 (x^{1} x^{6}) + 180 x^{6}) (x + 8) (x^{2} - 2 x + 1)$

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

$(5 x^{8} + 5 \cdot - 12 x^{1 + 6} + 180 x^{6}) (x + 8) (x^{2} - 2 x + 1)$

Add $1$ and $6$ .

$(5 x^{8} + 5 \cdot - 12 x^{7} + 180 x^{6}) (x + 8) (x^{2} - 2 x + 1)$

Multiply $5$ by $- 12$ .

$(5 x^{8} - 60 x^{7} + 180 x^{6}) (x + 8) (x^{2} - 2 x + 1)$

Expand $(5 x^{8} - 60 x^{7} + 180 x^{6}) (x + 8)$ by multiplying each term in the first expression by each term in the second expression.

$(5 x^{8} x + 5 x^{8} \cdot 8 - 60 x^{7} x - 60 x^{7} \cdot 8 + 180 x^{6} x + 180 x^{6} \cdot 8) (x^{2} - 2 x + 1)$

Simplify terms.

Simplify each term.

Multiply $x^{8}$ by $x$ by adding the exponents.

Move $x$ .

$(5 (x \cdot x^{8}) + 5 x^{8} \cdot 8 - 60 x^{7} x - 60 x^{7} \cdot 8 + 180 x^{6} x + 180 x^{6} \cdot 8) (x^{2} - 2 x + 1)$

Multiply $x$ by $x^{8}$ .

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

$(5 (x^{1} x^{8}) + 5 x^{8} \cdot 8 - 60 x^{7} x - 60 x^{7} \cdot 8 + 180 x^{6} x + 180 x^{6} \cdot 8) (x^{2} - 2 x + 1)$

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

$(5 x^{1 + 8} + 5 x^{8} \cdot 8 - 60 x^{7} x - 60 x^{7} \cdot 8 + 180 x^{6} x + 180 x^{6} \cdot 8) (x^{2} - 2 x + 1)$

Add $1$ and $8$ .

$(5 x^{9} + 5 x^{8} \cdot 8 - 60 x^{7} x - 60 x^{7} \cdot 8 + 180 x^{6} x + 180 x^{6} \cdot 8) (x^{2} - 2 x + 1)$

Multiply $8$ by $5$ .

$(5 x^{9} + 40 x^{8} - 60 x^{7} x - 60 x^{7} \cdot 8 + 180 x^{6} x + 180 x^{6} \cdot 8) (x^{2} - 2 x + 1)$

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

Move $x$ .

$(5 x^{9} + 40 x^{8} - 60 (x \cdot x^{7}) - 60 x^{7} \cdot 8 + 180 x^{6} x + 180 x^{6} \cdot 8) (x^{2} - 2 x + 1)$

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

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

$(5 x^{9} + 40 x^{8} - 60 (x^{1} x^{7}) - 60 x^{7} \cdot 8 + 180 x^{6} x + 180 x^{6} \cdot 8) (x^{2} - 2 x + 1)$

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

$(5 x^{9} + 40 x^{8} - 60 x^{1 + 7} - 60 x^{7} \cdot 8 + 180 x^{6} x + 180 x^{6} \cdot 8) (x^{2} - 2 x + 1)$

Add $1$ and $7$ .

$(5 x^{9} + 40 x^{8} - 60 x^{8} - 60 x^{7} \cdot 8 + 180 x^{6} x + 180 x^{6} \cdot 8) (x^{2} - 2 x + 1)$

Multiply $8$ by $- 60$ .

$(5 x^{9} + 40 x^{8} - 60 x^{8} - 480 x^{7} + 180 x^{6} x + 180 x^{6} \cdot 8) (x^{2} - 2 x + 1)$

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

Move $x$ .

$(5 x^{9} + 40 x^{8} - 60 x^{8} - 480 x^{7} + 180 (x \cdot x^{6}) + 180 x^{6} \cdot 8) (x^{2} - 2 x + 1)$

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

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

$(5 x^{9} + 40 x^{8} - 60 x^{8} - 480 x^{7} + 180 (x^{1} x^{6}) + 180 x^{6} \cdot 8) (x^{2} - 2 x + 1)$

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

$(5 x^{9} + 40 x^{8} - 60 x^{8} - 480 x^{7} + 180 x^{1 + 6} + 180 x^{6} \cdot 8) (x^{2} - 2 x + 1)$

Add $1$ and $6$ .

$(5 x^{9} + 40 x^{8} - 60 x^{8} - 480 x^{7} + 180 x^{7} + 180 x^{6} \cdot 8) (x^{2} - 2 x + 1)$

Multiply $8$ by $180$ .

$(5 x^{9} + 40 x^{8} - 60 x^{8} - 480 x^{7} + 180 x^{7} + 1440 x^{6}) (x^{2} - 2 x + 1)$

Simplify by adding terms.

Subtract $60 x^{8}$ from $40 x^{8}$ .

$(5 x^{9} - 20 x^{8} - 480 x^{7} + 180 x^{7} + 1440 x^{6}) (x^{2} - 2 x + 1)$

Add $- 480 x^{7}$ and $180 x^{7}$ .

$(5 x^{9} - 20 x^{8} - 300 x^{7} + 1440 x^{6}) (x^{2} - 2 x + 1)$

Expand $(5 x^{9} - 20 x^{8} - 300 x^{7} + 1440 x^{6}) (x^{2} - 2 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$5 x^{9} x^{2} + 5 x^{9} (- 2 x) + 5 x^{9} \cdot 1 - 20 x^{8} x^{2} - 20 x^{8} (- 2 x) - 20 x^{8} \cdot 1 - 300 x^{7} x^{2} - 300 x^{7} (- 2 x) - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

Simplify terms.

Simplify each term.

Multiply $x^{9}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$5 (x^{2} x^{9}) + 5 x^{9} (- 2 x) + 5 x^{9} \cdot 1 - 20 x^{8} x^{2} - 20 x^{8} (- 2 x) - 20 x^{8} \cdot 1 - 300 x^{7} x^{2} - 300 x^{7} (- 2 x) - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

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

$5 x^{2 + 9} + 5 x^{9} (- 2 x) + 5 x^{9} \cdot 1 - 20 x^{8} x^{2} - 20 x^{8} (- 2 x) - 20 x^{8} \cdot 1 - 300 x^{7} x^{2} - 300 x^{7} (- 2 x) - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

Add $2$ and $9$ .

$5 x^{11} + 5 x^{9} (- 2 x) + 5 x^{9} \cdot 1 - 20 x^{8} x^{2} - 20 x^{8} (- 2 x) - 20 x^{8} \cdot 1 - 300 x^{7} x^{2} - 300 x^{7} (- 2 x) - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

Rewrite using the commutative property of multiplication.

$5 x^{11} + 5 \cdot - 2 x^{9} x + 5 x^{9} \cdot 1 - 20 x^{8} x^{2} - 20 x^{8} (- 2 x) - 20 x^{8} \cdot 1 - 300 x^{7} x^{2} - 300 x^{7} (- 2 x) - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

Multiply $x^{9}$ by $x$ by adding the exponents.

Move $x$ .

$5 x^{11} + 5 \cdot - 2 (x \cdot x^{9}) + 5 x^{9} \cdot 1 - 20 x^{8} x^{2} - 20 x^{8} (- 2 x) - 20 x^{8} \cdot 1 - 300 x^{7} x^{2} - 300 x^{7} (- 2 x) - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

Multiply $x$ by $x^{9}$ .

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

$5 x^{11} + 5 \cdot - 2 (x^{1} x^{9}) + 5 x^{9} \cdot 1 - 20 x^{8} x^{2} - 20 x^{8} (- 2 x) - 20 x^{8} \cdot 1 - 300 x^{7} x^{2} - 300 x^{7} (- 2 x) - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

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

$5 x^{11} + 5 \cdot - 2 x^{1 + 9} + 5 x^{9} \cdot 1 - 20 x^{8} x^{2} - 20 x^{8} (- 2 x) - 20 x^{8} \cdot 1 - 300 x^{7} x^{2} - 300 x^{7} (- 2 x) - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

Add $1$ and $9$ .

$5 x^{11} + 5 \cdot - 2 x^{10} + 5 x^{9} \cdot 1 - 20 x^{8} x^{2} - 20 x^{8} (- 2 x) - 20 x^{8} \cdot 1 - 300 x^{7} x^{2} - 300 x^{7} (- 2 x) - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

Multiply $5$ by $- 2$ .

$5 x^{11} - 10 x^{10} + 5 x^{9} \cdot 1 - 20 x^{8} x^{2} - 20 x^{8} (- 2 x) - 20 x^{8} \cdot 1 - 300 x^{7} x^{2} - 300 x^{7} (- 2 x) - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

Multiply $5$ by $1$ .

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{8} x^{2} - 20 x^{8} (- 2 x) - 20 x^{8} \cdot 1 - 300 x^{7} x^{2} - 300 x^{7} (- 2 x) - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

Multiply $x^{8}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 (x^{2} x^{8}) - 20 x^{8} (- 2 x) - 20 x^{8} \cdot 1 - 300 x^{7} x^{2} - 300 x^{7} (- 2 x) - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

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

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{2 + 8} - 20 x^{8} (- 2 x) - 20 x^{8} \cdot 1 - 300 x^{7} x^{2} - 300 x^{7} (- 2 x) - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

Add $2$ and $8$ .

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{10} - 20 x^{8} (- 2 x) - 20 x^{8} \cdot 1 - 300 x^{7} x^{2} - 300 x^{7} (- 2 x) - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

Rewrite using the commutative property of multiplication.

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{10} - 20 \cdot - 2 x^{8} x - 20 x^{8} \cdot 1 - 300 x^{7} x^{2} - 300 x^{7} (- 2 x) - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

Multiply $x^{8}$ by $x$ by adding the exponents.

Move $x$ .

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{10} - 20 \cdot - 2 (x \cdot x^{8}) - 20 x^{8} \cdot 1 - 300 x^{7} x^{2} - 300 x^{7} (- 2 x) - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

Multiply $x$ by $x^{8}$ .

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

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{10} - 20 \cdot - 2 (x^{1} x^{8}) - 20 x^{8} \cdot 1 - 300 x^{7} x^{2} - 300 x^{7} (- 2 x) - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

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

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{10} - 20 \cdot - 2 x^{1 + 8} - 20 x^{8} \cdot 1 - 300 x^{7} x^{2} - 300 x^{7} (- 2 x) - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

Add $1$ and $8$ .

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{10} - 20 \cdot - 2 x^{9} - 20 x^{8} \cdot 1 - 300 x^{7} x^{2} - 300 x^{7} (- 2 x) - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

Multiply $- 20$ by $- 2$ .

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{10} + 40 x^{9} - 20 x^{8} \cdot 1 - 300 x^{7} x^{2} - 300 x^{7} (- 2 x) - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

Multiply $- 20$ by $1$ .

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{10} + 40 x^{9} - 20 x^{8} - 300 x^{7} x^{2} - 300 x^{7} (- 2 x) - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

Multiply $x^{7}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{10} + 40 x^{9} - 20 x^{8} - 300 (x^{2} x^{7}) - 300 x^{7} (- 2 x) - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

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

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{10} + 40 x^{9} - 20 x^{8} - 300 x^{2 + 7} - 300 x^{7} (- 2 x) - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

Add $2$ and $7$ .

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{10} + 40 x^{9} - 20 x^{8} - 300 x^{9} - 300 x^{7} (- 2 x) - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

Rewrite using the commutative property of multiplication.

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{10} + 40 x^{9} - 20 x^{8} - 300 x^{9} - 300 \cdot - 2 x^{7} x - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

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

Move $x$ .

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{10} + 40 x^{9} - 20 x^{8} - 300 x^{9} - 300 \cdot - 2 (x \cdot x^{7}) - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

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

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

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{10} + 40 x^{9} - 20 x^{8} - 300 x^{9} - 300 \cdot - 2 (x^{1} x^{7}) - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

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

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{10} + 40 x^{9} - 20 x^{8} - 300 x^{9} - 300 \cdot - 2 x^{1 + 7} - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

Add $1$ and $7$ .

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{10} + 40 x^{9} - 20 x^{8} - 300 x^{9} - 300 \cdot - 2 x^{8} - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

Multiply $- 300$ by $- 2$ .

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{10} + 40 x^{9} - 20 x^{8} - 300 x^{9} + 600 x^{8} - 300 x^{7} \cdot 1 + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

Multiply $- 300$ by $1$ .

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{10} + 40 x^{9} - 20 x^{8} - 300 x^{9} + 600 x^{8} - 300 x^{7} + 1440 x^{6} x^{2} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

Multiply $x^{6}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{10} + 40 x^{9} - 20 x^{8} - 300 x^{9} + 600 x^{8} - 300 x^{7} + 1440 (x^{2} x^{6}) + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

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

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{10} + 40 x^{9} - 20 x^{8} - 300 x^{9} + 600 x^{8} - 300 x^{7} + 1440 x^{2 + 6} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

Add $2$ and $6$ .

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{10} + 40 x^{9} - 20 x^{8} - 300 x^{9} + 600 x^{8} - 300 x^{7} + 1440 x^{8} + 1440 x^{6} (- 2 x) + 1440 x^{6} \cdot 1$

Rewrite using the commutative property of multiplication.

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{10} + 40 x^{9} - 20 x^{8} - 300 x^{9} + 600 x^{8} - 300 x^{7} + 1440 x^{8} + 1440 \cdot - 2 x^{6} x + 1440 x^{6} \cdot 1$

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

Move $x$ .

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{10} + 40 x^{9} - 20 x^{8} - 300 x^{9} + 600 x^{8} - 300 x^{7} + 1440 x^{8} + 1440 \cdot - 2 (x \cdot x^{6}) + 1440 x^{6} \cdot 1$

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

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

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{10} + 40 x^{9} - 20 x^{8} - 300 x^{9} + 600 x^{8} - 300 x^{7} + 1440 x^{8} + 1440 \cdot - 2 (x^{1} x^{6}) + 1440 x^{6} \cdot 1$

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

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{10} + 40 x^{9} - 20 x^{8} - 300 x^{9} + 600 x^{8} - 300 x^{7} + 1440 x^{8} + 1440 \cdot - 2 x^{1 + 6} + 1440 x^{6} \cdot 1$

Add $1$ and $6$ .

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{10} + 40 x^{9} - 20 x^{8} - 300 x^{9} + 600 x^{8} - 300 x^{7} + 1440 x^{8} + 1440 \cdot - 2 x^{7} + 1440 x^{6} \cdot 1$

Multiply $1440$ by $- 2$ .

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{10} + 40 x^{9} - 20 x^{8} - 300 x^{9} + 600 x^{8} - 300 x^{7} + 1440 x^{8} - 2880 x^{7} + 1440 x^{6} \cdot 1$

Multiply $1440$ by $1$ .

$5 x^{11} - 10 x^{10} + 5 x^{9} - 20 x^{10} + 40 x^{9} - 20 x^{8} - 300 x^{9} + 600 x^{8} - 300 x^{7} + 1440 x^{8} - 2880 x^{7} + 1440 x^{6}$

Simplify by adding terms.

Subtract $20 x^{10}$ from $- 10 x^{10}$ .

$5 x^{11} - 30 x^{10} + 5 x^{9} + 40 x^{9} - 20 x^{8} - 300 x^{9} + 600 x^{8} - 300 x^{7} + 1440 x^{8} - 2880 x^{7} + 1440 x^{6}$

Add $5 x^{9}$ and $40 x^{9}$ .

$5 x^{11} - 30 x^{10} + 45 x^{9} - 20 x^{8} - 300 x^{9} + 600 x^{8} - 300 x^{7} + 1440 x^{8} - 2880 x^{7} + 1440 x^{6}$

Subtract $300 x^{9}$ from $45 x^{9}$ .

$5 x^{11} - 30 x^{10} - 255 x^{9} - 20 x^{8} + 600 x^{8} - 300 x^{7} + 1440 x^{8} - 2880 x^{7} + 1440 x^{6}$

Add $- 20 x^{8}$ and $600 x^{8}$ .

$5 x^{11} - 30 x^{10} - 255 x^{9} + 580 x^{8} - 300 x^{7} + 1440 x^{8} - 2880 x^{7} + 1440 x^{6}$

Add $580 x^{8}$ and $1440 x^{8}$ .

$5 x^{11} - 30 x^{10} - 255 x^{9} + 2020 x^{8} - 300 x^{7} - 2880 x^{7} + 1440 x^{6}$

Subtract $2880 x^{7}$ from $- 300 x^{7}$ .

$5 x^{11} - 30 x^{10} - 255 x^{9} + 2020 x^{8} - 3180 x^{7} + 1440 x^{6}$

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.

$5 x^{11} \to 11$

$- 30 x^{10} \to 10$

$- 255 x^{9} \to 9$

$2020 x^{8} \to 8$

$- 3180 x^{7} \to 7$

$1440 x^{6} \to 6$

The largest exponent is the degree of the polynomial.

$11$

Step 3

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

$5 x^{11}$

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.

$5 x^{11}$

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

$5$

Step 5

List the results.

Polynomial Degree: $11$

Leading Term: $5 x^{11}$

Leading Coefficient: $5$

Find the Degree, Leading Term, and Leading Coefficient 5x^6(x-6)^2(x+8)(x^2-2x+1)

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