Find the Degree, Leading Term, and Leading Coefficient (x^6-1)^2(x^2+3)^3

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

Step 1

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

Rewrite ${(x^{6} - 1)}^{2}$ as $(x^{6} - 1) (x^{6} - 1)$ .

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

Expand $(x^{6} - 1) (x^{6} - 1)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

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

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

Add $6$ and $6$ .

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

Move $- 1$ to the left of $x^{6}$ .

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

Rewrite $- 1 x^{6}$ as $- x^{6}$ .

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

Rewrite $- 1 x^{6}$ as $- x^{6}$ .

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

Multiply $- 1$ by $- 1$ .

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

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

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

Use the Binomial Theorem.

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

Simplify each term.

Multiply the exponents in ${(x^{2})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Multiply $2$ by $3$ .

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

Multiply the exponents in ${(x^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Multiply $2$ by $2$ .

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

Multiply $3$ by $3$ .

$(x^{12} - 2 x^{6} + 1) (x^{6} + 9 x^{4} + 3 x^{2} \cdot 3^{2} + 3^{3})$

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

Move $3^{2}$ .

$(x^{12} - 2 x^{6} + 1) (x^{6} + 9 x^{4} + 3^{2} \cdot 3 x^{2} + 3^{3})$

Multiply $3^{2}$ by $3$ .

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

$(x^{12} - 2 x^{6} + 1) (x^{6} + 9 x^{4} + 3^{2} \cdot 3^{1} x^{2} + 3^{3})$

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

$(x^{12} - 2 x^{6} + 1) (x^{6} + 9 x^{4} + 3^{2 + 1} x^{2} + 3^{3})$

Add $2$ and $1$ .

$(x^{12} - 2 x^{6} + 1) (x^{6} + 9 x^{4} + 3^{3} x^{2} + 3^{3})$

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

$(x^{12} - 2 x^{6} + 1) (x^{6} + 9 x^{4} + 27 x^{2} + 3^{3})$

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

$(x^{12} - 2 x^{6} + 1) (x^{6} + 9 x^{4} + 27 x^{2} + 27)$

Expand $(x^{12} - 2 x^{6} + 1) (x^{6} + 9 x^{4} + 27 x^{2} + 27)$ by multiplying each term in the first expression by each term in the second expression.

$x^{12} x^{6} + x^{12} (9 x^{4}) + x^{12} (27 x^{2}) + x^{12} \cdot 27 - 2 x^{6} x^{6} - 2 x^{6} (9 x^{4}) - 2 x^{6} (27 x^{2}) - 2 x^{6} \cdot 27 + 1 x^{6} + 1 (9 x^{4}) + 1 (27 x^{2}) + 1 \cdot 27$

Simplify terms.

Simplify each term.

Multiply $x^{12}$ by $x^{6}$ by adding the exponents.

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

$x^{12 + 6} + x^{12} (9 x^{4}) + x^{12} (27 x^{2}) + x^{12} \cdot 27 - 2 x^{6} x^{6} - 2 x^{6} (9 x^{4}) - 2 x^{6} (27 x^{2}) - 2 x^{6} \cdot 27 + 1 x^{6} + 1 (9 x^{4}) + 1 (27 x^{2}) + 1 \cdot 27$

Add $12$ and $6$ .

$x^{18} + x^{12} (9 x^{4}) + x^{12} (27 x^{2}) + x^{12} \cdot 27 - 2 x^{6} x^{6} - 2 x^{6} (9 x^{4}) - 2 x^{6} (27 x^{2}) - 2 x^{6} \cdot 27 + 1 x^{6} + 1 (9 x^{4}) + 1 (27 x^{2}) + 1 \cdot 27$

Rewrite using the commutative property of multiplication.

$x^{18} + 9 x^{12} x^{4} + x^{12} (27 x^{2}) + x^{12} \cdot 27 - 2 x^{6} x^{6} - 2 x^{6} (9 x^{4}) - 2 x^{6} (27 x^{2}) - 2 x^{6} \cdot 27 + 1 x^{6} + 1 (9 x^{4}) + 1 (27 x^{2}) + 1 \cdot 27$

Multiply $x^{12}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$x^{18} + 9 (x^{4} x^{12}) + x^{12} (27 x^{2}) + x^{12} \cdot 27 - 2 x^{6} x^{6} - 2 x^{6} (9 x^{4}) - 2 x^{6} (27 x^{2}) - 2 x^{6} \cdot 27 + 1 x^{6} + 1 (9 x^{4}) + 1 (27 x^{2}) + 1 \cdot 27$

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

$x^{18} + 9 x^{4 + 12} + x^{12} (27 x^{2}) + x^{12} \cdot 27 - 2 x^{6} x^{6} - 2 x^{6} (9 x^{4}) - 2 x^{6} (27 x^{2}) - 2 x^{6} \cdot 27 + 1 x^{6} + 1 (9 x^{4}) + 1 (27 x^{2}) + 1 \cdot 27$

Add $4$ and $12$ .

$x^{18} + 9 x^{16} + x^{12} (27 x^{2}) + x^{12} \cdot 27 - 2 x^{6} x^{6} - 2 x^{6} (9 x^{4}) - 2 x^{6} (27 x^{2}) - 2 x^{6} \cdot 27 + 1 x^{6} + 1 (9 x^{4}) + 1 (27 x^{2}) + 1 \cdot 27$

Rewrite using the commutative property of multiplication.

$x^{18} + 9 x^{16} + 27 x^{12} x^{2} + x^{12} \cdot 27 - 2 x^{6} x^{6} - 2 x^{6} (9 x^{4}) - 2 x^{6} (27 x^{2}) - 2 x^{6} \cdot 27 + 1 x^{6} + 1 (9 x^{4}) + 1 (27 x^{2}) + 1 \cdot 27$

Multiply $x^{12}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$x^{18} + 9 x^{16} + 27 (x^{2} x^{12}) + x^{12} \cdot 27 - 2 x^{6} x^{6} - 2 x^{6} (9 x^{4}) - 2 x^{6} (27 x^{2}) - 2 x^{6} \cdot 27 + 1 x^{6} + 1 (9 x^{4}) + 1 (27 x^{2}) + 1 \cdot 27$

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

$x^{18} + 9 x^{16} + 27 x^{2 + 12} + x^{12} \cdot 27 - 2 x^{6} x^{6} - 2 x^{6} (9 x^{4}) - 2 x^{6} (27 x^{2}) - 2 x^{6} \cdot 27 + 1 x^{6} + 1 (9 x^{4}) + 1 (27 x^{2}) + 1 \cdot 27$

Add $2$ and $12$ .

$x^{18} + 9 x^{16} + 27 x^{14} + x^{12} \cdot 27 - 2 x^{6} x^{6} - 2 x^{6} (9 x^{4}) - 2 x^{6} (27 x^{2}) - 2 x^{6} \cdot 27 + 1 x^{6} + 1 (9 x^{4}) + 1 (27 x^{2}) + 1 \cdot 27$

Move $27$ to the left of $x^{12}$ .

$x^{18} + 9 x^{16} + 27 x^{14} + 27 \cdot x^{12} - 2 x^{6} x^{6} - 2 x^{6} (9 x^{4}) - 2 x^{6} (27 x^{2}) - 2 x^{6} \cdot 27 + 1 x^{6} + 1 (9 x^{4}) + 1 (27 x^{2}) + 1 \cdot 27$

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

Move $x^{6}$ .

$x^{18} + 9 x^{16} + 27 x^{14} + 27 x^{12} - 2 (x^{6} x^{6}) - 2 x^{6} (9 x^{4}) - 2 x^{6} (27 x^{2}) - 2 x^{6} \cdot 27 + 1 x^{6} + 1 (9 x^{4}) + 1 (27 x^{2}) + 1 \cdot 27$

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

$x^{18} + 9 x^{16} + 27 x^{14} + 27 x^{12} - 2 x^{6 + 6} - 2 x^{6} (9 x^{4}) - 2 x^{6} (27 x^{2}) - 2 x^{6} \cdot 27 + 1 x^{6} + 1 (9 x^{4}) + 1 (27 x^{2}) + 1 \cdot 27$

Add $6$ and $6$ .

$x^{18} + 9 x^{16} + 27 x^{14} + 27 x^{12} - 2 x^{12} - 2 x^{6} (9 x^{4}) - 2 x^{6} (27 x^{2}) - 2 x^{6} \cdot 27 + 1 x^{6} + 1 (9 x^{4}) + 1 (27 x^{2}) + 1 \cdot 27$

Rewrite using the commutative property of multiplication.

$x^{18} + 9 x^{16} + 27 x^{14} + 27 x^{12} - 2 x^{12} - 2 \cdot 9 x^{6} x^{4} - 2 x^{6} (27 x^{2}) - 2 x^{6} \cdot 27 + 1 x^{6} + 1 (9 x^{4}) + 1 (27 x^{2}) + 1 \cdot 27$

Multiply $x^{6}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$x^{18} + 9 x^{16} + 27 x^{14} + 27 x^{12} - 2 x^{12} - 2 \cdot 9 (x^{4} x^{6}) - 2 x^{6} (27 x^{2}) - 2 x^{6} \cdot 27 + 1 x^{6} + 1 (9 x^{4}) + 1 (27 x^{2}) + 1 \cdot 27$

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

$x^{18} + 9 x^{16} + 27 x^{14} + 27 x^{12} - 2 x^{12} - 2 \cdot 9 x^{4 + 6} - 2 x^{6} (27 x^{2}) - 2 x^{6} \cdot 27 + 1 x^{6} + 1 (9 x^{4}) + 1 (27 x^{2}) + 1 \cdot 27$

Add $4$ and $6$ .

$x^{18} + 9 x^{16} + 27 x^{14} + 27 x^{12} - 2 x^{12} - 2 \cdot 9 x^{10} - 2 x^{6} (27 x^{2}) - 2 x^{6} \cdot 27 + 1 x^{6} + 1 (9 x^{4}) + 1 (27 x^{2}) + 1 \cdot 27$

Multiply $- 2$ by $9$ .

$x^{18} + 9 x^{16} + 27 x^{14} + 27 x^{12} - 2 x^{12} - 18 x^{10} - 2 x^{6} (27 x^{2}) - 2 x^{6} \cdot 27 + 1 x^{6} + 1 (9 x^{4}) + 1 (27 x^{2}) + 1 \cdot 27$

Rewrite using the commutative property of multiplication.

$x^{18} + 9 x^{16} + 27 x^{14} + 27 x^{12} - 2 x^{12} - 18 x^{10} - 2 \cdot 27 x^{6} x^{2} - 2 x^{6} \cdot 27 + 1 x^{6} + 1 (9 x^{4}) + 1 (27 x^{2}) + 1 \cdot 27$

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

Move $x^{2}$ .

$x^{18} + 9 x^{16} + 27 x^{14} + 27 x^{12} - 2 x^{12} - 18 x^{10} - 2 \cdot 27 (x^{2} x^{6}) - 2 x^{6} \cdot 27 + 1 x^{6} + 1 (9 x^{4}) + 1 (27 x^{2}) + 1 \cdot 27$

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

$x^{18} + 9 x^{16} + 27 x^{14} + 27 x^{12} - 2 x^{12} - 18 x^{10} - 2 \cdot 27 x^{2 + 6} - 2 x^{6} \cdot 27 + 1 x^{6} + 1 (9 x^{4}) + 1 (27 x^{2}) + 1 \cdot 27$

Add $2$ and $6$ .

$x^{18} + 9 x^{16} + 27 x^{14} + 27 x^{12} - 2 x^{12} - 18 x^{10} - 2 \cdot 27 x^{8} - 2 x^{6} \cdot 27 + 1 x^{6} + 1 (9 x^{4}) + 1 (27 x^{2}) + 1 \cdot 27$

Multiply $- 2$ by $27$ .

$x^{18} + 9 x^{16} + 27 x^{14} + 27 x^{12} - 2 x^{12} - 18 x^{10} - 54 x^{8} - 2 x^{6} \cdot 27 + 1 x^{6} + 1 (9 x^{4}) + 1 (27 x^{2}) + 1 \cdot 27$

Multiply $27$ by $- 2$ .

$x^{18} + 9 x^{16} + 27 x^{14} + 27 x^{12} - 2 x^{12} - 18 x^{10} - 54 x^{8} - 54 x^{6} + 1 x^{6} + 1 (9 x^{4}) + 1 (27 x^{2}) + 1 \cdot 27$

Multiply $x^{6}$ by $1$ .

$x^{18} + 9 x^{16} + 27 x^{14} + 27 x^{12} - 2 x^{12} - 18 x^{10} - 54 x^{8} - 54 x^{6} + x^{6} + 1 (9 x^{4}) + 1 (27 x^{2}) + 1 \cdot 27$

Multiply $9 x^{4}$ by $1$ .

$x^{18} + 9 x^{16} + 27 x^{14} + 27 x^{12} - 2 x^{12} - 18 x^{10} - 54 x^{8} - 54 x^{6} + x^{6} + 9 x^{4} + 1 (27 x^{2}) + 1 \cdot 27$

Multiply $27 x^{2}$ by $1$ .

$x^{18} + 9 x^{16} + 27 x^{14} + 27 x^{12} - 2 x^{12} - 18 x^{10} - 54 x^{8} - 54 x^{6} + x^{6} + 9 x^{4} + 27 x^{2} + 1 \cdot 27$

Multiply $27$ by $1$ .

$x^{18} + 9 x^{16} + 27 x^{14} + 27 x^{12} - 2 x^{12} - 18 x^{10} - 54 x^{8} - 54 x^{6} + x^{6} + 9 x^{4} + 27 x^{2} + 27$

Simplify by adding terms.

Subtract $2 x^{12}$ from $27 x^{12}$ .

$x^{18} + 9 x^{16} + 27 x^{14} + 25 x^{12} - 18 x^{10} - 54 x^{8} - 54 x^{6} + x^{6} + 9 x^{4} + 27 x^{2} + 27$

Add $- 54 x^{6}$ and $x^{6}$ .

$x^{18} + 9 x^{16} + 27 x^{14} + 25 x^{12} - 18 x^{10} - 54 x^{8} - 53 x^{6} + 9 x^{4} + 27 x^{2} + 27$

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.

$x^{18} \to 18$

$9 x^{16} \to 16$

$27 x^{14} \to 14$

$25 x^{12} \to 12$

$- 18 x^{10} \to 10$

$- 54 x^{8} \to 8$

$- 53 x^{6} \to 6$

$9 x^{4} \to 4$

$27 x^{2} \to 2$

$27 \to 0$

The largest exponent is the degree of the polynomial.

$18$

Step 3

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

$x^{18}$

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.

$x^{18}$

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

$1$

Step 5

List the results.

Polynomial Degree: $18$

Leading Term: $x^{18}$

Leading Coefficient: $1$

Find the Degree, Leading Term, and Leading Coefficient (x^6-1)^2(x^2+3)^3

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