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

${(x^{5} - 2)}^{2} {(x^{2} + 3)}^{3}$

Step 1

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

Rewrite ${(x^{5} - 2)}^{2}$ as $(x^{5} - 2) (x^{5} - 2)$ .

$(x^{5} - 2) (x^{5} - 2) {(x^{2} + 3)}^{3}$

Expand $(x^{5} - 2) (x^{5} - 2)$ using the FOIL Method.

Apply the distributive property.

$(x^{5} (x^{5} - 2) - 2 (x^{5} - 2)) {(x^{2} + 3)}^{3}$

Apply the distributive property.

$(x^{5} x^{5} + x^{5} \cdot - 2 - 2 (x^{5} - 2)) {(x^{2} + 3)}^{3}$

Apply the distributive property.

$(x^{5} x^{5} + x^{5} \cdot - 2 - 2 x^{5} - 2 \cdot - 2) {(x^{2} + 3)}^{3}$

Simplify and combine like terms.

Simplify each term.

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

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

$(x^{5 + 5} + x^{5} \cdot - 2 - 2 x^{5} - 2 \cdot - 2) {(x^{2} + 3)}^{3}$

Add $5$ and $5$ .

$(x^{10} + x^{5} \cdot - 2 - 2 x^{5} - 2 \cdot - 2) {(x^{2} + 3)}^{3}$

Move $- 2$ to the left of $x^{5}$ .

$(x^{10} - 2 \cdot x^{5} - 2 x^{5} - 2 \cdot - 2) {(x^{2} + 3)}^{3}$

Multiply $- 2$ by $- 2$ .

$(x^{10} - 2 x^{5} - 2 x^{5} + 4) {(x^{2} + 3)}^{3}$

Subtract $2 x^{5}$ from $- 2 x^{5}$ .

$(x^{10} - 4 x^{5} + 4) {(x^{2} + 3)}^{3}$

Use the Binomial Theorem.

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

Simplify each term.

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

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

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

Multiply $2$ by $3$ .

$(x^{10} - 4 x^{5} + 4) (x^{6} + 3 {(x^{2})}^{2} \cdot 3 + 3 x^{2} \cdot 3^{2} + 3^{3})$

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

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

$(x^{10} - 4 x^{5} + 4) (x^{6} + 3 x^{2 \cdot 2} \cdot 3 + 3 x^{2} \cdot 3^{2} + 3^{3})$

Multiply $2$ by $2$ .

$(x^{10} - 4 x^{5} + 4) (x^{6} + 3 x^{4} \cdot 3 + 3 x^{2} \cdot 3^{2} + 3^{3})$

Multiply $3$ by $3$ .

$(x^{10} - 4 x^{5} + 4) (x^{6} + 9 x^{4} + 3 x^{2} \cdot 3^{2} + 3^{3})$

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

Move $3^{2}$ .

$(x^{10} - 4 x^{5} + 4) (x^{6} + 9 x^{4} + 3^{2} \cdot 3 x^{2} + 3^{3})$

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

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

$(x^{10} - 4 x^{5} + 4) (x^{6} + 9 x^{4} + 3^{2} \cdot 3^{1} x^{2} + 3^{3})$

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

$(x^{10} - 4 x^{5} + 4) (x^{6} + 9 x^{4} + 3^{2 + 1} x^{2} + 3^{3})$

Add $2$ and $1$ .

$(x^{10} - 4 x^{5} + 4) (x^{6} + 9 x^{4} + 3^{3} x^{2} + 3^{3})$

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

$(x^{10} - 4 x^{5} + 4) (x^{6} + 9 x^{4} + 27 x^{2} + 3^{3})$

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

$(x^{10} - 4 x^{5} + 4) (x^{6} + 9 x^{4} + 27 x^{2} + 27)$

Expand $(x^{10} - 4 x^{5} + 4) (x^{6} + 9 x^{4} + 27 x^{2} + 27)$ by multiplying each term in the first expression by each term in the second expression.

$x^{10} x^{6} + x^{10} (9 x^{4}) + x^{10} (27 x^{2}) + x^{10} \cdot 27 - 4 x^{5} x^{6} - 4 x^{5} (9 x^{4}) - 4 x^{5} (27 x^{2}) - 4 x^{5} \cdot 27 + 4 x^{6} + 4 (9 x^{4}) + 4 (27 x^{2}) + 4 \cdot 27$

Simplify terms.

Simplify each term.

Multiply $x^{10}$ by $x^{6}$ by adding the exponents.

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

$x^{10 + 6} + x^{10} (9 x^{4}) + x^{10} (27 x^{2}) + x^{10} \cdot 27 - 4 x^{5} x^{6} - 4 x^{5} (9 x^{4}) - 4 x^{5} (27 x^{2}) - 4 x^{5} \cdot 27 + 4 x^{6} + 4 (9 x^{4}) + 4 (27 x^{2}) + 4 \cdot 27$

Add $10$ and $6$ .

$x^{16} + x^{10} (9 x^{4}) + x^{10} (27 x^{2}) + x^{10} \cdot 27 - 4 x^{5} x^{6} - 4 x^{5} (9 x^{4}) - 4 x^{5} (27 x^{2}) - 4 x^{5} \cdot 27 + 4 x^{6} + 4 (9 x^{4}) + 4 (27 x^{2}) + 4 \cdot 27$

Rewrite using the commutative property of multiplication.

$x^{16} + 9 x^{10} x^{4} + x^{10} (27 x^{2}) + x^{10} \cdot 27 - 4 x^{5} x^{6} - 4 x^{5} (9 x^{4}) - 4 x^{5} (27 x^{2}) - 4 x^{5} \cdot 27 + 4 x^{6} + 4 (9 x^{4}) + 4 (27 x^{2}) + 4 \cdot 27$

Multiply $x^{10}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$x^{16} + 9 (x^{4} x^{10}) + x^{10} (27 x^{2}) + x^{10} \cdot 27 - 4 x^{5} x^{6} - 4 x^{5} (9 x^{4}) - 4 x^{5} (27 x^{2}) - 4 x^{5} \cdot 27 + 4 x^{6} + 4 (9 x^{4}) + 4 (27 x^{2}) + 4 \cdot 27$

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

$x^{16} + 9 x^{4 + 10} + x^{10} (27 x^{2}) + x^{10} \cdot 27 - 4 x^{5} x^{6} - 4 x^{5} (9 x^{4}) - 4 x^{5} (27 x^{2}) - 4 x^{5} \cdot 27 + 4 x^{6} + 4 (9 x^{4}) + 4 (27 x^{2}) + 4 \cdot 27$

Add $4$ and $10$ .

$x^{16} + 9 x^{14} + x^{10} (27 x^{2}) + x^{10} \cdot 27 - 4 x^{5} x^{6} - 4 x^{5} (9 x^{4}) - 4 x^{5} (27 x^{2}) - 4 x^{5} \cdot 27 + 4 x^{6} + 4 (9 x^{4}) + 4 (27 x^{2}) + 4 \cdot 27$

Rewrite using the commutative property of multiplication.

$x^{16} + 9 x^{14} + 27 x^{10} x^{2} + x^{10} \cdot 27 - 4 x^{5} x^{6} - 4 x^{5} (9 x^{4}) - 4 x^{5} (27 x^{2}) - 4 x^{5} \cdot 27 + 4 x^{6} + 4 (9 x^{4}) + 4 (27 x^{2}) + 4 \cdot 27$

Multiply $x^{10}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$x^{16} + 9 x^{14} + 27 (x^{2} x^{10}) + x^{10} \cdot 27 - 4 x^{5} x^{6} - 4 x^{5} (9 x^{4}) - 4 x^{5} (27 x^{2}) - 4 x^{5} \cdot 27 + 4 x^{6} + 4 (9 x^{4}) + 4 (27 x^{2}) + 4 \cdot 27$

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

$x^{16} + 9 x^{14} + 27 x^{2 + 10} + x^{10} \cdot 27 - 4 x^{5} x^{6} - 4 x^{5} (9 x^{4}) - 4 x^{5} (27 x^{2}) - 4 x^{5} \cdot 27 + 4 x^{6} + 4 (9 x^{4}) + 4 (27 x^{2}) + 4 \cdot 27$

Add $2$ and $10$ .

$x^{16} + 9 x^{14} + 27 x^{12} + x^{10} \cdot 27 - 4 x^{5} x^{6} - 4 x^{5} (9 x^{4}) - 4 x^{5} (27 x^{2}) - 4 x^{5} \cdot 27 + 4 x^{6} + 4 (9 x^{4}) + 4 (27 x^{2}) + 4 \cdot 27$

Move $27$ to the left of $x^{10}$ .

$x^{16} + 9 x^{14} + 27 x^{12} + 27 \cdot x^{10} - 4 x^{5} x^{6} - 4 x^{5} (9 x^{4}) - 4 x^{5} (27 x^{2}) - 4 x^{5} \cdot 27 + 4 x^{6} + 4 (9 x^{4}) + 4 (27 x^{2}) + 4 \cdot 27$

Multiply $x^{5}$ by $x^{6}$ by adding the exponents.

Move $x^{6}$ .

$x^{16} + 9 x^{14} + 27 x^{12} + 27 x^{10} - 4 (x^{6} x^{5}) - 4 x^{5} (9 x^{4}) - 4 x^{5} (27 x^{2}) - 4 x^{5} \cdot 27 + 4 x^{6} + 4 (9 x^{4}) + 4 (27 x^{2}) + 4 \cdot 27$

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

$x^{16} + 9 x^{14} + 27 x^{12} + 27 x^{10} - 4 x^{6 + 5} - 4 x^{5} (9 x^{4}) - 4 x^{5} (27 x^{2}) - 4 x^{5} \cdot 27 + 4 x^{6} + 4 (9 x^{4}) + 4 (27 x^{2}) + 4 \cdot 27$

Add $6$ and $5$ .

$x^{16} + 9 x^{14} + 27 x^{12} + 27 x^{10} - 4 x^{11} - 4 x^{5} (9 x^{4}) - 4 x^{5} (27 x^{2}) - 4 x^{5} \cdot 27 + 4 x^{6} + 4 (9 x^{4}) + 4 (27 x^{2}) + 4 \cdot 27$

Rewrite using the commutative property of multiplication.

$x^{16} + 9 x^{14} + 27 x^{12} + 27 x^{10} - 4 x^{11} - 4 \cdot 9 x^{5} x^{4} - 4 x^{5} (27 x^{2}) - 4 x^{5} \cdot 27 + 4 x^{6} + 4 (9 x^{4}) + 4 (27 x^{2}) + 4 \cdot 27$

Multiply $x^{5}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$x^{16} + 9 x^{14} + 27 x^{12} + 27 x^{10} - 4 x^{11} - 4 \cdot 9 (x^{4} x^{5}) - 4 x^{5} (27 x^{2}) - 4 x^{5} \cdot 27 + 4 x^{6} + 4 (9 x^{4}) + 4 (27 x^{2}) + 4 \cdot 27$

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

$x^{16} + 9 x^{14} + 27 x^{12} + 27 x^{10} - 4 x^{11} - 4 \cdot 9 x^{4 + 5} - 4 x^{5} (27 x^{2}) - 4 x^{5} \cdot 27 + 4 x^{6} + 4 (9 x^{4}) + 4 (27 x^{2}) + 4 \cdot 27$

Add $4$ and $5$ .

$x^{16} + 9 x^{14} + 27 x^{12} + 27 x^{10} - 4 x^{11} - 4 \cdot 9 x^{9} - 4 x^{5} (27 x^{2}) - 4 x^{5} \cdot 27 + 4 x^{6} + 4 (9 x^{4}) + 4 (27 x^{2}) + 4 \cdot 27$

Multiply $- 4$ by $9$ .

$x^{16} + 9 x^{14} + 27 x^{12} + 27 x^{10} - 4 x^{11} - 36 x^{9} - 4 x^{5} (27 x^{2}) - 4 x^{5} \cdot 27 + 4 x^{6} + 4 (9 x^{4}) + 4 (27 x^{2}) + 4 \cdot 27$

Rewrite using the commutative property of multiplication.

$x^{16} + 9 x^{14} + 27 x^{12} + 27 x^{10} - 4 x^{11} - 36 x^{9} - 4 \cdot 27 x^{5} x^{2} - 4 x^{5} \cdot 27 + 4 x^{6} + 4 (9 x^{4}) + 4 (27 x^{2}) + 4 \cdot 27$

Multiply $x^{5}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$x^{16} + 9 x^{14} + 27 x^{12} + 27 x^{10} - 4 x^{11} - 36 x^{9} - 4 \cdot 27 (x^{2} x^{5}) - 4 x^{5} \cdot 27 + 4 x^{6} + 4 (9 x^{4}) + 4 (27 x^{2}) + 4 \cdot 27$

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

$x^{16} + 9 x^{14} + 27 x^{12} + 27 x^{10} - 4 x^{11} - 36 x^{9} - 4 \cdot 27 x^{2 + 5} - 4 x^{5} \cdot 27 + 4 x^{6} + 4 (9 x^{4}) + 4 (27 x^{2}) + 4 \cdot 27$

Add $2$ and $5$ .

$x^{16} + 9 x^{14} + 27 x^{12} + 27 x^{10} - 4 x^{11} - 36 x^{9} - 4 \cdot 27 x^{7} - 4 x^{5} \cdot 27 + 4 x^{6} + 4 (9 x^{4}) + 4 (27 x^{2}) + 4 \cdot 27$

Multiply $- 4$ by $27$ .

$x^{16} + 9 x^{14} + 27 x^{12} + 27 x^{10} - 4 x^{11} - 36 x^{9} - 108 x^{7} - 4 x^{5} \cdot 27 + 4 x^{6} + 4 (9 x^{4}) + 4 (27 x^{2}) + 4 \cdot 27$

Multiply $27$ by $- 4$ .

$x^{16} + 9 x^{14} + 27 x^{12} + 27 x^{10} - 4 x^{11} - 36 x^{9} - 108 x^{7} - 108 x^{5} + 4 x^{6} + 4 (9 x^{4}) + 4 (27 x^{2}) + 4 \cdot 27$

Multiply $9$ by $4$ .

$x^{16} + 9 x^{14} + 27 x^{12} + 27 x^{10} - 4 x^{11} - 36 x^{9} - 108 x^{7} - 108 x^{5} + 4 x^{6} + 36 x^{4} + 4 (27 x^{2}) + 4 \cdot 27$

Multiply $27$ by $4$ .

$x^{16} + 9 x^{14} + 27 x^{12} + 27 x^{10} - 4 x^{11} - 36 x^{9} - 108 x^{7} - 108 x^{5} + 4 x^{6} + 36 x^{4} + 108 x^{2} + 4 \cdot 27$

Multiply $4$ by $27$ .

$x^{16} + 9 x^{14} + 27 x^{12} + 27 x^{10} - 4 x^{11} - 36 x^{9} - 108 x^{7} - 108 x^{5} + 4 x^{6} + 36 x^{4} + 108 x^{2} + 108$

Reorder.

Move $- 108 x^{5}$ .

$x^{16} + 9 x^{14} + 27 x^{12} + 27 x^{10} - 4 x^{11} - 36 x^{9} - 108 x^{7} + 4 x^{6} - 108 x^{5} + 36 x^{4} + 108 x^{2} + 108$

Move $27 x^{10}$ .

$x^{16} + 9 x^{14} + 27 x^{12} - 4 x^{11} + 27 x^{10} - 36 x^{9} - 108 x^{7} + 4 x^{6} - 108 x^{5} + 36 x^{4} + 108 x^{2} + 108$

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.

$x^{16} \to 16$

$9 x^{14} \to 14$

$27 x^{12} \to 12$

$- 4 x^{11} \to 11$

$27 x^{10} \to 10$

$- 36 x^{9} \to 9$

$- 108 x^{7} \to 7$

$4 x^{6} \to 6$

$- 108 x^{5} \to 5$

$36 x^{4} \to 4$

$108 x^{2} \to 2$

$108 \to 0$

The largest exponent is the degree of the polynomial.

$16$

Step 3

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

$x^{16}$

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.

$x^{16}$

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

$1$

Step 5

List the results.

Polynomial Degree: $16$

Leading Term: $x^{16}$

Leading Coefficient: $1$

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

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