Find the Degree, Leading Term, and Leading Coefficient (x^7-3)^2(x^2+2)^3

${(x^{7} - 3)}^{2} {(x^{2} + 2)}^{3}$

Step 1

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

Rewrite ${(x^{7} - 3)}^{2}$ as $(x^{7} - 3) (x^{7} - 3)$ .

$(x^{7} - 3) (x^{7} - 3) {(x^{2} + 2)}^{3}$

Expand $(x^{7} - 3) (x^{7} - 3)$ using the FOIL Method.

Apply the distributive property.

$(x^{7} (x^{7} - 3) - 3 (x^{7} - 3)) {(x^{2} + 2)}^{3}$

Apply the distributive property.

$(x^{7} x^{7} + x^{7} \cdot - 3 - 3 (x^{7} - 3)) {(x^{2} + 2)}^{3}$

Apply the distributive property.

$(x^{7} x^{7} + x^{7} \cdot - 3 - 3 x^{7} - 3 \cdot - 3) {(x^{2} + 2)}^{3}$

Simplify and combine like terms.

Simplify each term.

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

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

$(x^{7 + 7} + x^{7} \cdot - 3 - 3 x^{7} - 3 \cdot - 3) {(x^{2} + 2)}^{3}$

Add $7$ and $7$ .

$(x^{14} + x^{7} \cdot - 3 - 3 x^{7} - 3 \cdot - 3) {(x^{2} + 2)}^{3}$

Move $- 3$ to the left of $x^{7}$ .

$(x^{14} - 3 \cdot x^{7} - 3 x^{7} - 3 \cdot - 3) {(x^{2} + 2)}^{3}$

Multiply $- 3$ by $- 3$ .

$(x^{14} - 3 x^{7} - 3 x^{7} + 9) {(x^{2} + 2)}^{3}$

Subtract $3 x^{7}$ from $- 3 x^{7}$ .

$(x^{14} - 6 x^{7} + 9) {(x^{2} + 2)}^{3}$

Use the Binomial Theorem.

$(x^{14} - 6 x^{7} + 9) ({(x^{2})}^{3} + 3 {(x^{2})}^{2} \cdot 2 + 3 x^{2} \cdot 2^{2} + 2^{3})$

Simplify each term.

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

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

$(x^{14} - 6 x^{7} + 9) (x^{2 \cdot 3} + 3 {(x^{2})}^{2} \cdot 2 + 3 x^{2} \cdot 2^{2} + 2^{3})$

Multiply $2$ by $3$ .

$(x^{14} - 6 x^{7} + 9) (x^{6} + 3 {(x^{2})}^{2} \cdot 2 + 3 x^{2} \cdot 2^{2} + 2^{3})$

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

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

$(x^{14} - 6 x^{7} + 9) (x^{6} + 3 x^{2 \cdot 2} \cdot 2 + 3 x^{2} \cdot 2^{2} + 2^{3})$

Multiply $2$ by $2$ .

$(x^{14} - 6 x^{7} + 9) (x^{6} + 3 x^{4} \cdot 2 + 3 x^{2} \cdot 2^{2} + 2^{3})$

Multiply $2$ by $3$ .

$(x^{14} - 6 x^{7} + 9) (x^{6} + 6 x^{4} + 3 x^{2} \cdot 2^{2} + 2^{3})$

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

$(x^{14} - 6 x^{7} + 9) (x^{6} + 6 x^{4} + 3 x^{2} \cdot 4 + 2^{3})$

Multiply $4$ by $3$ .

$(x^{14} - 6 x^{7} + 9) (x^{6} + 6 x^{4} + 12 x^{2} + 2^{3})$

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

$(x^{14} - 6 x^{7} + 9) (x^{6} + 6 x^{4} + 12 x^{2} + 8)$

Expand $(x^{14} - 6 x^{7} + 9) (x^{6} + 6 x^{4} + 12 x^{2} + 8)$ by multiplying each term in the first expression by each term in the second expression.

$x^{14} x^{6} + x^{14} (6 x^{4}) + x^{14} (12 x^{2}) + x^{14} \cdot 8 - 6 x^{7} x^{6} - 6 x^{7} (6 x^{4}) - 6 x^{7} (12 x^{2}) - 6 x^{7} \cdot 8 + 9 x^{6} + 9 (6 x^{4}) + 9 (12 x^{2}) + 9 \cdot 8$

Simplify each term.

Multiply $x^{14}$ by $x^{6}$ by adding the exponents.

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

$x^{14 + 6} + x^{14} (6 x^{4}) + x^{14} (12 x^{2}) + x^{14} \cdot 8 - 6 x^{7} x^{6} - 6 x^{7} (6 x^{4}) - 6 x^{7} (12 x^{2}) - 6 x^{7} \cdot 8 + 9 x^{6} + 9 (6 x^{4}) + 9 (12 x^{2}) + 9 \cdot 8$

Add $14$ and $6$ .

$x^{20} + x^{14} (6 x^{4}) + x^{14} (12 x^{2}) + x^{14} \cdot 8 - 6 x^{7} x^{6} - 6 x^{7} (6 x^{4}) - 6 x^{7} (12 x^{2}) - 6 x^{7} \cdot 8 + 9 x^{6} + 9 (6 x^{4}) + 9 (12 x^{2}) + 9 \cdot 8$

Rewrite using the commutative property of multiplication.

$x^{20} + 6 x^{14} x^{4} + x^{14} (12 x^{2}) + x^{14} \cdot 8 - 6 x^{7} x^{6} - 6 x^{7} (6 x^{4}) - 6 x^{7} (12 x^{2}) - 6 x^{7} \cdot 8 + 9 x^{6} + 9 (6 x^{4}) + 9 (12 x^{2}) + 9 \cdot 8$

Multiply $x^{14}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$x^{20} + 6 (x^{4} x^{14}) + x^{14} (12 x^{2}) + x^{14} \cdot 8 - 6 x^{7} x^{6} - 6 x^{7} (6 x^{4}) - 6 x^{7} (12 x^{2}) - 6 x^{7} \cdot 8 + 9 x^{6} + 9 (6 x^{4}) + 9 (12 x^{2}) + 9 \cdot 8$

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

$x^{20} + 6 x^{4 + 14} + x^{14} (12 x^{2}) + x^{14} \cdot 8 - 6 x^{7} x^{6} - 6 x^{7} (6 x^{4}) - 6 x^{7} (12 x^{2}) - 6 x^{7} \cdot 8 + 9 x^{6} + 9 (6 x^{4}) + 9 (12 x^{2}) + 9 \cdot 8$

Add $4$ and $14$ .

$x^{20} + 6 x^{18} + x^{14} (12 x^{2}) + x^{14} \cdot 8 - 6 x^{7} x^{6} - 6 x^{7} (6 x^{4}) - 6 x^{7} (12 x^{2}) - 6 x^{7} \cdot 8 + 9 x^{6} + 9 (6 x^{4}) + 9 (12 x^{2}) + 9 \cdot 8$

Rewrite using the commutative property of multiplication.

$x^{20} + 6 x^{18} + 12 x^{14} x^{2} + x^{14} \cdot 8 - 6 x^{7} x^{6} - 6 x^{7} (6 x^{4}) - 6 x^{7} (12 x^{2}) - 6 x^{7} \cdot 8 + 9 x^{6} + 9 (6 x^{4}) + 9 (12 x^{2}) + 9 \cdot 8$

Multiply $x^{14}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$x^{20} + 6 x^{18} + 12 (x^{2} x^{14}) + x^{14} \cdot 8 - 6 x^{7} x^{6} - 6 x^{7} (6 x^{4}) - 6 x^{7} (12 x^{2}) - 6 x^{7} \cdot 8 + 9 x^{6} + 9 (6 x^{4}) + 9 (12 x^{2}) + 9 \cdot 8$

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

$x^{20} + 6 x^{18} + 12 x^{2 + 14} + x^{14} \cdot 8 - 6 x^{7} x^{6} - 6 x^{7} (6 x^{4}) - 6 x^{7} (12 x^{2}) - 6 x^{7} \cdot 8 + 9 x^{6} + 9 (6 x^{4}) + 9 (12 x^{2}) + 9 \cdot 8$

Add $2$ and $14$ .

$x^{20} + 6 x^{18} + 12 x^{16} + x^{14} \cdot 8 - 6 x^{7} x^{6} - 6 x^{7} (6 x^{4}) - 6 x^{7} (12 x^{2}) - 6 x^{7} \cdot 8 + 9 x^{6} + 9 (6 x^{4}) + 9 (12 x^{2}) + 9 \cdot 8$

Move $8$ to the left of $x^{14}$ .

$x^{20} + 6 x^{18} + 12 x^{16} + 8 \cdot x^{14} - 6 x^{7} x^{6} - 6 x^{7} (6 x^{4}) - 6 x^{7} (12 x^{2}) - 6 x^{7} \cdot 8 + 9 x^{6} + 9 (6 x^{4}) + 9 (12 x^{2}) + 9 \cdot 8$

Multiply $x^{7}$ by $x^{6}$ by adding the exponents.

Move $x^{6}$ .

$x^{20} + 6 x^{18} + 12 x^{16} + 8 x^{14} - 6 (x^{6} x^{7}) - 6 x^{7} (6 x^{4}) - 6 x^{7} (12 x^{2}) - 6 x^{7} \cdot 8 + 9 x^{6} + 9 (6 x^{4}) + 9 (12 x^{2}) + 9 \cdot 8$

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

$x^{20} + 6 x^{18} + 12 x^{16} + 8 x^{14} - 6 x^{6 + 7} - 6 x^{7} (6 x^{4}) - 6 x^{7} (12 x^{2}) - 6 x^{7} \cdot 8 + 9 x^{6} + 9 (6 x^{4}) + 9 (12 x^{2}) + 9 \cdot 8$

Add $6$ and $7$ .

$x^{20} + 6 x^{18} + 12 x^{16} + 8 x^{14} - 6 x^{13} - 6 x^{7} (6 x^{4}) - 6 x^{7} (12 x^{2}) - 6 x^{7} \cdot 8 + 9 x^{6} + 9 (6 x^{4}) + 9 (12 x^{2}) + 9 \cdot 8$

Rewrite using the commutative property of multiplication.

$x^{20} + 6 x^{18} + 12 x^{16} + 8 x^{14} - 6 x^{13} - 6 \cdot 6 x^{7} x^{4} - 6 x^{7} (12 x^{2}) - 6 x^{7} \cdot 8 + 9 x^{6} + 9 (6 x^{4}) + 9 (12 x^{2}) + 9 \cdot 8$

Multiply $x^{7}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$x^{20} + 6 x^{18} + 12 x^{16} + 8 x^{14} - 6 x^{13} - 6 \cdot 6 (x^{4} x^{7}) - 6 x^{7} (12 x^{2}) - 6 x^{7} \cdot 8 + 9 x^{6} + 9 (6 x^{4}) + 9 (12 x^{2}) + 9 \cdot 8$

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

$x^{20} + 6 x^{18} + 12 x^{16} + 8 x^{14} - 6 x^{13} - 6 \cdot 6 x^{4 + 7} - 6 x^{7} (12 x^{2}) - 6 x^{7} \cdot 8 + 9 x^{6} + 9 (6 x^{4}) + 9 (12 x^{2}) + 9 \cdot 8$

Add $4$ and $7$ .

$x^{20} + 6 x^{18} + 12 x^{16} + 8 x^{14} - 6 x^{13} - 6 \cdot 6 x^{11} - 6 x^{7} (12 x^{2}) - 6 x^{7} \cdot 8 + 9 x^{6} + 9 (6 x^{4}) + 9 (12 x^{2}) + 9 \cdot 8$

Multiply $- 6$ by $6$ .

$x^{20} + 6 x^{18} + 12 x^{16} + 8 x^{14} - 6 x^{13} - 36 x^{11} - 6 x^{7} (12 x^{2}) - 6 x^{7} \cdot 8 + 9 x^{6} + 9 (6 x^{4}) + 9 (12 x^{2}) + 9 \cdot 8$

Rewrite using the commutative property of multiplication.

$x^{20} + 6 x^{18} + 12 x^{16} + 8 x^{14} - 6 x^{13} - 36 x^{11} - 6 \cdot 12 x^{7} x^{2} - 6 x^{7} \cdot 8 + 9 x^{6} + 9 (6 x^{4}) + 9 (12 x^{2}) + 9 \cdot 8$

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

Move $x^{2}$ .

$x^{20} + 6 x^{18} + 12 x^{16} + 8 x^{14} - 6 x^{13} - 36 x^{11} - 6 \cdot 12 (x^{2} x^{7}) - 6 x^{7} \cdot 8 + 9 x^{6} + 9 (6 x^{4}) + 9 (12 x^{2}) + 9 \cdot 8$

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

$x^{20} + 6 x^{18} + 12 x^{16} + 8 x^{14} - 6 x^{13} - 36 x^{11} - 6 \cdot 12 x^{2 + 7} - 6 x^{7} \cdot 8 + 9 x^{6} + 9 (6 x^{4}) + 9 (12 x^{2}) + 9 \cdot 8$

Add $2$ and $7$ .

$x^{20} + 6 x^{18} + 12 x^{16} + 8 x^{14} - 6 x^{13} - 36 x^{11} - 6 \cdot 12 x^{9} - 6 x^{7} \cdot 8 + 9 x^{6} + 9 (6 x^{4}) + 9 (12 x^{2}) + 9 \cdot 8$

Multiply $- 6$ by $12$ .

$x^{20} + 6 x^{18} + 12 x^{16} + 8 x^{14} - 6 x^{13} - 36 x^{11} - 72 x^{9} - 6 x^{7} \cdot 8 + 9 x^{6} + 9 (6 x^{4}) + 9 (12 x^{2}) + 9 \cdot 8$

Multiply $8$ by $- 6$ .

$x^{20} + 6 x^{18} + 12 x^{16} + 8 x^{14} - 6 x^{13} - 36 x^{11} - 72 x^{9} - 48 x^{7} + 9 x^{6} + 9 (6 x^{4}) + 9 (12 x^{2}) + 9 \cdot 8$

Multiply $6$ by $9$ .

$x^{20} + 6 x^{18} + 12 x^{16} + 8 x^{14} - 6 x^{13} - 36 x^{11} - 72 x^{9} - 48 x^{7} + 9 x^{6} + 54 x^{4} + 9 (12 x^{2}) + 9 \cdot 8$

Multiply $12$ by $9$ .

$x^{20} + 6 x^{18} + 12 x^{16} + 8 x^{14} - 6 x^{13} - 36 x^{11} - 72 x^{9} - 48 x^{7} + 9 x^{6} + 54 x^{4} + 108 x^{2} + 9 \cdot 8$

Multiply $9$ by $8$ .

$x^{20} + 6 x^{18} + 12 x^{16} + 8 x^{14} - 6 x^{13} - 36 x^{11} - 72 x^{9} - 48 x^{7} + 9 x^{6} + 54 x^{4} + 108 x^{2} + 72$

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.

$x^{20} \to 20$

$6 x^{18} \to 18$

$12 x^{16} \to 16$

$8 x^{14} \to 14$

$- 6 x^{13} \to 13$

$- 36 x^{11} \to 11$

$- 72 x^{9} \to 9$

$- 48 x^{7} \to 7$

$9 x^{6} \to 6$

$54 x^{4} \to 4$

$108 x^{2} \to 2$

$72 \to 0$

The largest exponent is the degree of the polynomial.

$20$

Step 3

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

$x^{20}$

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.

$x^{20}$

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

$1$

Step 5

List the results.

Polynomial Degree: $20$

Leading Term: $x^{20}$

Leading Coefficient: $1$

Find the Degree, Leading Term, and Leading Coefficient (x^7-3)^2(x^2+2)^3

Related Questions

Support LCMGCF.com by disabling your adblocker.