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

${(2 x + y^{2})}^{5}$

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

Use the Binomial Theorem.

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

Simplify terms.

Simplify each term.

Apply the product rule to $2 x$ .

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

Raise $2$ to the power of $5$ .

$32 x^{5} + 5 {(2 x)}^{4} y^{2} + 10 {(2 x)}^{3} {(y^{2})}^{2} + 10 {(2 x)}^{2} {(y^{2})}^{3} + 5 (2 x) {(y^{2})}^{4} + {(y^{2})}^{5}$

Apply the product rule to $2 x$ .

$32 x^{5} + 5 (2^{4} x^{4}) y^{2} + 10 {(2 x)}^{3} {(y^{2})}^{2} + 10 {(2 x)}^{2} {(y^{2})}^{3} + 5 (2 x) {(y^{2})}^{4} + {(y^{2})}^{5}$

Raise $2$ to the power of $4$ .

$32 x^{5} + 5 (16 x^{4}) y^{2} + 10 {(2 x)}^{3} {(y^{2})}^{2} + 10 {(2 x)}^{2} {(y^{2})}^{3} + 5 (2 x) {(y^{2})}^{4} + {(y^{2})}^{5}$

Multiply $16$ by $5$ .

$32 x^{5} + 80 x^{4} y^{2} + 10 {(2 x)}^{3} {(y^{2})}^{2} + 10 {(2 x)}^{2} {(y^{2})}^{3} + 5 (2 x) {(y^{2})}^{4} + {(y^{2})}^{5}$

Apply the product rule to $2 x$ .

$32 x^{5} + 80 x^{4} y^{2} + 10 (2^{3} x^{3}) {(y^{2})}^{2} + 10 {(2 x)}^{2} {(y^{2})}^{3} + 5 (2 x) {(y^{2})}^{4} + {(y^{2})}^{5}$

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

$32 x^{5} + 80 x^{4} y^{2} + 10 (8 x^{3}) {(y^{2})}^{2} + 10 {(2 x)}^{2} {(y^{2})}^{3} + 5 (2 x) {(y^{2})}^{4} + {(y^{2})}^{5}$

Multiply $8$ by $10$ .

$32 x^{5} + 80 x^{4} y^{2} + 80 x^{3} {(y^{2})}^{2} + 10 {(2 x)}^{2} {(y^{2})}^{3} + 5 (2 x) {(y^{2})}^{4} + {(y^{2})}^{5}$

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

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

$32 x^{5} + 80 x^{4} y^{2} + 80 x^{3} y^{2 \cdot 2} + 10 {(2 x)}^{2} {(y^{2})}^{3} + 5 (2 x) {(y^{2})}^{4} + {(y^{2})}^{5}$

Multiply $2$ by $2$ .

$32 x^{5} + 80 x^{4} y^{2} + 80 x^{3} y^{4} + 10 {(2 x)}^{2} {(y^{2})}^{3} + 5 (2 x) {(y^{2})}^{4} + {(y^{2})}^{5}$

Apply the product rule to $2 x$ .

$32 x^{5} + 80 x^{4} y^{2} + 80 x^{3} y^{4} + 10 (2^{2} x^{2}) {(y^{2})}^{3} + 5 (2 x) {(y^{2})}^{4} + {(y^{2})}^{5}$

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

$32 x^{5} + 80 x^{4} y^{2} + 80 x^{3} y^{4} + 10 (4 x^{2}) {(y^{2})}^{3} + 5 (2 x) {(y^{2})}^{4} + {(y^{2})}^{5}$

Multiply $4$ by $10$ .

$32 x^{5} + 80 x^{4} y^{2} + 80 x^{3} y^{4} + 40 x^{2} {(y^{2})}^{3} + 5 (2 x) {(y^{2})}^{4} + {(y^{2})}^{5}$

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

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

$32 x^{5} + 80 x^{4} y^{2} + 80 x^{3} y^{4} + 40 x^{2} y^{2 \cdot 3} + 5 (2 x) {(y^{2})}^{4} + {(y^{2})}^{5}$

Multiply $2$ by $3$ .

$32 x^{5} + 80 x^{4} y^{2} + 80 x^{3} y^{4} + 40 x^{2} y^{6} + 5 (2 x) {(y^{2})}^{4} + {(y^{2})}^{5}$

Multiply $2$ by $5$ .

$32 x^{5} + 80 x^{4} y^{2} + 80 x^{3} y^{4} + 40 x^{2} y^{6} + 10 x {(y^{2})}^{4} + {(y^{2})}^{5}$

Multiply the exponents in ${(y^{2})}^{4}$ .

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

$32 x^{5} + 80 x^{4} y^{2} + 80 x^{3} y^{4} + 40 x^{2} y^{6} + 10 x y^{2 \cdot 4} + {(y^{2})}^{5}$

Multiply $2$ by $4$ .

$32 x^{5} + 80 x^{4} y^{2} + 80 x^{3} y^{4} + 40 x^{2} y^{6} + 10 x y^{8} + {(y^{2})}^{5}$

Multiply the exponents in ${(y^{2})}^{5}$ .

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

$32 x^{5} + 80 x^{4} y^{2} + 80 x^{3} y^{4} + 40 x^{2} y^{6} + 10 x y^{8} + y^{2 \cdot 5}$

Multiply $2$ by $5$ .

$32 x^{5} + 80 x^{4} y^{2} + 80 x^{3} y^{4} + 40 x^{2} y^{6} + 10 x y^{8} + y^{10}$

Simplify the expression.

Move $32 x^{5}$ .

$80 x^{4} y^{2} + 80 x^{3} y^{4} + 40 x^{2} y^{6} + 10 x y^{8} + y^{10} + 32 x^{5}$

Move $80 x^{4} y^{2}$ .

$80 x^{3} y^{4} + 40 x^{2} y^{6} + 10 x y^{8} + y^{10} + 80 x^{4} y^{2} + 32 x^{5}$

Move $80 x^{3} y^{4}$ .

$40 x^{2} y^{6} + 10 x y^{8} + y^{10} + 80 x^{3} y^{4} + 80 x^{4} y^{2} + 32 x^{5}$

Move $40 x^{2} y^{6}$ .

$10 x y^{8} + y^{10} + 40 x^{2} y^{6} + 80 x^{3} y^{4} + 80 x^{4} y^{2} + 32 x^{5}$

Reorder $10 x y^{8}$ and $y^{10}$ .

$y^{10} + 10 x y^{8} + 40 x^{2} y^{6} + 80 x^{3} y^{4} + 80 x^{4} y^{2} + 32 x^{5}$

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.

$y^{10} \to 10$

$10 x y^{8} \to 9$

$40 x^{2} y^{6} \to 8$

$80 x^{3} y^{4} \to 7$

$80 x^{4} y^{2} \to 6$

$32 x^{5} \to 5$

The largest exponent is the degree of the polynomial.

$10$

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

$y^{10}$

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.

$y^{10}$

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

$1$

List the results.

Polynomial Degree: $10$

Leading Term: $y^{10}$

Leading Coefficient: $1$

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

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