Find the Degree, Leading Term, and Leading Coefficient (x+2y)^4

${(x + 2 y)}^{4}$

Step 1

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

Use the Binomial Theorem.

$x^{4} + 4 x^{3} (2 y) + 6 x^{2} {(2 y)}^{2} + 4 x {(2 y)}^{3} + {(2 y)}^{4}$

Simplify each term.

Rewrite using the commutative property of multiplication.

$x^{4} + 4 \cdot 2 x^{3} y + 6 x^{2} {(2 y)}^{2} + 4 x {(2 y)}^{3} + {(2 y)}^{4}$

Multiply $4$ by $2$ .

$x^{4} + 8 x^{3} y + 6 x^{2} {(2 y)}^{2} + 4 x {(2 y)}^{3} + {(2 y)}^{4}$

Apply the product rule to $2 y$ .

$x^{4} + 8 x^{3} y + 6 x^{2} (2^{2} y^{2}) + 4 x {(2 y)}^{3} + {(2 y)}^{4}$

Rewrite using the commutative property of multiplication.

$x^{4} + 8 x^{3} y + 6 \cdot 2^{2} x^{2} y^{2} + 4 x {(2 y)}^{3} + {(2 y)}^{4}$

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

$x^{4} + 8 x^{3} y + 6 \cdot 4 x^{2} y^{2} + 4 x {(2 y)}^{3} + {(2 y)}^{4}$

Multiply $6$ by $4$ .

$x^{4} + 8 x^{3} y + 24 x^{2} y^{2} + 4 x {(2 y)}^{3} + {(2 y)}^{4}$

Apply the product rule to $2 y$ .

$x^{4} + 8 x^{3} y + 24 x^{2} y^{2} + 4 x (2^{3} y^{3}) + {(2 y)}^{4}$

Rewrite using the commutative property of multiplication.

$x^{4} + 8 x^{3} y + 24 x^{2} y^{2} + 4 \cdot 2^{3} x y^{3} + {(2 y)}^{4}$

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

$x^{4} + 8 x^{3} y + 24 x^{2} y^{2} + 4 \cdot 8 x y^{3} + {(2 y)}^{4}$

Multiply $4$ by $8$ .

$x^{4} + 8 x^{3} y + 24 x^{2} y^{2} + 32 x y^{3} + {(2 y)}^{4}$

Apply the product rule to $2 y$ .

$x^{4} + 8 x^{3} y + 24 x^{2} y^{2} + 32 x y^{3} + 2^{4} y^{4}$

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

$x^{4} + 8 x^{3} y + 24 x^{2} y^{2} + 32 x y^{3} + 16 y^{4}$

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.

$x^{4} \to 4$

$8 x^{3} y \to 4$

$24 x^{2} y^{2} \to 4$

$32 x y^{3} \to 4$

$16 y^{4} \to 4$

The largest exponent is the degree of the polynomial.

$4$

Step 3

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

$x^{4}$

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.

$x^{4}$

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

$1$

Step 5

List the results.

Polynomial Degree: $4$

Leading Term: $x^{4}$

Leading Coefficient: $1$

Find the Degree, Leading Term, and Leading Coefficient (x+2y)^4

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