Expand Using the Binomial Theorem (4x-5y)^2

${(4 x - 5 y)}^{2}$

Step 1

Use the binomial expansion theorem to find each term. The binomial theorem states ${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

$\sum_{k = 0}^{2} \frac{2!}{(2 - k)! k!} \cdot {(4 x)}^{2 - k} \cdot {(- 5 y)}^{k}$

Step 2

Expand the summation.

$\frac{2!}{(2 - 0)! 0!} \cdot {(4 x)}^{2 - 0} \cdot {(- 5 y)}^{0} + \frac{2!}{(2 - 1)! 1!} \cdot {(4 x)}^{2 - 1} \cdot {(- 5 y)}^{1} + \frac{2!}{(2 - 2)! 2!} \cdot {(4 x)}^{2 - 2} \cdot {(- 5 y)}^{2}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(4 x)}^{2} \cdot {(- 5 y)}^{0} + 2 \cdot {(4 x)}^{1} \cdot {(- 5 y)}^{1} + 1 \cdot {(4 x)}^{0} \cdot {(- 5 y)}^{2}$

Step 4

Simplify each term.

Multiply ${(4 x)}^{2}$ by $1$ .

${(4 x)}^{2} \cdot {(- 5 y)}^{0} + 2 \cdot {(4 x)}^{1} \cdot {(- 5 y)}^{1} + 1 \cdot {(4 x)}^{0} \cdot {(- 5 y)}^{2}$

Apply the product rule to $4 x$ .

$4^{2} x^{2} \cdot {(- 5 y)}^{0} + 2 \cdot {(4 x)}^{1} \cdot {(- 5 y)}^{1} + 1 \cdot {(4 x)}^{0} \cdot {(- 5 y)}^{2}$

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

$16 x^{2} \cdot {(- 5 y)}^{0} + 2 \cdot {(4 x)}^{1} \cdot {(- 5 y)}^{1} + 1 \cdot {(4 x)}^{0} \cdot {(- 5 y)}^{2}$

Apply the product rule to $- 5 y$ .

$16 x^{2} \cdot ({(- 5)}^{0} y^{0}) + 2 \cdot {(4 x)}^{1} \cdot {(- 5 y)}^{1} + 1 \cdot {(4 x)}^{0} \cdot {(- 5 y)}^{2}$

Rewrite using the commutative property of multiplication.

$16 \cdot {(- 5)}^{0} x^{2} y^{0} + 2 \cdot {(4 x)}^{1} \cdot {(- 5 y)}^{1} + 1 \cdot {(4 x)}^{0} \cdot {(- 5 y)}^{2}$

Anything raised to $0$ is $1$ .

$16 \cdot 1 x^{2} y^{0} + 2 \cdot {(4 x)}^{1} \cdot {(- 5 y)}^{1} + 1 \cdot {(4 x)}^{0} \cdot {(- 5 y)}^{2}$

Multiply $16$ by $1$ .

$16 x^{2} y^{0} + 2 \cdot {(4 x)}^{1} \cdot {(- 5 y)}^{1} + 1 \cdot {(4 x)}^{0} \cdot {(- 5 y)}^{2}$

Anything raised to $0$ is $1$ .

$16 x^{2} \cdot 1 + 2 \cdot {(4 x)}^{1} \cdot {(- 5 y)}^{1} + 1 \cdot {(4 x)}^{0} \cdot {(- 5 y)}^{2}$

Multiply $16$ by $1$ .

$16 x^{2} + 2 \cdot {(4 x)}^{1} \cdot {(- 5 y)}^{1} + 1 \cdot {(4 x)}^{0} \cdot {(- 5 y)}^{2}$

Simplify.

$16 x^{2} + 2 \cdot (4 x) \cdot {(- 5 y)}^{1} + 1 \cdot {(4 x)}^{0} \cdot {(- 5 y)}^{2}$

Multiply $4$ by $2$ .

$16 x^{2} + 8 x \cdot {(- 5 y)}^{1} + 1 \cdot {(4 x)}^{0} \cdot {(- 5 y)}^{2}$

Simplify.

$16 x^{2} + 8 x \cdot (- 5 y) + 1 \cdot {(4 x)}^{0} \cdot {(- 5 y)}^{2}$

Rewrite using the commutative property of multiplication.

$16 x^{2} + 8 \cdot - 5 x y + 1 \cdot {(4 x)}^{0} \cdot {(- 5 y)}^{2}$

Multiply $8$ by $- 5$ .

$16 x^{2} - 40 x y + 1 \cdot {(4 x)}^{0} \cdot {(- 5 y)}^{2}$

Multiply ${(4 x)}^{0}$ by $1$ .

$16 x^{2} - 40 x y + {(4 x)}^{0} \cdot {(- 5 y)}^{2}$

Apply the product rule to $4 x$ .

$16 x^{2} - 40 x y + 4^{0} x^{0} \cdot {(- 5 y)}^{2}$

Anything raised to $0$ is $1$ .

$16 x^{2} - 40 x y + 1 x^{0} \cdot {(- 5 y)}^{2}$

Multiply $x^{0}$ by $1$ .

$16 x^{2} - 40 x y + x^{0} \cdot {(- 5 y)}^{2}$

Anything raised to $0$ is $1$ .

$16 x^{2} - 40 x y + 1 \cdot {(- 5 y)}^{2}$

Multiply ${(- 5 y)}^{2}$ by $1$ .

$16 x^{2} - 40 x y + {(- 5 y)}^{2}$

Apply the product rule to $- 5 y$ .

$16 x^{2} - 40 x y + {(- 5)}^{2} y^{2}$

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

$16 x^{2} - 40 x y + 25 y^{2}$

Expand Using the Binomial Theorem (4x-5y)^2

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