Expand Using the Binomial Theorem (3x^2-7)^2

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

Step 1

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

$\sum_{k = 0}^{2} \frac{2!}{(2 - k)! k!} \cdot {(3 x^{2})}^{2 - k} \cdot {(- 7)}^{k}$

Step 2

Expand the summation.

$\frac{2!}{(2 - 0)! 0!} \cdot {(3 x^{2})}^{2 - 0} \cdot {(- 7)}^{0} + \frac{2!}{(2 - 1)! 1!} \cdot {(3 x^{2})}^{2 - 1} \cdot {(- 7)}^{1} + \frac{2!}{(2 - 2)! 2!} \cdot {(3 x^{2})}^{2 - 2} \cdot {(- 7)}^{2}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(3 x^{2})}^{2} \cdot {(- 7)}^{0} + 2 \cdot {(3 x^{2})}^{1} \cdot {(- 7)}^{1} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 7)}^{2}$

Step 4

Simplify each term.

Multiply ${(3 x^{2})}^{2}$ by $1$ .

${(3 x^{2})}^{2} \cdot {(- 7)}^{0} + 2 \cdot {(3 x^{2})}^{1} \cdot {(- 7)}^{1} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 7)}^{2}$

Apply the product rule to $3 x^{2}$ .

$3^{2} {(x^{2})}^{2} \cdot {(- 7)}^{0} + 2 \cdot {(3 x^{2})}^{1} \cdot {(- 7)}^{1} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 7)}^{2}$

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

$9 {(x^{2})}^{2} \cdot {(- 7)}^{0} + 2 \cdot {(3 x^{2})}^{1} \cdot {(- 7)}^{1} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 7)}^{2}$

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

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

$9 x^{2 \cdot 2} \cdot {(- 7)}^{0} + 2 \cdot {(3 x^{2})}^{1} \cdot {(- 7)}^{1} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 7)}^{2}$

Multiply $2$ by $2$ .

$9 x^{4} \cdot {(- 7)}^{0} + 2 \cdot {(3 x^{2})}^{1} \cdot {(- 7)}^{1} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 7)}^{2}$

Anything raised to $0$ is $1$ .

$9 x^{4} \cdot 1 + 2 \cdot {(3 x^{2})}^{1} \cdot {(- 7)}^{1} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 7)}^{2}$

Multiply $9$ by $1$ .

$9 x^{4} + 2 \cdot {(3 x^{2})}^{1} \cdot {(- 7)}^{1} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 7)}^{2}$

Simplify.

$9 x^{4} + 2 \cdot (3 x^{2}) \cdot {(- 7)}^{1} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 7)}^{2}$

Multiply $3$ by $2$ .

$9 x^{4} + 6 x^{2} \cdot {(- 7)}^{1} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 7)}^{2}$

Evaluate the exponent.

$9 x^{4} + 6 x^{2} \cdot - 7 + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 7)}^{2}$

Multiply $- 7$ by $6$ .

$9 x^{4} - 42 x^{2} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 7)}^{2}$

Multiply ${(3 x^{2})}^{0}$ by $1$ .

$9 x^{4} - 42 x^{2} + {(3 x^{2})}^{0} \cdot {(- 7)}^{2}$

Apply the product rule to $3 x^{2}$ .

$9 x^{4} - 42 x^{2} + 3^{0} {(x^{2})}^{0} \cdot {(- 7)}^{2}$

Anything raised to $0$ is $1$ .

$9 x^{4} - 42 x^{2} + 1 {(x^{2})}^{0} \cdot {(- 7)}^{2}$

Multiply ${(x^{2})}^{0}$ by $1$ .

$9 x^{4} - 42 x^{2} + {(x^{2})}^{0} \cdot {(- 7)}^{2}$

Multiply the exponents in ${(x^{2})}^{0}$ .

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

$9 x^{4} - 42 x^{2} + x^{2 \cdot 0} \cdot {(- 7)}^{2}$

Multiply $2$ by $0$ .

$9 x^{4} - 42 x^{2} + x^{0} \cdot {(- 7)}^{2}$

Anything raised to $0$ is $1$ .

$9 x^{4} - 42 x^{2} + 1 \cdot {(- 7)}^{2}$

Multiply ${(- 7)}^{2}$ by $1$ .

$9 x^{4} - 42 x^{2} + {(- 7)}^{2}$

Raise $- 7$ to the power of $2$ .

$9 x^{4} - 42 x^{2} + 49$

Expand Using the Binomial Theorem (3x^2-7)^2

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