Expand Using Pascal's Triangle (2x+3)^4

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

Step 1

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

The triangle can be used to calculate the coefficients of the expansion of ${(a + b)}^{n}$ by taking the exponent $n$ and adding $1$ . The coefficients will correspond with line $n + 1$ of the triangle. For ${(2 x + 3)}^{4}$ , $n = 4$ so the coefficients of the expansion will correspond with line $5$ .

Step 2

The expansion follows the rule ${(a + b)}^{n} = c_{0} a^{n} b^{0} + c_{1} a^{n - 1} b^{1} + c_{n - 1} a^{1} b^{n - 1} + c_{n} a^{0} b^{n}$ . The values of the coefficients, from the triangle, are $1 - 4 - 6 - 4 - 1$ .

$1 a^{4} b^{0} + 4 a^{3} b + 6 a^{2} b^{2} + 4 a b^{3} + 1 a^{0} b^{4}$

Step 3

Substitute the actual values of $a$ $2 x$ and $b$ $3$ into the expression.

$1 {(2 x)}^{4} {(3)}^{0} + 4 {(2 x)}^{3} {(3)}^{1} + 6 {(2 x)}^{2} {(3)}^{2} + 4 {(2 x)}^{1} {(3)}^{3} + 1 {(2 x)}^{0} {(3)}^{4}$

Step 4

Simplify each term.

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

${(2 x)}^{4} {(3)}^{0} + 4 {(2 x)}^{3} {(3)}^{1} + 6 {(2 x)}^{2} {(3)}^{2} + 4 {(2 x)}^{1} {(3)}^{3} + 1 {(2 x)}^{0} {(3)}^{4}$

Apply the product rule to $2 x$ .

$2^{4} x^{4} {(3)}^{0} + 4 {(2 x)}^{3} {(3)}^{1} + 6 {(2 x)}^{2} {(3)}^{2} + 4 {(2 x)}^{1} {(3)}^{3} + 1 {(2 x)}^{0} {(3)}^{4}$

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

$16 x^{4} {(3)}^{0} + 4 {(2 x)}^{3} {(3)}^{1} + 6 {(2 x)}^{2} {(3)}^{2} + 4 {(2 x)}^{1} {(3)}^{3} + 1 {(2 x)}^{0} {(3)}^{4}$

Anything raised to $0$ is $1$ .

$16 x^{4} \cdot 1 + 4 {(2 x)}^{3} {(3)}^{1} + 6 {(2 x)}^{2} {(3)}^{2} + 4 {(2 x)}^{1} {(3)}^{3} + 1 {(2 x)}^{0} {(3)}^{4}$

Multiply $16$ by $1$ .

$16 x^{4} + 4 {(2 x)}^{3} {(3)}^{1} + 6 {(2 x)}^{2} {(3)}^{2} + 4 {(2 x)}^{1} {(3)}^{3} + 1 {(2 x)}^{0} {(3)}^{4}$

Apply the product rule to $2 x$ .

$16 x^{4} + 4 (2^{3} x^{3}) {(3)}^{1} + 6 {(2 x)}^{2} {(3)}^{2} + 4 {(2 x)}^{1} {(3)}^{3} + 1 {(2 x)}^{0} {(3)}^{4}$

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

$16 x^{4} + 4 (8 x^{3}) {(3)}^{1} + 6 {(2 x)}^{2} {(3)}^{2} + 4 {(2 x)}^{1} {(3)}^{3} + 1 {(2 x)}^{0} {(3)}^{4}$

Multiply $8$ by $4$ .

$16 x^{4} + 32 x^{3} {(3)}^{1} + 6 {(2 x)}^{2} {(3)}^{2} + 4 {(2 x)}^{1} {(3)}^{3} + 1 {(2 x)}^{0} {(3)}^{4}$

Evaluate the exponent.

$16 x^{4} + 32 x^{3} \cdot 3 + 6 {(2 x)}^{2} {(3)}^{2} + 4 {(2 x)}^{1} {(3)}^{3} + 1 {(2 x)}^{0} {(3)}^{4}$

Multiply $3$ by $32$ .

$16 x^{4} + 96 x^{3} + 6 {(2 x)}^{2} {(3)}^{2} + 4 {(2 x)}^{1} {(3)}^{3} + 1 {(2 x)}^{0} {(3)}^{4}$

Apply the product rule to $2 x$ .

$16 x^{4} + 96 x^{3} + 6 (2^{2} x^{2}) {(3)}^{2} + 4 {(2 x)}^{1} {(3)}^{3} + 1 {(2 x)}^{0} {(3)}^{4}$

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

$16 x^{4} + 96 x^{3} + 6 (4 x^{2}) {(3)}^{2} + 4 {(2 x)}^{1} {(3)}^{3} + 1 {(2 x)}^{0} {(3)}^{4}$

Multiply $4$ by $6$ .

$16 x^{4} + 96 x^{3} + 24 x^{2} {(3)}^{2} + 4 {(2 x)}^{1} {(3)}^{3} + 1 {(2 x)}^{0} {(3)}^{4}$

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

$16 x^{4} + 96 x^{3} + 24 x^{2} \cdot 9 + 4 {(2 x)}^{1} {(3)}^{3} + 1 {(2 x)}^{0} {(3)}^{4}$

Multiply $9$ by $24$ .

$16 x^{4} + 96 x^{3} + 216 x^{2} + 4 {(2 x)}^{1} {(3)}^{3} + 1 {(2 x)}^{0} {(3)}^{4}$

Simplify.

$16 x^{4} + 96 x^{3} + 216 x^{2} + 4 (2 x) {(3)}^{3} + 1 {(2 x)}^{0} {(3)}^{4}$

Multiply $2$ by $4$ .

$16 x^{4} + 96 x^{3} + 216 x^{2} + 8 x {(3)}^{3} + 1 {(2 x)}^{0} {(3)}^{4}$

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

$16 x^{4} + 96 x^{3} + 216 x^{2} + 8 x \cdot 27 + 1 {(2 x)}^{0} {(3)}^{4}$

Multiply $27$ by $8$ .

$16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 1 {(2 x)}^{0} {(3)}^{4}$

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

$16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + {(2 x)}^{0} {(3)}^{4}$

Apply the product rule to $2 x$ .

$16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 2^{0} x^{0} {(3)}^{4}$

Anything raised to $0$ is $1$ .

$16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 1 x^{0} {(3)}^{4}$

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

$16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + x^{0} {(3)}^{4}$

Anything raised to $0$ is $1$ .

$16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 1 {(3)}^{4}$

Multiply ${(3)}^{4}$ by $1$ .

$16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + {(3)}^{4}$

Raise $3$ to the power of $4$ .

$16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81$

Expand Using Pascal's Triangle (2x+3)^4

Related Questions

Support LCMGCF.com by disabling your adblocker.