Expand Using Pascal's Triangle (x+2)^8

${(x + 2)}^{8}$

Step 1

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$\dots$

$1 - 6 - 15 - 20 - 15 - 6 - 1$

$1 - 7 - 21 - 35 - 35 - 21 - 7 - 1$

$1 - 8 - 28 - 56 - 70 - 56 - 28 - 8 - 1$

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 ${(x + 2)}^{8}$ , $n = 8$ so the coefficients of the expansion will correspond with line $9$ .

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 $1 - 8 - 28 - 56 - 70 - 56 - 28 - 8 - 1$ .

$1 a^{8} b^{0} + 8 a^{7} b + 28 a^{6} b^{2} + 56 a^{5} b^{3} + 70 a^{4} b^{4} + 56 a^{3} b^{5} + 28 a^{2} b^{6} + 8 a b^{7} + 1 a^{0} b^{8}$

Step 3

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

$1 {(x)}^{8} {(2)}^{0} + 8 {(x)}^{7} {(2)}^{1} + 28 {(x)}^{6} {(2)}^{2} + 56 {(x)}^{5} {(2)}^{3} + 70 {(x)}^{4} {(2)}^{4} + 56 {(x)}^{3} {(2)}^{5} + 28 {(x)}^{2} {(2)}^{6} + 8 {(x)}^{1} {(2)}^{7} + 1 {(x)}^{0} {(2)}^{8}$

Step 4

Simplify each term.

Multiply ${(x)}^{8}$ by $1$ .

${(x)}^{8} {(2)}^{0} + 8 {(x)}^{7} {(2)}^{1} + 28 {(x)}^{6} {(2)}^{2} + 56 {(x)}^{5} {(2)}^{3} + 70 {(x)}^{4} {(2)}^{4} + 56 {(x)}^{3} {(2)}^{5} + 28 {(x)}^{2} {(2)}^{6} + 8 {(x)}^{1} {(2)}^{7} + 1 {(x)}^{0} {(2)}^{8}$

Anything raised to $0$ is $1$ .

$x^{8} \cdot 1 + 8 {(x)}^{7} {(2)}^{1} + 28 {(x)}^{6} {(2)}^{2} + 56 {(x)}^{5} {(2)}^{3} + 70 {(x)}^{4} {(2)}^{4} + 56 {(x)}^{3} {(2)}^{5} + 28 {(x)}^{2} {(2)}^{6} + 8 {(x)}^{1} {(2)}^{7} + 1 {(x)}^{0} {(2)}^{8}$

Multiply $x^{8}$ by $1$ .

$x^{8} + 8 {(x)}^{7} {(2)}^{1} + 28 {(x)}^{6} {(2)}^{2} + 56 {(x)}^{5} {(2)}^{3} + 70 {(x)}^{4} {(2)}^{4} + 56 {(x)}^{3} {(2)}^{5} + 28 {(x)}^{2} {(2)}^{6} + 8 {(x)}^{1} {(2)}^{7} + 1 {(x)}^{0} {(2)}^{8}$

Evaluate the exponent.

$x^{8} + 8 x^{7} \cdot 2 + 28 {(x)}^{6} {(2)}^{2} + 56 {(x)}^{5} {(2)}^{3} + 70 {(x)}^{4} {(2)}^{4} + 56 {(x)}^{3} {(2)}^{5} + 28 {(x)}^{2} {(2)}^{6} + 8 {(x)}^{1} {(2)}^{7} + 1 {(x)}^{0} {(2)}^{8}$

Multiply $2$ by $8$ .

$x^{8} + 16 x^{7} + 28 {(x)}^{6} {(2)}^{2} + 56 {(x)}^{5} {(2)}^{3} + 70 {(x)}^{4} {(2)}^{4} + 56 {(x)}^{3} {(2)}^{5} + 28 {(x)}^{2} {(2)}^{6} + 8 {(x)}^{1} {(2)}^{7} + 1 {(x)}^{0} {(2)}^{8}$

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

$x^{8} + 16 x^{7} + 28 x^{6} \cdot 4 + 56 {(x)}^{5} {(2)}^{3} + 70 {(x)}^{4} {(2)}^{4} + 56 {(x)}^{3} {(2)}^{5} + 28 {(x)}^{2} {(2)}^{6} + 8 {(x)}^{1} {(2)}^{7} + 1 {(x)}^{0} {(2)}^{8}$

Multiply $4$ by $28$ .

$x^{8} + 16 x^{7} + 112 x^{6} + 56 {(x)}^{5} {(2)}^{3} + 70 {(x)}^{4} {(2)}^{4} + 56 {(x)}^{3} {(2)}^{5} + 28 {(x)}^{2} {(2)}^{6} + 8 {(x)}^{1} {(2)}^{7} + 1 {(x)}^{0} {(2)}^{8}$

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

$x^{8} + 16 x^{7} + 112 x^{6} + 56 x^{5} \cdot 8 + 70 {(x)}^{4} {(2)}^{4} + 56 {(x)}^{3} {(2)}^{5} + 28 {(x)}^{2} {(2)}^{6} + 8 {(x)}^{1} {(2)}^{7} + 1 {(x)}^{0} {(2)}^{8}$

Multiply $8$ by $56$ .

$x^{8} + 16 x^{7} + 112 x^{6} + 448 x^{5} + 70 {(x)}^{4} {(2)}^{4} + 56 {(x)}^{3} {(2)}^{5} + 28 {(x)}^{2} {(2)}^{6} + 8 {(x)}^{1} {(2)}^{7} + 1 {(x)}^{0} {(2)}^{8}$

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

$x^{8} + 16 x^{7} + 112 x^{6} + 448 x^{5} + 70 x^{4} \cdot 16 + 56 {(x)}^{3} {(2)}^{5} + 28 {(x)}^{2} {(2)}^{6} + 8 {(x)}^{1} {(2)}^{7} + 1 {(x)}^{0} {(2)}^{8}$

Multiply $16$ by $70$ .

$x^{8} + 16 x^{7} + 112 x^{6} + 448 x^{5} + 1120 x^{4} + 56 {(x)}^{3} {(2)}^{5} + 28 {(x)}^{2} {(2)}^{6} + 8 {(x)}^{1} {(2)}^{7} + 1 {(x)}^{0} {(2)}^{8}$

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

$x^{8} + 16 x^{7} + 112 x^{6} + 448 x^{5} + 1120 x^{4} + 56 x^{3} \cdot 32 + 28 {(x)}^{2} {(2)}^{6} + 8 {(x)}^{1} {(2)}^{7} + 1 {(x)}^{0} {(2)}^{8}$

Multiply $32$ by $56$ .

$x^{8} + 16 x^{7} + 112 x^{6} + 448 x^{5} + 1120 x^{4} + 1792 x^{3} + 28 {(x)}^{2} {(2)}^{6} + 8 {(x)}^{1} {(2)}^{7} + 1 {(x)}^{0} {(2)}^{8}$

Raise $2$ to the power of $6$ .

$x^{8} + 16 x^{7} + 112 x^{6} + 448 x^{5} + 1120 x^{4} + 1792 x^{3} + 28 x^{2} \cdot 64 + 8 {(x)}^{1} {(2)}^{7} + 1 {(x)}^{0} {(2)}^{8}$

Multiply $64$ by $28$ .

$x^{8} + 16 x^{7} + 112 x^{6} + 448 x^{5} + 1120 x^{4} + 1792 x^{3} + 1792 x^{2} + 8 {(x)}^{1} {(2)}^{7} + 1 {(x)}^{0} {(2)}^{8}$

Simplify.

$x^{8} + 16 x^{7} + 112 x^{6} + 448 x^{5} + 1120 x^{4} + 1792 x^{3} + 1792 x^{2} + 8 x {(2)}^{7} + 1 {(x)}^{0} {(2)}^{8}$

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

$x^{8} + 16 x^{7} + 112 x^{6} + 448 x^{5} + 1120 x^{4} + 1792 x^{3} + 1792 x^{2} + 8 x \cdot 128 + 1 {(x)}^{0} {(2)}^{8}$

Multiply $128$ by $8$ .

$x^{8} + 16 x^{7} + 112 x^{6} + 448 x^{5} + 1120 x^{4} + 1792 x^{3} + 1792 x^{2} + 1024 x + 1 {(x)}^{0} {(2)}^{8}$

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

$x^{8} + 16 x^{7} + 112 x^{6} + 448 x^{5} + 1120 x^{4} + 1792 x^{3} + 1792 x^{2} + 1024 x + {(x)}^{0} {(2)}^{8}$

Anything raised to $0$ is $1$ .

$x^{8} + 16 x^{7} + 112 x^{6} + 448 x^{5} + 1120 x^{4} + 1792 x^{3} + 1792 x^{2} + 1024 x + 1 {(2)}^{8}$

Multiply ${(2)}^{8}$ by $1$ .

$x^{8} + 16 x^{7} + 112 x^{6} + 448 x^{5} + 1120 x^{4} + 1792 x^{3} + 1792 x^{2} + 1024 x + {(2)}^{8}$

Raise $2$ to the power of $8$ .

$x^{8} + 16 x^{7} + 112 x^{6} + 448 x^{5} + 1120 x^{4} + 1792 x^{3} + 1792 x^{2} + 1024 x + 256$

Expand Using Pascal's Triangle (x+2)^8

Related Questions

Support LCMGCF.com by disabling your adblocker.