# Use Pascal's triangle to expand the binomial (8v + s)^5.

## Question:

Use Pascal's triangle to expand the binomial {eq}(8v + s)^5 {/eq}

## Pascal's Triangle:

The coefficients used in binomial expansion are arranged within Pascal's triangle. The coefficients used are determined by the exponent the binomial is raised to, for the value of the exponent correlates with a row on Pascal's triangle.

The expanded binomial {eq}(8v + s)^5 {/eq} is:

{eq}32768v^{5} + 20480v^{4}s + 5120v^{3}s^{2} + 640v^{2}s^{3} + 40vs^{4} + s^{5} {/eq}

Since the binomial is raised to the 5th power, the coefficients that will be used to expand the binomial are located along the 5th row from the tip of the triangle.

{eq}0| 1\\ 1| 1, 1\\ 2| 1, 2, 1\\ 3| 1, 3, 3, 1\\ 4| 1, 4, 6, 4, 1\\ 5| 1, 5, 10, 10, 5, 1\\ {/eq}

Plug in the coefficient values into the expansion.

{eq}1((a)^5(b)^0) + 5((a)^4(b)^1) + 10((a)^3(b)^2) + 10((a)^2(b)^3) + 5((a)^1(b)^4) + 1((a)^0(b)^5) {/eq}

Substitute {eq}8v {/eq} for {eq}a {/eq}.

{eq}1((8v)^5(b)^0) + 5((8v)^4(b)^1) + 10((8v)^3(b)^2) + 10((8v)^2(b)^3) + 5((8v)^1(b)^4) + 1((8v)^0(b)^5) {/eq}

Substitute {eq}s {/eq} for {eq}b {/eq}.

{eq}1((8v)^5(s)^0) + 5((8v)^4(s)^1) + 10((8v)^3(s)^2) + 10((8v)^2(s)^3) + 5((8v)^1(s)^4) + 1((8v)^0(s)^5) {/eq}

Simplify.

{eq}1((32768v^5)(1)) + 5((4096v^4)(s)) + 10((512v^3)(s^2)) + 10((64v^2)(s^3)) + 5((8v)(s^4)) + 1((1)(s^5))\\~\\ 32768(v^5) + 20480(v^4)(s) + 5120(v^3)(s^2) + 640(v^2)(s^3) + 40(v)(s^4) + (s^5)\\~\\ 32768v^5 + 20480v^{4}s + 5120v^{3}s^{2} + 640v^{2}s^{3} + 40vs^4 + s^5 {/eq} 