Question

In Exercises 43–48, use Pascal’s Triangle to expand the binomial. \((g+2)^5\)

Short answer

The result of the binomial expansion of \((g + 2)^5\) is \(g^5 + 10*g^4 + 40*g^3 + 80*g^2 + 80*g + 32\).

Step-by-step solution

Recalling Pascal’s Triangle for n=5

The n-th row of Pascal's triangle gives the coefficients of the expanded terms of a binomial raised to the n-th power. The 5th row of Pascal's triangle (starting with n=0 as the first row) is 1, 5, 10, 10, 5, 1.

Applying these coefficients to the binomial

These coefficients are applied in a decreasing order of power of the first term (g) and increasing order of power of the second term (2). So, the expanded expression becomes: \(1*g^5*2^0 + 5*g^4*2^1+ 10*g^3*2^2 + 10*g^2*2^3 + 5*g^1*2^4 + 1*g^0*2^5\).

Simplify the expression

Simplify the expression by calculating the actual figures for each term: \(g^5 + 5*g^4*2 + 10*g^3*4 + 10*g^2*8 + 5*g*16 + 32\). After that, simplify it again to achieve the final answer: \(g^5 + 10*g^4 + 40*g^3 + 80*g^2 + 80*g + 32\).