The expression 81 p^4 + 108p^3r^3 + 54p^2r^6 + 12pr^9 + r^12 is the expansion of which binomial?
My answer: (3p + r^3)^4
yes
coefs of row 4
1 4 6 4 1
so
a^4 + 4 a^3 b + 6 a^2 b^2 + 4 a b^3 + b^4
(3p+r^3)^4
a = 3p
b = r^3
a^4 = 81 p^4 check
4 a^3 b = 4*27 or 108 p^3 r^3 check
6 a^2 b^2 = 6*9 or 54 p^2 r^6 check
etc
To determine the expansion of the binomial, we can factor out common terms from each term in the expression.
Let's break down the given expression:
81p^4 + 108p^3r^3 + 54p^2r^6 + 12pr^9 + r^12
We notice that the coefficients of each term are in a specific pattern. The pattern is based on the binomial expansion formula, which is (a + b)^n.
In this case, we have p and r as our variables, so we can rewrite the expression as follows:
(3p)^4 + 3(3p)^3(r^3) + 3(3p)^2(r^3)^2 + 3(3p)(r^3)^3 + (r^3)^4
Now we can see that the pattern fits the binomial expansion formula.
Using the formula, we know that the coefficients in each term are determined by combinations. In this case, the coefficient is determined by the combination of choosing the number of p's (3p) raised to a power, and the number of r^3 raised to a power.
So, using the formula (a + b)^n, where a = 3p and b = r^3, we can determine the expansion.
The expansion is given by:
(3p + r^3)^4
And that is the answer to the question.