Find the constant term in the expansion of [x + (3/x squared)]^9
Find the coefficient of (x)^4(y)^9 in the expansion of (x + (2)(y)^3)^7
for the first one, first of all, find the general term.
I got
C(7,r)x^(7-r)(3/x^2)^r
=c(7,r)(3)^r x^(9-3r)
for that to be a constant the exponent of the x terms must be zero,so
9-3r=0
r=3
so the constant term is C(7,3)(3^3) = 2268
for the second question, to have the x^4, it must be term number 3.
You work this one out, let me know what you got
correction, whenever you see a 7, it should be a 9
so here it is again
for the first one, first of all, find the general term.
I got
C(9,r)x^(9-r)(3/x^2)^r
=c(9,r)(3)^r x^(9-3r)
for that to be a constant the exponent of the x terms must be zero,so
9-3r=0
r=3
so the constant term is C(9,3)(3^3) = 2268
To find the constant term in the expansion of [x + (3/x^2)]^9, we first need to understand the pattern of an expanded binomial using the binomial theorem.
The binomial theorem states that when you expand (a + b)^n (where a and b are constants and n is a positive integer), the terms in the expansion will be of the form:
C(n, k) * (a)^(n - k) * (b)^k
Here, C(n, k) represents the binomial coefficient, which can be calculated using the formula:
C(n, k) = (n!)/(k!(n - k)!)
From the given expression, [x + (3/x^2)]^9, we can see that a = x and b = (3/x^2). Since we are looking for the constant term, we are interested in the term obtained when k (the power of b) is zero.
In the binomial expansion, when k is zero, the term will simplify to:
C(n, 0) * (a)^(n - 0) * (b)^0
The term (b)^0 is equal to 1, so we can ignore it in our calculation.
Now, let's calculate the value of the constant term for this expression:
C(9, 0) * (x)^(9 - 0) * (3/x^2)^0
= C(9, 0) * x^9 * 1
= C(9, 0) * x^9
= 1 * x^9
= x^9
Therefore, the constant term in the expansion of [x + (3/x^2)]^9 is x^9.
Moving on to the second question:
To find the coefficient of (x)^4(y)^9 in the expansion of (x + 2y^3)^7, we can again use the binomial theorem.
In this case, a = x and b = 2y^3. We are interested in the term obtained when the exponent of a (x) is 4 (k = 4) and the exponent of b (2y^3) is 9 (n - k = 9).
The term will be:
C(7, 4) * (x)^(7 - 4) * (2y^3)^9
Now, let's evaluate this expression:
C(7, 4) * (x)^(7 - 4) * (2y^3)^9
= C(7, 4) * x^3 * (2)^9 * (y^3)^9
= C(7, 4) * x^3 * 2^9 * y^(3 * 9)
= C(7, 4) * x^3 * 512 * y^27
Here, C(7, 4) represents the binomial coefficient:
C(7, 4) = (7!)/(4!(7 - 4)!) = (7!)/(4! * 3!) = (7 * 6 * 5)/(3 * 2 * 1) = 35
Substituting this value, we get:
35 * x^3 * 512 * y^27
= 179,200 * x^3 * y^27
Therefore, the coefficient of (x)^4(y)^9 in the expansion of (x + 2y^3)^7 is 179,200.