Hi, can somebody help me with this math problem I am stuck on? How do I use binomial theorem to expand(2x – 5y)^3. I need to show my word so any help you can provide would be greatly appreciated.
Expand
(A + B)^3 = A^3 + 3 A B^2 + 3 A^2B + B^3
And then in this expansion, substitute
A = 2x
B = -5 y
Thank you for your help.
Of course! Let me help you with that.
To expand the expression (2x – 5y)^3 using the binomial theorem, follow these steps:
Step 1: Identify the values of a and b in the binomial expression (2x – 5y).
In this case, a = 2x and b = -5y.
Step 2: Determine the exponent in the expanded form.
The exponent in this case is 3, as indicated by (2x – 5y)^3.
Step 3: Apply the binomial theorem formula to find the expanded form:
The binomial theorem formula states that (a + b)^n = C(n,0)a^n b^0 + C(n,1)a^(n-1) b^1 + C(n,2)a^(n-2) b^2 + ... + C(n,n-1)a^1 b^(n-1) + C(n,n)a^0 b^n, where C(n,k) represents the binomial coefficient, also known as “n choose k.”
Step 4: Calculate the values of the binomial coefficients C(n,k) for each term.
In this case, C(3,0) = 1, C(3,1) = 3, C(3,2) = 3, and C(3,3) = 1.
Step 5: Substitute the values into the binomial theorem formula.
The expanded form of (2x – 5y)^3 is:
1 * (2x)^3 * (-5y)^0 + 3 * (2x)^2 * (-5y)^1 + 3 * (2x)^1 * (-5y)^2 + 1 * (2x)^0 * (-5y)^3
Step 6: Simplify each term.
Simplifying each term, we get:
8x^3 + (-60x^2y) + 150xy^2 + (-125y^3)
So, the expanded form of (2x – 5y)^3 is:
8x^3 - 60x^2y + 150xy^2 - 125y^3.
I hope this helps! Let me know if you have any other questions.