How would i do this?
Use the binomial theorem to expand (2x – 5y)^3
Seems like this one came through recently
(2x-5y)^3 =
(2x)^3 - 3(2x)^2)(5y) + 3(2x)(5y)^2 - (5y)^3
= 8x^3 - 60x^2y + 150xy^2 - 125y^3
thank you, Steve :)
To expand the expression (2x – 5y)^3 using the binomial theorem, follow these steps:
Step 1: Determine the exponents.
The exponent of the binomial is 3, so we'll have three terms in the expansion.
Step 2: Write the first term.
The first term is obtained by raising the first term of the binomial (2x) to the power of 3:
(2x)^3 = 8x^3
Step 3: Write the second term.
The second term will have two factors of the first term (2x) and one factor of the second term (-5y):
(2x)^2 * (-5y) = 4x^2 * (-5y) = -20x^2y
Step 4: Write the third term.
The third term will have one factor of the first term (2x) and two factors of the second term (-5y):
(2x) * (-5y)^2 = 2x * 25y^2 = 50xy^2
Step 5: Combine the terms.
The expanded form of (2x – 5y)^3 is:
8x^3 - 20x^2y + 50xy^2
To expand the expression (2x - 5y)^3 using the binomial theorem, you can follow these steps:
Step 1: Determine the values of n and i.
Here, n = 3 since the expression is raised to the power of 3. i will represent the index of each term as it varies from 0 to 3.
Step 2: Use the formula of the binomial theorem.
The formula for the binomial theorem is:
(a + b)^n = C(n,0) * a^(n-0) * b^0 + C(n,1) * a^(n-1) * b^1 + C(n,2) * a^(n-2) * b^2 + ... + C(n,n-1) * a^(n -(n-1)) * b^(n-1) + C(n,n) * a^(n-n) * b^n
In this formula, C(n,i) represents the binomial coefficient, which can be calculated as C(n,i) = n! / (i!(n-i)!), where n! stands for n factorial.
Step 3: Calculate each term of the expansion.
Using the formula above, calculate each term of the expansion by substituting the values of a and b from the expression (2x - 5y), and the values of n and i from Step 1.
The expanded form of (2x - 5y)^3 will include four terms, since i can range from 0 to 3. Simplify each term, and write them side by side to get the final expanded form.
For example, the first term would be: C(3,0) * (2x)^(3-0) * (-5y)^0
Step 4: Simplify the terms.
Simplify each term by evaluating the binomial coefficients and exponents, then combine like terms if applicable.
By following these steps, you will be able to expand the expression (2x - 5y)^3 using the binomial theorem.