Explain the steps of binomial expansion, I just learned about binomial theorem but i'm having trouble understanding the expansion process. Please help!
in the general expansion of (a+b)^n
1. If n is a whole number:
(a+b)^1 = a+b
(a+b)^2 = a^2 + 2ab + b^2
(a+b)^3 = a^3 + 3a^2 b + 3 a b^2 + b^3
(a+b)^4 = a^4 + 4a^3 b + 6a^2 b^2 + 4 a b^3 + b^4
...
notice that the coefficients are the terms of rows in Pascal's Triangle, you must
absolutely familiarize yourself with that
in general for whole numbers of n
(a+b)^n = C(n,n)a^n + C(n,n-1) a^(n-1) b + C(n,n-2) a^(n-2) b^2 + .... +C(n,0) b^n
e.g.
(2x - 3y)^4 = C(4,4)(2x)^4 (-3y)^0 + C(4,3)(2x)^3 (-3y)^1 + C(4,2)(2x)^2 (-3y)^2 + C(4,1)(2x)^1 (-3y)^3 + C(4,0)(2x)^0 (-3y)^4
= 1(16x^4)(1) + 4(8x^3)(-3y) + 6(4x^2)(9y^2) + 4(2x)(-27y^3) + (1)(1)(81y^4)
= 16x^4 - 96x^3 y + 216x^2 y^2 - 35x y^3 + 81y^4
2. if n is not a whole number, but could be any real number, then
(a+b)^r = a^r + r(a^(r-1) b + r(r-1)/2! (a^(r-2)) (b^2) + r(r-1)(r-2)/3! (a^(r-3))(b^3) + r(r-1)(r-2)(r-3)/4! (a^(r-4))(b^4) + ....
Can you see the pattern ?
e.g. (2x - 3y)^-5
= (2x)^-5 + (-5)(2x)^-6 (-3y) + (-5)(-6)/2! (2x)^-7 (-3y)^2 + (-5)(-6)(-7)/3! (2x)^-8 (-3y)^3 + ... never ends
= (1/32)(1/x^5) + (15/64)(y/x^6) + (135/128)(y^2 / x^7) + (945/256)(y^3/x^8) + ....
In the "olden days" we would use this to find roots of numbers.
e.g. find cube root of 30 to 3 decimals
30^(1/3) < find the closest perfect cube to 30 which is 27
= (27 + 3)^(1/3)
= 27^(1/3) (1 + 3/27)^(1/3) <<< factored out 27^(1/3), which I know
= 3 (1 + 1/9)^(1/3) <---- now expand
= 3[1^(1/3) + (1/3)(1)(1/9) + (1/3)(1/3-1)/2! (1)(1/9)^2 + (1/3)(-2/3)(-5/3)/3! (1) (1/9)^3 + ...]
=3 [ 1 + 1/27 - 1/729 + 5/59049 - .... ]
= 3[ 1 + .03703... - .00137.. + .0000846 ..... ]
= 3[1.03574..]
= 3.10723..
my calculator gave me 3.107232506
Hey, not bad, eh?
Sure! The binomial expansion is a way to expand a binomial expression raised to a power. To explain the process, let's start with the binomial theorem, which states:
For any positive integer n, the expansion of (a + b)^n can be written as:
(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
Now, let me explain the steps involved in the binomial expansion:
Step 1: Determine the values of a, b, and n.
- "a" and "b" are the terms of the binomial expression, such as (2x + 3y).
- "n" represents the power to which the binomial is raised.
Step 2: Identify the coefficients.
- In the binomial theorem formula, the coefficients C(n, k) determine the number of ways to choose k terms from the n terms. These coefficients can be calculated using the binomial coefficient formula: C(n, k) = n! / (k! * (n-k)!).
Step 3: Write out the expansion.
- Starting from the left side of the binomial theorem formula, replace the variables and coefficients with their respective values, according to each term.
- Multiply the variables "a" and "b" with their respective powers.
Step 4: Simplify each term.
- Evaluate the powers of the variables by performing the necessary calculations.
- Simplify the coefficients by calculating the factorial values.
Step 5: Add all the terms together.
- Combine all the simplified terms to obtain the expanded form of the binomial expression.
By following these steps, you should be able to expand any binomial expression using the binomial theorem. Remember to pay attention to the coefficients and the power of the variables in each term.