how do you expand the following as a series of ascending powers of x upto x^2 using binomial expansion
for this function (1+x)^2(1-5x)^14
(5x+4)^3
To expand the function (1+x)^2(1-5x)^14 as a series of ascending powers of x up to x^2 using binomial expansion, we can use the binomial theorem.
The binomial theorem states that for any real numbers a and b, and any non-negative integer n, the expansion of (a + b)^n can be found using the formula:
(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-2) * a^2 * b^(n-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, defined as:
C(n, k) = n! / (k! * (n-k)!)
Let's use this formula to expand (1+x)^2(1-5x)^14.
Step 1: Expand (1+x)^2
Using the binomial expansion, we have:
(1 + x)^2 = C(2, 0) * 1^2 * x^0 + C(2, 1) * 1^1 * x^1 + C(2, 2) * 1^0 * x^2
= 1 * 1 * x^0 + 2 * 1 * x^1 + 1 * 1 * x^2
= 1 + 2x + x^2
Step 2: Expand (1-5x)^14
Using the binomial expansion, we have:
(1 - 5x)^14 = C(14, 0) * 1^14 * (-5x)^0 + C(14, 1) * 1^13 * (-5x)^1 + C(14, 2) * 1^12 * (-5x)^2 + ... + C(14, 13) * 1^1 * (-5x)^13 + C(14, 14) * 1^0 * (-5x)^14
The terms with powers higher than x^2 can be ignored since we are expanding the series up to x^2. We will focus only on the terms with powers up to x^2.
Step 3: Simplify the terms up to x^2
Let's simplify the terms from (1 - 5x)^14 up to x^2:
C(14, 0) * 1^14 * (-5x)^0 = 1 * 1 * (-5x)^0 = 1
C(14, 1) * 1^13 * (-5x)^1 = 14 * 1 * (-5x)^1 = -70x
C(14, 2) * 1^12 * (-5x)^2 = 91 * 1 * (-5x)^2 = 455x^2
Step 4: Multiply the two expansions
Now, let's multiply the expanded forms of (1+x)^2 and (1-5x)^14:
(1 + 2x + x^2)(1 - 70x + 455x^2)
Multiplying the terms, we get:
1 + 2x + x^2 - 70x - 140x^2 + 70x^3 + 455x^2 - 910x^3 + 455x^4
Simplifying and combining like terms, we finally get the expansion of the given function as a series of ascending powers of x up to x^2:
1 - 68x - 595x^2 + 70x^3 + 455x^4
Therefore, the expansion of the function (1+x)^2(1-5x)^14 as a series of ascending powers of x up to x^2 using binomial expansion is 1 - 68x - 595x^2 + 70x^3 + 455x^4.