Use the binomial theorem to expand [2-1/4 x]^5
You should know that the row from Pascal's triangle needed here are
1 5 1010 5
Then (2 - x/4)^5
= 1(2)^5 - 5(2^4)(x/4) + 10(2^3)(x/4)^2 - 10(2^2)(x/4)^3 + 5(2)(x/4)^4 - (x/4)^5
= 32 - 20x + 5x^2 - (5/8)x^3 + (5/128)x^4 - (1/1024)x^5
Sure, let me put on my mathematician nose and clown wig to help you out with this one!
Using the binomial theorem, we can expand [2 - (1/4)x]^5. Here's how it goes:
1st term: (2)^5 = 32
2nd term: (5 choose 1) * (2)^4 * (-(1/4)x)^1 = -40x
3rd term: (5 choose 2) * (2)^3 * (-(1/4)x)^2 = 30x^2
4th term: (5 choose 3) * (2)^2 * (-(1/4)x)^3 = -10x^3
5th term: (5 choose 4) * (2)^1 * (-(1/4)x)^4 = (5/8)x^4
6th term: (5 choose 5) * (2)^0 * (-(1/4)x)^5 = -(1/32)x^5
Summing all of these terms up, we get the expanded form of [2 - (1/4)x]^5:
32 - 40x + 30x^2 - 10x^3 + (5/8)x^4 - (1/32)x^5
Voila! Hope that brings some mathematical giggles to your day!
To expand the expression (2 - 1/4x)^5 using the binomial theorem, follow these steps:
Step 1: Determine the values of n, a, and b.
In this case, n = 5, a = 2, and b = -1/4x.
Step 2: Write down the binomial theorem.
The binomial theorem 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) is the binomial coefficient.
Step 3: Find the binomial coefficients.
Using the formula for binomial coefficients, C(n, k) = n! / (k! * (n - k)!), we can find the values of C(n, k) for each term.
C(5, 0) = 5! / (0! * (5 - 0)!) = 1
C(5, 1) = 5! / (1! * (5 - 1)!) = 5
C(5, 2) = 5! / (2! * (5 - 2)!) = 10
C(5, 3) = 5! / (3! * (5 - 3)!) = 10
C(5, 4) = 5! / (4! * (5 - 4)!) = 5
C(5, 5) = 5! / (5! * (5 - 5)!) = 1
Step 4: Substitute the values into the binomial theorem.
Now we can substitute the values of the binomial coefficients into the binomial theorem:
(2 - 1/4x)^5 = 1(2^5)(-1/4x)^0 + 5(2^4)(-1/4x)^1 + 10(2^3)(-1/4x)^2 + 10(2^2)(-1/4x)^3 + 5(2^1)(-1/4x)^4 + 1(2^0)(-1/4x)^5
Step 5: Simplify each term.
Simplify each term by applying the power rule and distributing:
= 1(32)(1) + 5(16)(-1/4x) + 10(8)(1/16x^2) + 10(4)(-1/64x^3) + 5(2)(1/256x^4) + 1(1)(-1/1024x^5)
= 32 - 20x + 5/2x^2 - 5/16x^3 + 5/128x^4 - 1/1024x^5
Thus, the expansion of (2 - 1/4x)^5 using the binomial theorem is:
32 - 20x + 5/2x^2 - 5/16x^3 + 5/128x^4 - 1/1024x^5.
To expand the expression [2 - (1/4)x]^5 using the binomial theorem, we need to identify the values of n, a, b, and x in the general formula of the binomial theorem:
(x + y)^n = C(n, 0) * x^0 * y^n + C(n, 1) * x^1 * y^(n-1) + ... + C(n, r) * x^r * y^(n-r) + ... + C(n, n) * x^n * y^0
where:
- C(n, r) is the binomial coefficient, equal to n! / (r! * (n - r)!).
- n is the exponent that the binomial is raised to.
- x and y are constants or variables.
- r is the power of x.
In our case, the expression is [2 - (1/4)x]^5, so we have:
- n = 5
- a = 2
- b = -(1/4)x
Now, we can substitute these values into the formula and expand the expression:
[2 - (1/4)x]^5
= C(5, 0) * (2)^0 * (-(1/4)x)^5
+ C(5, 1) * (2)^1 * (-(1/4)x)^(5 - 1)
+ C(5, 2) * (2)^2 * (-(1/4)x)^(5 - 2)
+ C(5, 3) * (2)^3 * (-(1/4)x)^(5 - 3)
+ C(5, 4) * (2)^4 * (-(1/4)x)^(5 - 4)
+ C(5, 5) * (2)^5 * (-(1/4)x)^(5 - 5)
Now we simplify each term and evaluate the binomial coefficients:
= C(5, 0) * (2)^0 * (-(1/4)x)^5
+ C(5, 1) * (2)^1 * (-(1/4)x)^4
+ C(5, 2) * (2)^2 * (-(1/4)x)^3
+ C(5, 3) * (2)^3 * (-(1/4)x)^2
+ C(5, 4) * (2)^4 * (-(1/4)x)^1
+ C(5, 5) * (2)^5 * (-(1/4)x)^0
Now, we compute each term using the binomial coefficients:
= C(5, 0) * (2)^0 * (-(1/4)x)^5
+ C(5, 1) * (2)^1 * (-(1/4)x)^4
+ C(5, 2) * (2)^2 * (-(1/4)x)^3
+ C(5, 3) * (2)^3 * (-(1/4)x)^2
+ C(5, 4) * (2)^4 * (-(1/4)x)^1
+ C(5, 5) * (2)^5 * (-(1/4)x)^0
Calculating each term using the binomial coefficients:
= C(5, 0) * 1 * (1/1024)x^5
+ C(5, 1) * 2 * (-1/256)x^4
+ C(5, 2) * 4 * (1/64)x^3
+ C(5, 3) * 8 * (-1/16)x^2
+ C(5, 4) * 16 * (1/4)x^1
+ C(5, 5) * 32 * 1
Simplifying each term:
= 1 * (1/1024)x^5
- 5 * (1/256)x^4
+ 10 * (1/64)x^3
- 10 * (1/16)x^2
+ 5 * (1/4)x
+ 1 * 32
Combining like terms:
= (1/1024)x^5
- (5/256)x^4
+ (10/64)x^3
- (10/16)x^2
+ (5/4)x
+ 32
Therefore, the expanded form of [2 - (1/4)x]^5 is (1/1024)x^5 - (5/256)x^4 + (10/64)x^3 - (10/16)x^2 + (5/4)x + 32.