1) expand (1+3x)^4 using the Binomial Theorem.
2) Use Pascal's Triangle to expand(x+2)^5
3)What is the third term of (a+b)^11?
Use the rows of Pascal's Triangle:
4: 1 4 6 4 1
5: 1 5 10 10 5 1
11: 1 11 55 ...
use those as coefficients of your powers.
For example using row 6,
1 6 15 20 15 6 1
(2x - 3)^6 =
1*(2x^6) + 6*(2x)^5(-3^1) + + 15*(2x^4)*(-3)^2 + ...
= 64x^6 - 576x^5 + 2160x^4 - ...
1) To expand (1+3x)^4 using the Binomial Theorem, you can follow these steps:
Step 1: Identify the values of n and k.
In this case, n represents the power (4) and k represents the term number.
Step 2: Write down the Binomial Theorem formula.
The Binomial Theorem formula states that (a + b)^n can be expanded as follows:
(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) * a^0 * b^n
Step 3: Use Pascal's Triangle to determine the binomial coefficients.
The binomial coefficients can be obtained from Pascal's Triangle. The row number of the triangle corresponds to the value of n, while the term number within the row corresponds to the value of k.
For the fourth power (n = 4), the respective row of Pascal's Triangle is: 1 4 6 4 1.
So, the binomial coefficients for n = 4 are: 1, 4, 6, 4, 1.
Step 4: Substitute the values into the formula.
For (1+3x)^4, you have:
(1+3x)^4 = C(4, 0) * 1^4 * (3x)^0 + C(4, 1) * 1^3 * (3x)^1 + C(4, 2) * 1^2 * (3x)^2 + C(4, 3) * 1^1 * (3x)^3 + C(4, 4) * 1^0 * (3x)^4
Simplifying the equation above will give you the expanded form of (1+3x)^4.
2) To expand (x+2)^5 using Pascal's Triangle, follow these steps:
Step 1: Identify the power.
In this case, n represents the power (5).
Step 2: Locate the corresponding row in Pascal's Triangle.
The row for n = 5 in Pascal's Triangle is 1 5 10 10 5 1.
Step 3: Write down the expanded form.
(x+2)^5 = C(5, 0) * x^5 * 2^0 + C(5, 1) * x^4 * 2^1 + C(5, 2) * x^3 * 2^2 + C(5, 3) * x^2 * 2^3 + C(5, 4) * x^1 * 2^4 + C(5, 5) * x^0 * 2^5
Using the binomial coefficients from the respective row in Pascal's Triangle and substituting them into the formula above will give you the expanded form of (x+2)^5.
3) To find the third term of (a+b)^11, you can use the Binomial Theorem. Here's how:
Step 1: Write down the Binomial Theorem 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) * a^0 * b^n
Step 2: Identify the values of n and k.
In this case, n represents the power (11) and k represents the term number.
Step 3: Use Pascal's Triangle to determine the binomial coefficients.
For the eleventh power (n = 11), the respective row of Pascal's Triangle is: 1 11 55 165 330 462 462 330 165 55 11 1.
So, the binomial coefficients for n = 11 are: 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1.
Step 4: Substitute the values into the formula.
For (a+b)^11, the third term would be:
C(11, 2) * a^9 * b^2 = 55 * a^9 * b^2
Therefore, the third term of (a+b)^11 is 55 * a^9 * b^2.