13. Write the binomial expansion of (2a – 3b)4.
(2a - 3b)^4 =
(2a - 3b)^2 (2a - 3b)^2 =
(2a^2 - 12ab + 9b^2)(2a^2-12ab +9b^2)=
4a^4 - 24a^3b + 18a^2b^2 -
24a^3b + 144a^2b^2 - 108ab^3 +
18a^2b^2 - 108ab^3 + 81b^4.
Combine like-terms:
4a^4 - 48a^3b +180a^2b^2 - 216ab^3 +
81b^4.
The number of terms in the final results is always 1 greater than
thee exponent, and every other
term is positive.
The binomial expansion of (2a – 3b)^4 can be calculated using the Binomial Theorem. The general formula for the binomial theorem is:
(x + y)^n = C(n, 0) * x^n * y^0 + C(n, 1) * x^(n-1) * y^1 + C(n, 2) * x^(n-2) * y^2 + ... + C(n, n-1) * x^1 * y^(n-1) + C(n, n) * x^0 * y^n
where C(n, r) represents the combination of n objects taken r at a time.
For the given expression (2a – 3b)^4, we can substitute x = 2a and y = -3b.
Using the binomial theorem, the expansion becomes:
(2a – 3b)^4 = C(4, 0) * (2a)^4 * (-3b)^0 + C(4, 1) * (2a)^3 * (-3b)^1 + C(4, 2) * (2a)^2 * (-3b)^2 + C(4, 3) * (2a)^1 * (-3b)^3 + C(4, 4) * (2a)^0 * (-3b)^4
Simplifying the terms with combinations, we have:
(2a – 3b)^4 = 1 * (2a)^4 * 1 + 4 * (2a)^3 * (-3b) + 6 * (2a)^2 * (-3b)^2 + 4 * (2a)^1 * (-3b)^3 + 1 * (-3b)^4
Expanding each term, we get:
(2a – 3b)^4 = 16a^4 - 96a^3b + 216a^2b^2 - 216ab^3 + 81b^4
Therefore, the binomial expansion of (2a – 3b)^4 is 16a^4 - 96a^3b + 216a^2b^2 - 216ab^3 + 81b^4.
To write the binomial expansion of (2a - 3b)^4, we can use the Binomial Theorem. The Binomial Theorem states that for any positive integer n, the expansion of (x + y)^n can be found using the formula:
(x + y)^n = C(n, 0)x^n y^0 + C(n, 1)x^(n-1) y^1 + C(n, 2)x^(n-2) y^2 + ... + C(n, n-1)x^1 y^(n-1) + C(n, n)x^0 y^n
Where:
- C(n, k) represents the binomial coefficient or "n choose k," which is the number of ways to choose k items from a set of n distinct items.
- x^n represents x raised to the power of n.
- y^k represents y raised to the power of k.
In our case, the expression (2a - 3b)^4 can be expanded using the Binomial Theorem. We have:
n = 4, x = 2a, and y = -3b.
Let's calculate the coefficients for each term:
C(4, 0) = 1
C(4, 1) = 4
C(4, 2) = 6
C(4, 3) = 4
C(4, 4) = 1
Now, we can write the binomial expansion of (2a - 3b)^4:
(2a - 3b)^4 = 1(2a)^4 (-3b)^0 + 4(2a)^3 (-3b)^1 + 6(2a)^2 (-3b)^2 + 4(2a)^1 (-3b)^3 + 1(2a)^0 (-3b)^4
Simplifying each term using the properties of exponents:
= 1(16a^4)(1) + 4(8a^3)(-3b) + 6(4a^2)(9b^2) + 4(2a)(-27b^3) + 1(1)(81b^4)
= 16a^4 - 96a^3b + 216a^2b^2 - 216ab^3 + 81b^4
Therefore, the binomial expansion of (2a - 3b)^4 is 16a^4 - 96a^3b + 216a^2b^2 - 216ab^3 + 81b^4.