Final answer:
To expand the expression (r - t)^4 using the binomial theorem, we can use the formula (a + b)^n = C(n, 0)a^nb^0 + C(n, 1)a^(n-1)b^1 + C(n, 2)a^(n-2)b^2 + ... + C(n, n-1)a^1b^(n-1) + C(n, n)a^0b^n. Plugging in the values for a, b, and n, we can simplify the expression to (r - t)^4 = r^4 + 4r^3(-t) + 6r^2(-t)^2 + 4r(-t)^3 + (-t)^4.
Step-by-step explanation:
To expand the expression (r - t)^4 using the binomial theorem, we can use the formula:
(a + b)^n = C(n, 0)a^nb^0 + C(n, 1)a^(n-1)b^1 + C(n, 2)a^(n-2)b^2 + ... + C(n, n-1)a^1b^(n-1) + C(n, n)a^0b^n
In this case, a = r and b = -t, and we're expanding to the fourth power, so n = 4. Plugging these values into the formula, we get:
(r - t)^4 = C(4, 0)r^4(-t)^0 + C(4, 1)r^3(-t)^1 + C(4, 2)r^2(-t)^2 + C(4, 3)r^1(-t)^3 + C(4, 4)r^0(-t)^4
Expanding the binomial coefficients, we have:
(r - t)^4 = r^4 + 4r^3(-t) + 6r^2(-t)^2 + 4r(-t)^3 + (-t)^4
To expand the expression (r - t)^4 using the binomial theorem, we apply the theorem which is generally written as:
(a + b)^n = a^n + n * a^(n-1) * b + n(n-1)/2! * a^(n-2) * b^2 + n(n-1)(n-2)/3! * a^(n-3) * b^3 + ...
In this case, a is r, b is -t, and n is 4. The expansion is then:
(r - t)^4 = r^4 + 4 * r^3 * (-t) + 6 * r^2 * (-t)^2 + 4 * r * (-t)^3 + (-t)^4
Which simplifies to:
r^4 - 4r^3t + 6r^2t^2 - 4rt^3 + t^4.