Final answer:
To complete the expansion of (2q-r)³ using Pascal's Triangle, you can use the binomial theorem and the values from Pascal's Triangle to determine the coefficients. The expansion will result in 8q³ - 12q²r + 6qr² - r³.
Step-by-step explanation:
To complete the expansion of (2q-r)³ using Pascal's Triangle, we can use the binomial theorem. The binomial theorem states that (a + b)ⁿ can be expanded as the sum of terms of the form C(n, r) * a^(n-r) * b^r, where C(n, r) is the binomial coefficient. In this case, (2q-r)³ can be expanded as:
(2q-r)³ = C(3, 0) * (2q)³ * (-r)^0 + C(3, 1) * (2q)² * (-r)^1 + C(3, 2) * (2q)¹ * (-r)^2 + C(3, 3) * (2q)⁰ * (-r)^3
Using Pascal's Triangle, we can determine the values of the binomial coefficients:
C(3, 0) = 1, C(3, 1) = 3, C(3, 2) = 3, C(3, 3) = 1
Plugging these values into the expanded form, we get:
(2q-r)³ = 1 * (2q)³ * (-r)^0 + 3 * (2q)² * (-r)^1 + 3 * (2q)¹ * (-r)^2 + 1 * (2q)⁰ * (-r)^3
Simplifying further, we have:
(2q-r)³ = 8q³ - 12q²r + 6qr² - r³