Find the LCM

t^3+8t^2+16,t^2-8t

t^2 - 8t = t(t-8)

Now, a cubic is not easy to factor, but since you're looking for a LCM, try dividing by (t-8).

No luck there. So, since the two polynomials have no common factors, the LCM is just

(t^3+8t^2+16)(t^2-8t)

Katie, shouldn't the 16 be negative? Check it out.

To find the Least Common Multiple (LCM) of the given expressions, we need to factor each expression completely first.

Let's start with the first expression, t^3 + 8t^2 + 16.

1. First, notice that we can factor out a common factor of t^2 from each term: t^2(t + 8) + 16.

2. Now, factor the expression inside the parentheses: t^2(t + 8) + 16 = t^2(t + 8) + 4^2.

So, the first expression completely factored is: t^2(t + 8) + 4^2.

Next, let's factor the second expression, t^2 - 8t.

1. Factor out a common factor of t: t(t - 8).

Therefore, the second expression completely factored is: t(t - 8).

Now that we have factored both expressions, we can find their LCM.

The LCM of two or more expressions is the product of their highest powers of each unique factor.

From the first expression, we have two unique factors: t^2 and (t + 8)^2.

From the second expression, we have two unique factors: t and (t - 8).

To find the LCM, we take the highest power of each unique factor: t^2 * (t + 8)^2 * (t - 8).

So, the LCM of the expressions t^3 + 8t^2 + 16 and t^2 - 8t is t^2 * (t + 8)^2 * (t - 8).