Find LCM

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

To find the least common multiple (LCM) of two polynomial expressions, we need to factorize each expression and identify their common factors.

Let's start with the first expression, t^3 + 8t^2 + 16. The expression cannot be easily factored using common methods like factoring by grouping or difference of squares. However, we can use a special factorization formula called the sum of cubes identity.

The equation t^3 + 8t^2 + 16 can be rewritten as (t + 2)^3. This is because the expression follows the sum of cubes pattern: a^3 + b^3 = (a + b)(a^2 - ab + b^2), where a = t and b = 2.

Next, let's factorize the second expression, t^2 - 8t. We can factor out a common factor of t, which gives us t(t - 8).

Now, let's determine the LCM by finding the highest power of each factor. Since we have (t+2)^3 as a factor, the highest power of (t+2) is 3. Similarly, the highest power of (t-8) is 1 (since it's not squared or cubed).

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

Note that you can also multiply the expressions out completely and then take the common factors, but it is generally preferred to factorize them partially unless specifically asked to do so.