Factorizing y³ - 2y² - 29y - 52 using Factor Theorem:



1. Factor Theorem:

The Factor Theorem states that if a polynomial f(x) has a factor (x - a), then f(a) = 0.



2. Given Polynomial:

y³ - 2y² - 29y - 52



3. Using Factor Theorem:

Let's assume that (y - k) is a factor of the given polynomial.

Now, according to the Factor Theorem, if (y - k) is a factor, then the polynomial will be evenly divisible by (y - k) when f(k) = 0.

Therefore, f(k) = k³ - 2k² - 29k - 52 = 0



4. Finding the Root:

By trial and error, we can find that k = 4 is a root of the polynomial as f(4) = 4³ - 2(4)² - 29(4) - 52 = 0.



5. Dividing the Polynomial:

Now, we can divide the given polynomial by (y - 4) using synthetic division or polynomial division to factorize it completely.

After dividing, we get: (y - 4)(y² + 2y + 13) = 0



6. Final Factorization:

Therefore, the factorization of y³ - 2y² - 29y - 52 is:

(y - 4)(y² + 2y + 13)