complete factorization

y^4-25y^2+144
(^) denotes following number is Power

y^4 - 25y^2 + 144

You might try replacing y^2 with x. Therefore your statement becomes:

x^2 - 25x + 144

...which is easier to factor.

To factorize the expression y^4 - 25y^2 + 144, we can follow these steps:

Step 1: Replace y^2 with x. This substitution will make the expression easier to factor.
x^2 - 25x + 144

Step 2: Factorize the quadratic expression x^2 - 25x + 144. To do this, we need to find two factors of 144 that add up to -25 (the coefficient of x).

The factors of 144 are:
1, 144
2, 72
3, 48
4, 36
6, 24
8, 18
9, 16
12

From these factors, we can observe that -16 and -9 are two numbers that add up to -25. So, we can rewrite -25x as -16x - 9x:

x^2 - 16x - 9x + 144

Step 3: Group the terms and factor by grouping. Group the first two terms and the last two terms:

(x^2 - 16x) + (-9x + 144)

Step 4: Factor out the common factor from each group:

x(x - 16) - 9(x - 16)

Step 5: We can see that both terms have a common factor of (x - 16). Factor that common factor out:

(x - 16)(x - 9)

Step 6: Substitute y^2 back in for x:

(y^2 - 16)(y^2 - 9)

So, the complete factorization of the expression y^4 - 25y^2 + 144 is (y^2 - 16)(y^2 - 9).