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).