To find the completely factored form of the expression x4y + 4x2y + 5y, we can start by factoring out the common terms.
1. Identify Common Factors:
The expression has a common factor of y.
2. Factor Out ‘y’:
By factoring out ‘y’, we get:
y(x4 + 4x2 + 5)
3. Next, we need to focus on the expression inside the parentheses: x4 + 4x2 + 5. This is a quadratic in terms of x2. Let’s substitute u = x2, transforming our expression into:
u2 + 4u + 5
4. Complete the Square (if applicable):
To factor this further, we can check if it can be factored over the real numbers. We calculate the discriminant:
- Discriminant = b2 – 4ac
- In our case, a = 1, b = 4, c = 5
- Discriminant = 42 – 4(1)(5) = 16 – 20 = -4
Since the discriminant is negative, u2 + 4u + 5 cannot be factored into real linear factors.
5. Final Factored Form:
Thus, the completely factored form of the original expression is:
y(x4 + 4x2 + 5)
Since we have determined that x4 + 4x2 + 5 does not factor further over the reals, we conclude that:
y(x4 + 4x2 + 5) is the completely factored form of the expression x4y + 4x2y + 5y.