which one is correct:
factor completely:
y^4+3y^3-5y^2-15y
I ended up with two different ways but i am not sure which one is correct.
Is it
(A) y(y^2-5)(y+3)
or
(B) (y^3-5y)(y+3)
Personally i think its A but then when i do foil to both of them i get the same answer.
Both are the same, but neither is factored completely.
y(y+sqrt5)(y-sqrt5)(y+3)
To determine which answer is correct, let's begin by factoring the given polynomial completely.
The polynomial we have is y^4 + 3y^3 - 5y^2 - 15y.
To factor a polynomial completely, we need to look for common factors, factor out any possible common factors, and then proceed with further factoring if necessary.
Step 1: Look for common factors.
In this case, there are no common factors among all the terms.
Step 2: Factor out any possible common factors.
We can start by factoring out the greatest common factor (GCF) of all the terms, which is y.
By factoring out y, we get the expression y(y^3 + 3y^2 - 5y - 15).
Now, let's focus on factoring the expression inside the parentheses, y^3 + 3y^2 - 5y - 15.
Step 3: Factor the expression further.
To factor this cubic polynomial, we can use various techniques such as factoring by grouping, synthetic division, or applying the rational root theorem. In this case, we can observe that -3 is a root of the polynomial (also known as a zero or a solution) by synthetic division.
Performing synthetic division with -3 as the root, we get:
-3 | 1 3 -5 -15
|___ -3 0 15
1 0 -5 0
The resulting quotient polynomial is y^2 + 0y - 5, which can be further factored as (y + sqrt(5))(y - sqrt(5)). Therefore, our completely factored expression is:
y(y^2 + 0y - 5)(y + 3).
Now, let's compare this with the given options (A) and (B):
(A) y(y^2 - 5)(y + 3)
(B) (y^3 - 5y)(y + 3)
Comparing their forms with our completely factored expression, we can see that both options (A) and (B) are incorrect.
The correct fully factored form of the given polynomial y^4 + 3y^3 - 5y^2 - 15y is:
y(y^2 + 0y - 5)(y + 3).
Note: It's important to carefully check the factored form and make sure all possible factors are included to ensure completeness.