Explanation:
To check Olivia's work, we can multiply the factors she obtained to see if they result in the original expression. Let's perform the multiplication:
(x + 3y)(x^2 - 1) = x(x^2 - 1) + 3y(x^2 - 1)
Distributing the terms:
= x * x^2 - x * 1 + 3y * x^2 - 3y * 1
= x^3 - x + 3yx^2 - 3y
As we compare this with the original expression:
x^3 - x + 3x^2y = 3y
We can see that Olivia's factored expression, (x + 3y)(x^2 - 1), does not match the original expression. Olivia made a mistake in the step where she distributed the terms.
To correct the mistake, we need to distribute the terms correctly. Let's go through the factoring process again:
Starting with the original expression: x^3 - x + 3x^2y = 3y
Rearranging the terms: x^3 + 3x^2y - x - 3y = 0
Now, we can factor by grouping:
x^2(x + 3y) - 1(x + 3y) = 0
Notice that we have a common factor of (x + 3y). Factoring it out:
(x + 3y)(x^2 - 1) = 0
Now we have the correct factored expression.