Andrew factored the expression 20x^3-12x^2+8x as 4x(5x^2-12x^2+8x). But when Melissa applied the distributive law and multiplied out 4x(5x^2-12x^2+8x),she got 20x^3-48x^3+32x^2; thus, Andrew's solution does not appear to check. Why is that? Please help Andrew to understand this better. Explain your reasoning and correctly factor the original expression, if possible. If the expression is prime, state this.
Andrew messed up, should have been
= 4x(5x^2 - 3x + 2)
To help Andrew understand why his solution does not appear to check, let's go through the process step by step.
When Andrew factored the expression 20x^3 - 12x^2 + 8x as 4x(5x^2 - 12x^2 + 8x), he correctly pulled out the greatest common factor, which is 4x. However, the second part of his factorization, (5x^2 - 12x^2 + 8x), is incorrect.
To correctly factor the expression, we need to factor the trinomial (5x^2 - 12x^2 + 8x) further. Let's break it down:
5x^2 - 12x^2 + 8x
First, notice that 5x^2 and 12x^2 have the same variable (x^2), so we can combine them:
-7x^2 + 8x
Now, we can see that both terms have a common factor of x, so we can factor that out as well:
x(-7x + 8)
Therefore, the correct factored form of the expression is 4x(-7x + 8).
To check if this is the correct factorization, let's use the distributive law and multiply out 4x(-7x + 8):
4x(-7x + 8) = (-7x)(4x) + 8(4x) = -28x^2 + 32x
As we can see, this does not match the original expression 20x^3 - 12x^2 + 8x. Therefore, it seems like the original expression cannot be factored further using integer coefficients.
In conclusion, the factored form of the original expression 20x^3 - 12x^2 + 8x is 4x(5x^2 - 12x^2 + 8x), but it cannot be factored further using integer coefficients.