Andrew factored the expression 54x^3-30x^2+15x as 5x(9x^2-6x+3) but when Melissa applied the distributive law and multiplied out 5x(9x^2-6x+3). she got 45x^3-30x^2+15x. thus andrew's solution does not appear to check. why is this? eplain your reasoning and correctly factor the original expression if possible.If the expression is a prime state so
3x(18x^2-10x-5)
To understand why Andrew's solution does not seem to check, let's go step by step.
Andrew factored the expression 54x^3 - 30x^2 + 15x as 5x(9x^2 - 6x + 3). To determine whether his factoring is correct, we need to apply the distributive law and multiply it back out.
When Melissa multiplied 5x(9x^2 - 6x + 3), she obtained 45x^3 - 30x^2 + 15x, which does not match the original expression. So, Andrew's factoring does not seem to be correct.
To find the correct factorization, we need to consider the factors of each term in the expression. Let's break down each term:
54x^3: The prime factors of 54 are 2 and 3, and x^3 is a common factor of x.
30x^2: The prime factors of 30 are 2, 3, and 5, and x^2 is a common factor of x^2.
15x: The prime factors of 15 are 3 and 5, and x is a common factor of x.
Now, let's see if we can derive a common factor from these prime factors.
First, we notice that 3 is a common factor for all three terms. We can factor it out as follows:
3(18x^3 - 10x^2 + 5x)
Next, we can derive the common factor of x:
3x(18x^2 - 10x + 5)
So, the correct factorization of the expression 54x^3 - 30x^2 + 15x is 3x(18x^2 - 10x + 5).
Since we were able to factor the expression further, it is not a prime state.