Andrew factored the expression -12x^3-28x^2-3x as -4x (3x^2-7x+9). But when Melissa applied the distributive law and multiplied out -4x (3x^2-7x+9), she got -12x^3+28x^2-36x; 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, so state.
Given: -12X^3 - 28X^2 - 3X.
Andrew made 2 mistakes:
1. You can't factor out -4x, because -3
is not divisible by -4.
2.The 2nd term in the parenthesis should be +7X.
You can only factor out an X:
X(-12X^2 - 28X - -3).
To understand why Andrew's solution does not appear to check, let's go through the process of factoring the original expression and then multiplying it out to verify if it matches Andrew's factored form.
The original expression is:
-12x^3 - 28x^2 - 3x
To factor this expression, we can look for common factors first. In this case, the common factor is -1x or simply -x. Factoring out -x, we get:
-x(12x^2 + 28x + 3)
Now we need to factor the quadratic expression within the parentheses, 12x^2 + 28x + 3. This can be done by either factoring by grouping or using the quadratic formula. Let's use factoring by grouping:
12x^2 + 28x + 3
First, multiply the leading coefficient (12) and the constant term (3), which gives us 36.
Next, we need to find two numbers that add up to the middle coefficient (28) and multiply to give us 36. Those numbers are 6 and 6.
We can now split the middle term using these two numbers:
12x^2 + 6x + 22x + 3
Now, we can factor by grouping by taking out the common factors from the first two terms and the last two terms:
6x(2x + 1) + 3(2x + 1)
We can see that (2x + 1) appears in both terms, so we can factor it out:
(6x + 3)(2x + 1)
Now we have factored the quadratic expression: 12x^2 + 28x + 3 = (6x + 3)(2x + 1)
Combining this with the common factor -x, the fully factored expression becomes:
-x(6x + 3)(2x + 1)
If we expand this factored form using the distributive law, we get:
-x(6x)(2x) - x(6x)(1) - x(3)(2x) - x(3)(1)
-12x^3 - 6x^2 - 6x^2 - 3x - 6x^2 - 3x - 3
Simplifying this expression, we get:
-12x^3 - 28x^2 - 3x
We can see that the expanded expression matches the original expression, which means our factored form is correct.
Therefore, Andrew's original factored form, -4x (3x^2-7x+9), is incorrect. The correct factored form is -x(6x + 3)(2x + 1).