To identify which equation demonstrates that you can divide polynomials by recognizing division as the inverse operation of multiplication, we need to look for an equation where division is shown as a multiplication of a polynomial by a factor that should equal the left side.
We are given the equation formats:
- \(8x^2 - 4x + 12 = (-14x)(8x^2 - 4x + 12)\)
- \(8x^2 - 4x + 12 = (-4x)(8x^2 - 4x + 12)\)
- \(8x^2 - 4x + 12 = (14x)(8x^2 - 4x + 12)\)
- \(8x^2 - 4x + 12 = (4x)(8x^2 - 4x + 12)\)
In each option, we see the polynomial \(8x^2 - 4x + 12\) presented as equal to the product of another polynomial and a multiple of \(x\).
To divide \(8x^2 - 4x + 12\) by the polynomial on the right effectively indicates that both sides must yield the same value when appropriately simplified.
To show division clearly, we'd typically see a situation where we can isolate the polynomial on one side against established coefficients. We want to express, for example with \(k\):
If \( k = \frac{8x^2 - 4x + 12}{\text{other polynomial}} \)
None of the provided equations clearly follow that rule.
For division to be mirrored with multiplication here, we'd expect something like this:
\[ \frac{8x^2 - 4x + 12}{4x} = 2x - 1 + \frac{3}{4x} \]
But since that is not one of the options you provided and instead looks like they're just multiplying forms together without supporting any division clarity, let’s review them for correctness in terms of identifying the appropriate factor:
- Correct assessment: If a number multiplied by something yields another number, using inversion would imply division of that number gives back the original polynomial.
Thus, the right analogy can be noticed in the last choice: \[ 8x^2 - 4x + 12 = (4x)(8x^2 - 4x + 12) \]
This statement, although incorrectly equated, at least suggests a factor that relates back to the polynomial through multiplication division.
Final choice would be: \(8x^2 - 4x + 12 = (4x)(8x^2 - 4x + 12)\) as it suggests inverse relation to initial polynomial and presents an intuitive idea of what division would mean in terms of balancing the multiplication. Thus, mathematically providing insight into polynomial behavior (albeit mistakenly, in terms of strict equality).
3 answers