Why am I getting the wrong answer? Simplification of $(1-\beta x)(-\beta x)^2 + \beta x(1-\beta x)^2$?

Question

This is driving me crazy, I thought this would be quite straightforward, but it doesn't appear to be, I have:

$$(1-\beta x)(-\beta x)^2 + \beta x(1-\beta x)^2$$ Divide by $(1-\beta x)$ and we get,

$$(-\beta x)^ 2+ \beta x(1-\beta x)$$

Then divide by $\beta x$

$$\beta x + (1-\beta x)$$

But the answer is $(1-\beta x)\beta x$

Thanks

Answer 1

It's $$\beta^2x^2-\beta^3x^2+\beta x-2\beta^2x^2+\beta^3x^2=\beta x-\beta^2x^2=\beta x(1-\beta x).$$

Answer 2

You lost a factor of $1-\beta x$ when you divided the whole expression by $1-\beta x$, and then you lost a factor of $\beta x$ when you divided through by $\beta x$. Restore those missing factors, and you get the correct answer, $\beta x(1-\beta x)\big(\beta x+(1-\beta x)\big)=\beta x(1-\beta x)$. Here is the calculation done properly:

$$\begin{align*} (1-\beta x)(-\beta x)^2+\beta x(1-\beta x)^2&=(1-\beta x)(\beta x)^2+\beta x(1-\beta x)^2\\ &=\beta x(1-\beta x)(\beta x+1-\beta x)\\ &=\beta x(1-\beta x) \end{align*}$$

What you did is analogous to ‘simplifying’ $2\cdot(-3)^2+3\cdot 2^2=30$ by dividing through by $2$ to get (-3)^2+3\cdot2 and then by $3$ to get $3+2=5$. The divisions lost a whole factor of $2\cdot 3=6.

Answer 3

Diving out two factors gives you $1$, so the original was their product, which is what you've been told.

Answer 4

What you are doing is not simplifying $$(1-\beta x)(-\beta x)^2 + \beta x(1-\beta x)^2$$. You are simplifying $$(1-\beta x)(-\beta x)^2 + \beta x(1-\beta x)^2=0$$. which are not the same; the first is a polynomial, and the second is a equation.

To simplify $(1-\beta x)(-\beta x)^2 + \beta x(1-\beta x)^2$, you need to do $$(1-\beta x)(-\beta x)^2 + \beta x(1-\beta x)^2 = (1-\beta x)((-\beta x)^2+\beta x(1-\beta x)) = (1-\beta x)\beta x$$

Answer 5

$(1−\beta x)(−\beta x)^2+\beta x(1−\beta x)^2$ Divide by $(1−\beta x)$

and we get,

$(−\beta x)^2+\beta x(1−\beta x)$

Then divide by $\beta x$

$\beta x+(1−\beta x)= 1$ Now, since u divided by $\beta x$ and by $(1-\beta x)$ you have to multiply $1$ by each of those terms and you get $(1−\beta x)\beta x$ ...basically u divided by two terms,but to preserve the equation you have to multiply the numerator by those same terms

But the answer is $(1−\beta x)\beta x$