1. Solve the equations $2 \log _{3}(x+2)-\log _{3} x=2$ and $25^{x}=0.008$.

a) Solve the equations.

b) The solutions of the equations are the zeros of a polynomial $p$ of the third degree. The graph of this polynomial also passes through the point (2,7). Write the formula of the polynomial $p$ in its factored form.
1. The logarithmic equation is transformed into $\frac{(x+2)^{2}}{x}=3^{2}$, rearranged into $\frac{x^{2}+4 x+4}{x}=9$ or $x^{2}-5 x+4=0$ and factored as $(x-4)(x-1)=0$. The solutions are $x_{1}=4$ and $x_{2}=1$. Then the exponential equation is rearranged into $5^{2 x}=5^{-3}$. The solution to this equation is $x=-\frac{3}{2}$. From all the solutions of the given equations, we write the polynomial $p(x)=a(x-4)(x-1)\left(x+\frac{3}{2}\right)$. Using the coordinates of the point $(2,7)$, we calculate the leading coefficient $a=-1$. Then we write the polynomial in its zero form as $p(x)=-(x-4)(x-1)\left(x+\frac{3}{2}\right)$.

Transforming the logarithmic equation to $x^{2}-5 x+4=0 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

![](https://cdn.mathpix.com/cropped/2024_06_07_59669f975316a35884bfg-13.jpg?height=52&width=1642&top_left_y=702&top_left_x=204)

![](https://cdn.mathpix.com/cropped/2024_06_07_59669f975316a35884bfg-13.jpg?height=57&width=1639&top_left_y=748&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_59669f975316a35884bfg-13.jpg?height=66&width=1639&top_left_y=795&top_left_x=206)

Writing the polynomial $p(x)=a(x-4)(x-1)\left(x+\frac{3}{2}\right) \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

Writing the polynomial $p(x)=-(x-4)(x-1)\left(x+\frac{3}{2}\right) \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$...................................