2. Given the polynomial $p(x)=x^{4}-3 x^{3}+a x^{2}$ and the line $y=b x+20$. The graphs of both functions intersect at points with abscissas $x=5$ and $x=-2$. Determine the coefficients $a$ and $b$ and write both functions.
2. To find the x-coordinates of the intersection points of the graphs of both functions, we solve the equation $x^{4}-3 x^{3}+a x^{2}=b x+20$ or $x^{4}-3 x^{3}+a x^{2}-b x-20=0$. Since the solutions are $x=5$ and $x=-2$, we have $625-375+25 a-5 b-20=0$ and $16+24+4 a+2 b-20=0$. Rearranging the equations gives $25 a-5 b+230=0$ and $4 a+2 b+20=0$. The system has the solution $a=-8$ and $b=6$. Therefore, the functions are $p(x)=x^{4}-3 x^{3}-8 x^{2}$ and $y=6 x+20$.

![](https://cdn.mathpix.com/cropped/2024_06_07_40efadbcfacda9b65264g-12.jpg?height=57&width=1642&top_left_y=1114&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_40efadbcfacda9b65264g-12.jpg?height=57&width=1639&top_left_y=1162&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_40efadbcfacda9b65264g-12.jpg?height=57&width=1642&top_left_y=1214&top_left_x=207)

Correct solution of the system .................................................................................................................

![](https://cdn.mathpix.com/cropped/2024_06_07_40efadbcfacda9b65264g-12.jpg?height=52&width=1639&top_left_y=1316&top_left_x=206)

The functions $p(x)=x^{4}-3 x^{3}-8 x^{2}$ and $y=6 x+20 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$ point