What are the coordinates of the points where the graphs of $f(x)=x^3 + x^2 - 3x + 5$ and $g(x)=x^4-7x^3+5x^2-18x+17$ intersect?

Give your answer as a list of points separated by commas, with the points ordered such that their $x$-coordinates are in increasing order. (So "(1,-3), (2,3), (5,-7)" - without the quotes - is a valid answer format.)

bader Dec 2, 2023

#1**0 **

By the quadratic formula, the roots of x2−27x+413 are [ \frac{7\pm\sqrt{\left(\frac{7}{2}\right)^2-4\cdot1\cdot\frac{13}{4}}}{2}=\frac{7\pm\sqrt{1}}{2}=\frac{7}{2}\pm\frac{\sqrt{1}}{2} ]

Therefore, the values of x at which the graphs of f and g intersect are x=2, x=27+1 and x=27−1.

We can plug these x values back into f(x) to get that f(2)=21, f(27+1)=825+1 and f(27−1)=825−1, so the points of intersection are [ \left(2,21\right),\left(\frac{7+\sqrt1}{2},\frac{25+\sqrt{1}}{8}\right),\left(\frac{7-\sqrt1}{2},\frac{25-\sqrt{1}}{8}\right). ]

parmen Dec 2, 2023