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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.)

 Dec 9, 2023
 #1
avatar+222 
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We want to find the x-values for which f(x)=g(x). Therefore, we must solve the equation [x^4 - 7x^3 + 5x^2 - 18x + 17 = x^3 + x^2 - 3x + 5.]

This simplifies to [x^4 - 8x^3 + 4x^2 - 15x + 12 = 0.]

 

By factoring, we get [(x^3 - 7x^2)(x - 12) = 0.]

 

Therefore, x=0,12,4,3. Since f(4)=f(3) and g(4)=g(3), the points of intersection are (0,5), (3,39), (4,53), and (12,−67). Thus, our answer is (0,5),(3,39),(4,53),(12,−67)​.

 Dec 9, 2023

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