Polynomial Question (1 Viewer)

[michaeljennings]

The graph of has turning points at x=2 and x=-1. Find the values of k such that the equation has 3 real and distinct roots

How do you do this question?

 

[Drongoski]

The graph of has turning points at x=2 and x=-1. Find the values of k such that the equation has 3 real and distinct roots

How do you do this question?

Imagine you have a (rigid wire) graph of

and move it up k unit (or down k units if k is negative. You get 3 roots only when the repositioned graph cuts the x-axis at 3 distinct points.

 

[michaeljennings]

Imagine you have a (rigid wire) graph of

and move it up k unit (or down k units if k is negative. You get 3 roots only when the repositioned graph cuts the x-axis at 3 distinct points.

ohhh right but how do you approach this question, like how do you begin?

 

[Drongoski]

ohhh right but how do you approach this question, like how do you begin?

You then do as suggested by b3kh1t. You will find the turning points at x = -1 (a max) and x=2 (a min). For







That means: You should not push up the graph of y = f(x) more than 20 nor down more than 7 units.

i.e. -7 < k < 20.

 

[b3kh1t]

sorry I did not read the question properly, there is another method to do it.

 

[michaeljennings]

You then do as suggested by b3kh1t.

i dont understand how to do itttttttttt

 

[math man]

If you find the first and second derivative you find x=-1 is a max turning point, and x=2 is a min turning point.

If i sub x=2 into y, i get:



so that means (2, k-20) is my mim turning point

If i sub x=-1 into y, i get:



therefore (-1, k+7) is my max turning point.

If i plot these two points on a number plane and sketch the curve i see that when k=20 i will only have 2 distinct roots, so therefore k<20,

also, when k = -7 i will only have two distinct roots, so k>-7.

Therefore, for y to have 3 distinct real roots :
-7< k < 20

 

[HyperComplexxx]

you let a line be y = k
then you draw it onto the polynomial, finding the region in which y = k intercepts the graph

 

[b3kh1t]

sorry with this question you form two equations; and
then you sketch the graph for and find where the line will intersect three times with the function

 

[michaeljennings]

i considered the graph y = 2x^3-3x^2-12x by itself and found the y-values of the TPs. then for 3 real and distinct roots the graph must be cut by the line y = k at three different points. therefore you get -20 to k to 7

alrighty thanks makes sense

edit: thanks to everyone else as well

 

[apollo1]

alrighty thanks makes sense

edit: thanks to everyone else as well

i made a mistake with the actual answer. math mans answers the right one.

 

[michaeljennings]

If you find the first and second derivative you find x=-1 is a max turning point, and x=2 is a min turning point.

If i sub x=2 into y, i get:



so that means (2, k-20) is my mim turning point

If i sub x=-1 into y, i get:



therefore (-1, k+7) is my max turning point.

If i plot these two points on a number plane and sketch the curve i see that when k=20 i will only have 2 distinct roots, so therefore k<20,

also, when k = -7 i will only have two distinct roots, so k>-7.

Therefore, for y to have 3 distinct real roots :
-7< k < 20

thanks i would rep you but i cant rep you til i share it with others

 

[math man]

thanks i would rep you but i cant rep you til i share it with others

lol thxs