Polynomials Question (1 Viewer)

[cutemouse]

Could someone please help me with this question?

The polynomial P(x) is given by P(x)=2x<sup class="moz-txt-sup">3</sup> - 9x<sup class="moz-txt-sup">2</sup> + 12x - k (where k is real). Find the range of values for k for which P(x)=0 has 3 real roots.

Thanks

 

[Iruka]

Find the x values of the two stationary points.

Find conditions on k that ensure that the two stationary points have y-values that are opposite in sign.

(Drawing a picture might help.)

 

[Timothy.Siu]

Could someone please help me with this question?

The polynomial P(x) is given by P(x)=2x<sup class="moz-txt-sup">3</sup> - 9x<sup class="moz-txt-sup">2</sup> + 12x - k (where k is real). Find the range of values for k for which P(x)=0 has 3 real roots.

Thanks

P'(x )=6x²-18x+12
=6(x²-3x+2)
=6(x-2)(x-1)
=0 when x=2 or 1

(just to check)
P''(x )=6(2x-3)
when x=2 P''(x)=6 , minimum turning point.
when x=1 P''(x)=-6, max. turning point.

ok.
x=1 P(x)=5-k
x=2 P(x)=4-k

i'm not sure if this is the best way to do it.
5-k>0
and 4-k<0
(since we want 3 real roots)
5>k and 4<k
therefore 4<k<5
i think.

 

[lyounamu]

ye i reckon timmy's approach is right =)

well done

 

[cutemouse]

lol, I'm lost... Could anyone else explain it?

I mean, I could do this graphically I guess... ie. by sketching y = 2x<sup class="moz-txt-sup">3</sup> - 9x<sup class="moz-txt-sup">2</sup> + 12x (showing all stat points) against y=k and then by noting where the graphs intesect 3 times... But IMO that's a waste of time and there should be a different/faster way to do it.

 

[clintmyster]

P'(x )=6x²-18x+12
=6(x²-3x+2)
=6(x-2)(x-1)
=0 when x=2 or 1

(just to check)
P''(x )=6(2x-3)
when x=2 P''(x)=6 , minimum turning point.
when x=1 P''(x)=-6, max. turning point.

ok.
x=1 P(x)=5-k
x=2 P(x)=4-k

i'm not sure if this is the best way to do it.
5-k>0
and 4-k<0
(since we want 3 real roots)
5>k and 4<k>
therefore 4<k><5
i think.

graphically that makes sense. Can you just use graphs for this question saything that it has to be between those two values as otherwise you will get a double root and another root?

</k></k>

lol, I'm lost... Could anyone else explain it?

I mean, I could do this graphically I guess... ie. by sketching y = 2x3 - 9x2 + 12x (showing all stat points) against y=k and then by noting where the graphs intesect 3 times... But IMO that's a waste of time and there should be a different/faster way to do it.

well tim essentially could sketch it really quickly using what he found for the turning points. if you realise like i did that 3 roots occurs between the values for k where theres a double root and another single root, its quite easy. If you can see it visually in your mind you could do a very very rough sketch and save time.

 

[azureus88]

if it has 3 real roots, then the y-values of the 2 turning points must be opposite in sign. So:

P(1).P(2)<0
(5-k)(4-k)<0

4 < k < 5

 

[cutemouse]

Yeah I figured it out... Thanks

 

[gurmies]

Love these questions

 

[omniscience]

1+1 = 2

 

[cutemouse]

1+1 = 2

Yep. 1+1=Your IQ

 

[Iruka]

graphically that makes sense. Can you just use graphs for this question saything that it has to be between those two values as otherwise you will get a double root and another root?

</k></k>



well tim essentially could sketch it really quickly using what he found for the turning points. if you realise like i did that 3 roots occurs between the values for k where theres a double root and another single root, its quite easy. If you can see it visually in your mind you could do a very very rough sketch and save time.

There is a question in the 2001 2u HSC paper (Q6(c)) which uses a similar idea, but with a different cubic.