Help - POLYNOMIALS Question (revision) (1 Viewer)

[bmn]

They are basic but im stuck...

1) If the sum of two roots of x^4 + 2x^3 -8x^2 - 18x + 9 =0 is 0, find the roots of the question
2) The sum of two roots of x^4 - 7x^3 + 5x^2 - x + 3 = 0 is 3. Find the sum of the other 2 roots

 

[Timothy.Siu]

They are basic but im stuck...

1) If the sum of two roots of x^4 + 2x^3 -8x^2 - 18x + 9 =0 is 0, find the roots of the question
2) The sum of two roots of x^4 - 7x^3 + 5x^2 - x + 3 = 0 is 3. Find the sum of the other 2 roots

1) If the sum of two roots of x^4 + 2x^3 -8x^2 - 18x + 9 =0 is 0, find the roots of the question

let roots be a,b,c,d let a+b=0
then,
a+b+c+d=-2
c+d=-2
ab+ac+ad+bc+bd+cd=-8 .....(2)
abc+abd+acd+bcd=18 .....(3)
abcd=9
from (2)
a(c+d)+b(c+d)+ab+cd=-8
(a+b)(c+d)+ab+cd=-8
ab+cd=-8
from (3)
ab(c+d)+cd(a+b)=18
-2ab=18
ab=-9
a+b=0 a=-b sub into ab=-9
-b^2=-9
b=+-3 a=-+3

cd=1 c+d=-2
c=1/d d^2+2d+1=0
(d+1)^2=0 d=-1 c=-1

wait,,edit: made a mistake,hmm i have conflicting statements...oh wells cant find the error

2) The sum of two roots of x^4 - 7x^3 + 5x^2 - x + 3 = 0 is 3. Find the sum of the other 2 root
let roots be a,b,c,d a+b=3
a+b+c+d=7
c+d=4

 

[Trebla]

Are you sure you typed the equation in 1) correctly? A sketch of the graph shows that there are no roots equal but opposite in sign. I suspect the constant should be - 9 rather than + 9 perhaps? (in which case the roots are 3, - 3 and -1 as a double root)

 

[bmn]

1) If the sum of two roots of x^4 + 2x^3 -8x^2 - 18x + 9 =0 is 0, find the roots of the question

let roots be a,b,c,d let a+b=0
then,
a+b+c+d=-2
c+d=-2
ab+ac+ad+bc+bd+cd=-8 .....(2)
abc+abd+acd+bcd=18 .....(3)
abcd=9
from (2)
a(c+d)+b(c+d)+ab+cd=-8
(a+b)(c+d)+ab+cd=-8
ab+cd=-8
from (3)
ab(c+d)+cd(a+b)=18
-2ab=18
ab=-9
a+b=0 a=-b sub into ab=-9
-b^2=-9
b=+-3 a=-+3

cd=1 c+d=-2
c=1/d d^2+2d+1=0
(d+1)^2=0 d=-1 c=-1

wait,,edit: made a mistake,hmm i have conflicting statements...oh wells cant find the error

2) The sum of two roots of x^4 - 7x^3 + 5x^2 - x + 3 = 0 is 3. Find the sum of the other 2 root
let roots be a,b,c,d a+b=3
a+b+c+d=7
c+d=4

Well, the answers to the first one was x = -1 +/- 3, which you have

Thanks again...

Also, I'm stuck on a basic thing in paramatic equations of the parabola

3) Find the equation of the tangent to the parabola x = 4t, y =2t^2 at the point where t = 3... I know a = 2, x = 12 and y = 18.... and x^2 = 8y... Mind blank kind of :/

 

[bmn]

Are you sure you typed the equation in 1) correctly? A sketch of the graph shows that there are no roots equal but opposite in sign. I suspect the constant should be - 9 rather than + 9 perhaps? (in which case the roots are 3, - 3 and -1 as a double root)

and you are right... 2nd time ive typed one wrong in 2 lots of asking...

 

[kaz1]

Well, the answers to the first one was x = -1 +/- 3, which you have

Thanks again...

Also, I'm stuck on a basic thing in paramatic equations of the parabola

3) Find the equation of the tangent to the parabola x = 4t, y =2t^2 at the point where t = 3... I know a = 2, x = 12 and y = 18.... and x^2 = 8y... Mind blank kind of :/

Since it is known that the parameter is the gradient.

y-18=3(x-12)
y-18=3x-36
3x-y-18=0

 

[Trebla]

1) If the sum of two roots of x^4 + 2x^3 -8x^2 - 18x + 9 =0 is 0, find the roots of the question

let roots be a,b,c,d let a+b=0
then,
a+b+c+d=-2
c+d=-2
ab+ac+ad+bc+bd+cd=-8 .....(2)
abc+abd+acd+bcd=18 .....(3)
abcd=9
from (2)
a(c+d)+b(c+d)+ab+cd=-8
(a+b)(c+d)+ab+cd=-8
ab+cd=-8
from (3)
ab(c+d)+cd(a+b)=18
-2ab=18
ab=-9
a+b=0 a=-b sub into ab=-9
-b^2=-9
b=+-3 a=-+3

cd=1 c+d=-2
c=1/d d^2+2d+1=0
(d+1)^2=0 d=-1 c=-1

It can seen that if a = 3, b = -3 and c = d = -1, then abcd = - 9, not 9. However, since the incorrect result abcd = 9 was never actually used, the mistake in the question was undetected lol.

 

[Timothy.Siu]

It can seen that if a = 3, b = -3 and c = d = -1, then abcd = - 9, not 9. However, since the incorrect result abcd = 9 was never actually used, the mistake in the question was undetected lol.

yeah thought something was fishy