polynomial help :( (1 Viewer)

[AnandDNA]

1. Find the roots of the equation 4x^3-8x^2-3x+9=0 given that two of the roots are equal

2. Find the zeroes of the equation 8x^3-14x^2+7x-1=0 if the zeroes form a geometric sequence

 

[tommykins]

回复: polynomial help

1. Find the roots of the equation 4x^3-8x^2-3x+9=0 given that two of the roots are equal

2. Find the zeroes of the equation 8x^3-14x^2+7x-1=0 if the zeroes form a geometric sequence

1. Let roots be a, a, b.

2a+b = 2
a^2*b = -9/4

Solve simulataneously. Or you can just sub in factors of 9 and find a root then polynomial division.

2. Let roots be a/r, a, ar

(a/r)*(a)*(ar) = a^3 = 1/8

a = 1/2

sum of roots is a/r + a + ar = 14/8

sub in a = 1/2 and solve for r.

factorise the polynomial with your roots.

 

[lyounamu]

1. Find the roots of the equation 4x^3-8x^2-3x+9=0 given that two of the roots are equal

2. Find the zeroes of the equation 8x^3-14x^2+7x-1=0 if the zeroes form a geometric sequence

1. Let the roots be a, a and b

a + a + b = -b/a = 8/4 = 2 ...(1)
a^2 + ab + ab = c/a = -3/4 ...(2)
a^2 . b = -d/a = -9/4

Then from (1), you can find that b = 2-2a
Then substitute that to (2)
So you get a^2 + 4a - 4a^2 = - 3/4
3a^2 - 4a - 3/4 = 0 (through manipulation of the equation)
Then 12a^2 - 16a - 3 = 0
(6a+1)(2a-3) = 0
so a = -1/6 or a = 3/2
Substitute that to equation (1) to get value for b.

b = -1 (when a = 3/2) or 7/3 (when a = -1/6)
2. As tommykins said.

 

[AnandDNA]

b=2-2a (1)
a^2*b=-9/4 (2)

sub 1 into 2
a^2(2-2a)=-9/4
expand
8a^3-8a^2-9=0
What do i do now and is this method also okay or is this more of a nuisance

 

[lyounamu]

b=2-2a (1)
a^2*b=-9/4 (2)

sub 1 into 2
a^2(2-2a)=-9/4
expand
8a^3-8a^2-9=0
What do i do now and is this method also okay or is this more of a nuisance

Look at my method. I got that sorted out.

 

[AnandDNA]

yea but i wanted to know what was wrong with my method

 

[lyounamu]

yea but i wanted to know what was wrong with my method

Well, your method is not wrong but by doing that you complicated it. Look at the last line of your working out, in that case, you will find it difficult to find the value of a because it is in a cubic form. It is relatively easier when you have the normal quadratic equation where you can either use quadratic formula or work out.

 

[tommykins]

回复: Re: polynomial help

yea but i wanted to know what was wrong with my method

Nothing, it just makes the process of getting a longer and more tedious (as you get another cubic)

 

[AnandDNA]

Re: 回复: Re: polynomial help

If i didnt know the sum of the roots two at a time then where should i have subbed in the b=2-2a

 

[lyounamu]

Re: 回复: Re: polynomial help

If i didnt know the sum of the roots two at a time then where should i have subbed in the b=2-2a

Um, there are not many options available then. You will end up with that cubic and then you will have to solve it from there which will be quite difficult. You will probably end up trying to solve by via "trial and error" but it won't work as you have already seen, the answers are in fraction or decimal. But you will still get 1 mark out of 3. (if it was out of 3 marks)

 

[krazedrealitii]

Re: 回复: Re: polynomial help

If i didnt know the sum of the roots two at a time then where should i have subbed in the b=2-2a

I think you would have to know since (I assume...) you are given the equation of the polynomial and therefore it's always possible to work it out?