Polynomials Have Turned Me Emo. HELP! (1 Viewer)

[nimrod_dookie]

Hi, I am having some problems with some polynomials questions which I was absent for when they were covered :burn: and now my teacher has been away ill and hence, has been unable to help me. If anyone could help me I would greatly appreciate it.

1. Find real numbers a and b such that x^4-4x^2+8x-4=(x^2+ax+2)(x^2+bx-2) and hence solve the equation x^4+8x=4x^2+4.

2. 3z+2 is a factor of 3z^3-z^2+(a+1)z+a. Find a and all zeros of the cubic.

3. x+3 and 2x-1 are factors of 2x^4+ax^3+bx^2+ax+3. Find a and b and hence determine all zeros of the quartic.

4. 3x^3+4x^2-x+m has two identical linear factors. Find m and the zeros of the polynomial in all possible cases.

 

[pLuvia]

3. Since x+3 and 2x-1 are factors then
P(-3)=0
P(1/2)=0
Subbing those values in you should get two equations, then simultaneously solve them for a and b.
Then when you find a and b, (x+3)(2x-1) is a factor, so then use long division to get the zeros

4. 3x³+4x²-x+m
Two identical means double roots
P'(x)=9x²+8x-1=0
x=1/9,-1
If P(-1)=-3+4+1+m=2+m=0
m=-2
If P(1/9)=m-14/243=0
m=14/243

If m=-2
3³+4x²-x-2=0
(x+1) is a factor
Using long division you should get the other factors
If m=14/243
3³+4x²-x+14/243=0
(x-1/9) is a factor
Using long division you should get the other factors and the roots

 

[pLuvia]

2. Since 3z+2 is a factor then P(-2/3)=0 which means
P(-2/3)=-8/9-4/9+(a+1)(2/3)+a=0
=-2/3+5/3a=0
a=2/5

P(x)=3x³-z²+z+4/5=0
=15x³-5x²+5z+4=0

Since (3z+2) is a factor using long division you can get the other factors hence the zeroes

 

[nimrod_dookie]

Thanks .

 

[pLuvia]

1. x⁴-4x²+8x-4=(x²+ax+2)(x²+bx-2)
Sub in a value for x
I.e
Let x=1
1=(a+3)(b-1)
1=ab-a+3b-3
ab-a+3b-4=0 --- (1)

Let x=-1
-15=(-a+3)(-b-1)
-15=ab+a-3b-3
ab+a-3b+12=0 --- (2)

Then just simultaneously solve

Then the next part just use the answer to the first question, factorise the equation and you should get answers

Hope all that helped

 

[LoneShadow]

1. Expand and evaluate coefficients of eaxh power of x. I got a=-2 and b=2. So x^4-4x^2+8x-4=(x^2-2x+2)(x^2+2x-2)=(x-(1+i))(x-(1+i))(x-(1-sqrt(3))(x-(1+sqrt(3))=0.

2. If 3z+2 is a factor of 3z^3-z^2+(a+1)z+a, the the long division of 3z^3-z^2+(a+1)z+a by 3z+2 should have a remainder of zero. After long division there will be a term with a's in it. let it equal to zero and you should get a=6. So 3z^3-z^2+7z+6 = (3z+2)(z^2-z+3) = (3z+2)(z-0.5(1-i*sqrt(11)))(z-0.5(1+i*sqrt(11))) = 0.


3. Do the same as 2. Here you'll get 2 equations in a and b. Solve simultaneously to get a and b.

4. 3x^3+4x^2-x+m = (x+a)(x+a)(3x+b). Expand and evaluate vaules of a and b. Then sub to get m. [or do the shorter method of pLuvia ]

Edit: I've been beaten to answer.

 

[nimrod_dookie]

Thankyou so much. That has helped me immensly. :wave: You have saved my life. I was ready to go and get a razorblade.

 

[Rax]

Thankyou so much. That has helped me immensly. :wave: You have saved my life. I was ready to go and get a razorblade.

Ouch, I hope that was a joke. Polynomials can get a bit tricky, but still nothin on that damn Algebra bash of doom that is Parametric Form Conics.

Good that you aint suiciding

GG

 

[housemouse]

Yeah I hate polynomials. Too much theorems and stuff about roots and zeroes. I didnt understand a thing in tutor.

 

[pLuvia]

Yeah I hate polynomials. Too much theorems and stuff about roots and zeroes. I didnt understand a thing in tutor.

Theorems? Not many

 

[Riviet]

There are only about 3 MX2 polynomial theorems that we need know, other than factor/remainder theorems which are covered in 3 unit as well.

 

[Rax]

And which theorems would these be?.......

I don't remember to much of polynomial theorem's at the moment

*Rushes to old MXII Book*

 

[Riviet]

The three that I can think of from the top of my head are the complex conjugate root theorem, integer root theorem, and the multiple root theorem. I've been told by my teacher that we need to know how to derive these.

 

[housemouse]

I dont even know when to apply the theoerms

 

[Raginsheep]

Whats the multiple root theorem again?