Hey guys, how do you solve these Polynomials? (1 Viewer)

[jamesfirst]

I didn't focus on Polynomials last year so I'm struggling. I can do all the easier ones like Factor, remainder theorems, long devisions, etc etc.


1.Show that 3x^4 + 7x^3 - 3x^2 + x = 0 has a zero root
2.What property is common to all equations of the for x^3 + mx + k = 0
3.Solve the equation x^3 - 12x^2 + 39x - 28 = 0 if the roots are in arithmetic prog.
4.Solve the equation x^3 - 14x^2 + 56x - 64 = 0 if the roots are in geometric prog.
5.Given that the equation x^3 - 7x^2 +16x +k = 0 has two equal and integral roots find the value of k.


I did these questions before, like last term... but I forgot how to do most of it.

Can you guys give me some hints also as to how to solve these questions? Like what I should be aware of etc etc. Seems like you have to know heaps about roots...

 

[Deep Blue]

1. Obviously you can take x out as a factor and hence x=0 is a root of the equation.
2. They all have a point of inflexion but no other turning points. ie, any polynomial of degree 3 without an x^2 term has only a point of inflexion.

 

[slyhunter]

^

3. Let the roots be . Use sum and product of roots to solve.

4. Same as above but let roots be

5. Let roots be and solve simultaneously to get k.

 

[hscishard]

^ continuing the chain
5. You can't use what slyhunter said I think. You're given that the equation has a root of multiplicity of 2, differentiate and the stationary point where the x value is an integer will be the double root. Then use sum of the thing.

 

[xV1P3R]

2. They all have a point of inflexion but no other turning points. ie, any polynomial of degree 3 without an x^2 term has only a point of inflexion.

Differentiating his given equation:

y = x³ + mx + k
y' = 3x² + m

You would have turning points for m<0...

 

[Deep Blue]

Differentiating his given equation:

y = x³ + mx + k
y' = 3x² + m

You would have turning points for m<0...

Ha, you would too. Thanks for picking me up on that. What's the answer to the question then?

 

[jamesfirst]

For question 1. Do you just factorise the x and say x=0 hence it has zero roots??? Is that 'enough' working out?


I kinda understand the arit and geo prog. ones. but still don't get 2 and 5....

 

[xV1P3R]

I'm thinking 2) is just one point of inflexion and at least one root.

 

[gurmies]

I'm thinking 2) is just one point of inflexion and at least one root.

Real root.

 

[xV1P3R]

Real root.

Thanks