Maths Ext 1- Polynomials 1 Prelim 2014 (1 Viewer)

[jkerr138]

Hey guys and girls,

Currently doing Maths Ext 1 Prelim course.
Have an exam this week, and have a feeling on the polynomials, I will be assessed on the quartic equation.
Here is the question, I am having an issue on finding alpha beta gamma in part c of the question, as I don't believe there is a value for -d/a?


If alpha and beta are the roots of x^2-5x-5=0, find;
a) 1/alpha + 1/beta
b) alpha^2 + beta^2
c) alpha^3 + beta^3

Also, could I please have someone show how to work out alpha^2 + beta^2 and alpha^3 + beta^3 rather than just stating soln.
Thanks in advance.
Josh

 

[dunjaaa]

a=alpha, b=beta
(b) (a+b)^2=a^2+b^2+2ab
Therefore, a^2+b^2=(a+b)^2-2ab
= 25 + 10 = 35
(c) Multiply both sides by x, you are allowed to because the additional root you've created is = 0 -> x^3-5x^2-5x=0 (Note a and b are still roots to this new polynomial)
Since a and b are the roots to this polynomial, substituting a and b in separately we obtain;
a^3-5a^2-5a=0 and b^3-5b^2-5b=0
Rearranging each one, a^3=5a^2+5a, b^3=5b^2+5b
Therefore, a^3+b^3=5(a^2+b^2)+5(a+b)
= 175 + 25 = 200
When you get good at this you can literally go straight to this step -> a^3+b^3=5(a^2+b^2)+5(a+b)

 

[jkerr138]

I solved b) and c) and they weren't the same as yours

Let Alpha= A and B= Beta
B) a^2+b^2 = (a+b)^2 - 2ab
= 5^2 - 2 * -5
= 25+10
= 35

c) (A + B)^3 = (A+B)^3 - 3ab (a+b)
= 5^3 - 3 * -5 * 5
= 125+ 75
=200

I also checked the back of the book to verify the soln answer's and they were correct.

 

[dunjaaa]

i got the same, made a typo in the initial post

 

[jkerr138]

Yeah when you edited it. Well you tried anyway thanks.

 

[Kurosaki]

Hey guys and girls,

Currently doing Maths Ext 1 Prelim course.
Have an exam this week, and have a feeling on the polynomials, I will be assessed on the quartic equation.
Here is the question, I am having an issue on finding alpha beta gamma in part c of the question, as I don't believe there is a value for -d/a?


If alpha and beta are the roots of x^2-5x-5=0, find;
a) 1/alpha + 1/beta
b) alpha^2 + beta^2
c) alpha^3 + beta^3

Also, could I please have someone show how to work out alpha^2 + beta^2 and alpha^3 + beta^3 rather than just stating soln.
Thanks in advance.
Josh

Another way to do part C is using:

 

[jkerr138]

Another way to do part C is using:

 

[dunjaaa]

Yes, I'm just so use to 4U polynomials lol

 

[Kurosaki]

Another way to do part C is using: , and you can quickly compute by product of roots.

 

[dunjaaa]

Just a tip, when you have polynomials with degree 3 or 4, my technique will be of great use especially for finding say a^3+b^3+c^3 or a^4+b^4+c^4 etc