Can You Use the Factor Theorem to Solve for Coefficients Using Known Roots?

In summary, the roots theorem is a mathematical principle that states that the roots of a polynomial equation can be found by setting the equation equal to zero and solving for the variable. It is closely related to the factor theorem, which states that if a polynomial equation has a root, then the corresponding linear factor is a factor of the polynomial. The roots and factor theorem have many practical applications in fields such as engineering, economics, and physics. To solve equations using these theorems, one must first identify the polynomial equation and its factors, and then use the roots theorem to find the possible solutions. These theorems can be applied to all polynomial equations with real coefficients, but for higher degree polynomials, more advanced techniques may be required.
  • #1
safat
14
0

Homework Statement


knowing a,b and c are roots 3x^3-x^2-10x+8=0
show that:
1) 1/a+1/b+1/c=5/4
2)a^2+b^2+c^2=61/9

Homework Equations


factor theorem --> (x-a)(x-b)(x-c)

The Attempt at a Solution


can only use factor theorem:
therefore (x-a)(x-b)(x-c)--> up to: x^3-x^2(a+b+c)+x(ac+bc+ab)-abc
no idea where to go now!
 
Last edited:
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  • #2
welcome to pf!

hi safat! welcome to pf! :smile:

(try using the X2 button just above the Reply box :wink:)

(1) = what / what ?

(2) = (a+b+c)2 minus … ? :wink:
 
  • #3
thanks tiny-tim.

that's what I got by confrontating the original equation (for the record:3(x3-1/3x2-10/3x+8/3)=0 )
and x3-x2(a+b+c)+x(ac+bc+ab)-abc=0

1/a+1/b+1/c=(ac+bc+ab)/abc=(-10/3)/(-8/3)=5/4

About part two:
should I use the quadratic formula or can I go by guessing?
I got (a+b+c)2-2(ac+bc+ab)=(-1/3)2-2(-10/3)=61/9
 
  • #4
hi safat! :smile:

(btw, no such word as "confrontate" … i think you mean "compare"! :wink:)
safat said:
thanks tiny-tim.

that's what I got by confrontating the original equation (for the record:3(x3-1/3x2-10/3x+8/3)=0 )
and x3-x2(a+b+c)+x(ac+bc+ab)-abc=0

1/a+1/b+1/c=(ac+bc+ab)/abc=(-10/3)/(-8/3)=5/4

yup! :biggrin:
About part two:
should I use the quadratic formula or can I go by guessing?
I got (a+b+c)2-2(ac+bc+ab)=(-1/3)2-2(-10/3)=61/9

(i'm not sure what you mean by the quadratic formula, but anyway …)

yes, guessing is fine, so long as you check that your guess works! :smile:
 
  • #5
thanks again tiny-tim!
you also provide a really efficient proof reading service! :cool:

About the second part..
tbh, I have done it by guessing. But it looks like the great majority of my peers used the quadratic formula (x=-b+-(√b2-4ac)/2a)
I was wondering how it could be done that way.
 
  • #6
safat said:
… tbh, I have done it by guessing. But it looks like the great majority of my peers used the quadratic formula (x=-b+-(√b2-4ac)/2a)
I was wondering how it could be done that way.

beats me! :rolleyes:

your way is better :smile:​
 
  • #7
safat said:
About the second part..
tbh, I have done it by guessing. But it looks like the great majority of my peers used the quadratic formula (x=-b+-(√b2-4ac)/2a)
I was wondering how it could be done that way.
It's not really guessing. You have made use of some simple relationships between the roots and the coefficients (which you derived and which can be fairly easily memorized). The object of this type of question is generally to solve it without explicitly finding the roots.

Of course if it happens that you can explicitly find all the roots then that is a viable option. In this particular example the roots turned out to be easy to find, but what if the question was slightly different, say [itex]3 x^3 - x^2 -10x +7 = 0[/itex]. Your method would still work just as easily, but your class mates would presumably find the going much tougher.
 

FAQ: Can You Use the Factor Theorem to Solve for Coefficients Using Known Roots?

What is the roots theorem?

The roots theorem, also known as the zero or root finding theorem, is a mathematical principle that states that for any polynomial equation, the roots (or solutions) of the equation can be found by setting the equation equal to zero and solving for the variable.

How is the roots theorem related to the factor theorem?

The roots theorem and the factor theorem are closely related as they both involve finding the roots of a polynomial equation. The factor theorem states that if a polynomial equation has a root, then the corresponding linear factor is a factor of the polynomial. In other words, if a value satisfies the roots theorem, it will also be a factor of the polynomial.

What is the significance of the roots and factor theorem?

The roots and factor theorem are important concepts in mathematics and have many practical applications. They are used in solving polynomial equations, graphing polynomial functions, and understanding the behavior of polynomial expressions. They also have applications in fields such as engineering, economics, and physics.

How can the roots and factor theorem be used to solve equations?

To solve equations using the roots and factor theorem, you first need to identify the polynomial equation and its factors. Then, you can use the roots theorem to find the possible solutions or roots of the equation. Once you have identified the roots, you can use them to factor the polynomial and solve the equation.

Can the roots and factor theorem be applied to all polynomial equations?

Yes, the roots and factor theorem can be applied to all polynomial equations as long as the equation is in standard form and has real coefficients. However, for higher degree polynomials, finding the roots and factors can be more complex and may require the use of advanced algebraic techniques.

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