Relationship between roots and coefficients

In summary: For (ii), from what you've said, it sounds like you're trying to find the roots of a fourth degree polynomial. This is impossible to do in general- a fourth degree polynomial has 4 roots, which are alpha and beta double roots.
  • #1
aanandpatel
16
0

Homework Statement



SCAN0341.jpg


Homework Equations



Sum of roots taken one at a time is -b/a
Sum of roots taken two at a time is c/a
three at a time is -d/a
four at a time is e/a

The Attempt at a Solution


I did part one by solving the two equations simultaneously.
For part two, I said that it has those roots because that is where the two curves touch
I'm stuck on part three - tried to solve it by applying the above equations and eliminating [itex]\gamma[/itex] and [itex]\delta[/itex] since they are equal to [itex]\alpha[/itex] and [itex]\beta[/itex] respectively but this did not work.
Help would be greatly appreciated :)
 
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  • #2
aanandpatel said:

Homework Statement



Homework Equations



Sum of roots taken one at a time is -b/a
Sum of roots taken two at a time is c/a
three at a time is -d/a
four at a time is e/a

The Attempt at a Solution


I did part one by solving the two equations simultaneously.
For part two, I said that it has those roots because that is where the two curves touch
I'm stuck on part three - tried to solve it by applying the above equations and eliminating [itex]\gamma[/itex] and [itex]\delta[/itex] since they are equal to [itex]\alpha[/itex] and [itex]\beta[/itex] respectively but this did not work.
Help would be greatly appreciated :)

For (i). Solving the equations simultaneously only means that the points satisfying that equation are on both the circle and one hyperbola. It doesn't mean that such points occur where the curves are tangent to each other.

Since this is in the pre-calculus section, I ask, do you know how to show that the points of intersection are points of tangency ?
 
  • #3
The question says that the curves touch at the points A and B so I assumed they were tangential to each other at those points. Not sure how I would prove it otherwise seeing as I only have an x value for the points.
 
  • #4
It's possible for these curves to intersect in as many as 4 points. The fact that they intersect (touch) at only two points is a hint to answering question ii .

How many real roots can a degree 4 polynomial have in general ?
 
  • #5
a fourth degree polynomial has 4 roots therefore alpha and beta are double roots?
 

FAQ: Relationship between roots and coefficients

What is the relationship between roots and coefficients?

The relationship between roots and coefficients is that roots are the values that make the polynomial equation equal to zero, while coefficients are the numbers that multiply each term in the polynomial. The roots and coefficients work together to determine the shape and behavior of the polynomial graph.

How are roots and coefficients connected in a polynomial equation?

In a polynomial equation, the roots and coefficients are connected through the fundamental theorem of algebra. This theorem states that the number of roots in a polynomial equation is equal to the degree of the polynomial, and the roots can be found by factoring the equation and setting each factor equal to zero.

Can the roots of a polynomial equation be negative?

Yes, the roots of a polynomial equation can be negative. In fact, a polynomial equation can have both positive and negative roots. The sign of the roots depends on the coefficients of the polynomial and their relationship to each other.

How do the roots and coefficients affect the graph of a polynomial equation?

The roots and coefficients play a crucial role in determining the shape and behavior of a polynomial graph. The roots are the points where the graph crosses the x-axis, and the coefficients determine the steepness and direction of the graph at different points. For example, a positive leading coefficient will result in an upward opening graph, while a negative leading coefficient will result in a downward opening graph.

Is there a way to find the roots of a polynomial equation without factoring?

Yes, there are other methods to find the roots of a polynomial equation, such as using the quadratic formula or using synthetic division. However, factoring is often the most efficient and reliable method to find the roots of a polynomial equation.

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