How Are the Roots of t^2-t-2 Related to the Solutions of y''-y'-2y=0?

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In summary, this conversation discusses verifying solutions to a given differential equation and finding connections between the roots of a polynomial and the solutions of the differential equation. The concept of "roots" is also clarified in this context.
  • #1
gigi9
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Someone please help me to do this problem below. Thanks
1)a) Verify that y= e^(-x) and y= e^(2x)are both solutions of the differential equation y"-y'-2y=0
b) Factor the polynomial t^2-t-2. What connection do you see between the roots of this polynomial and the solutions of y"-y'-2y=0 ?---What does "roots" means in this case??..I don't understand.
*** t^2-t-2=(t-2)(t+1)
 
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  • #2
Once again, this really should be in homework (its a territorial matter!) and you really should show what you have tried.

However I will give you a hint: To show that y= e-x and y= e2x are solutions to the differential equation y"-y'-2y=0 means to find y' and y" for each function, put them into the given equation and see what happens!

") Factor the polynomial t^2-t-2. What connection do you see between the roots of this polynomial and the solutions of y"-y'-2y=0 ?---What does "roots" means in this case??..I don't understand."

I started to write "exactly what it means in algebra" but you do have a point (perhaps inadverdant): Strictly speaking to talk about "roots" you must have equation, not just a polynomial. This problem should have said either "the roots of the equation t2- t- 2= 0" or "the zeroes of the polynomial t2-t-2".

Okay, you have factored t2- t- 2= (t+ 1)(t- 2). Now WHAT are the roots of t2- t- 2= (t+ 1)(t- 2)= 0 and what possible connection could that have with e-x and e2x?
 
  • #3


To factor the polynomial t^2-t-2, we can use the standard method of grouping or the quadratic formula. Let's use the grouping method:

Step 1: Write the polynomial in the form t^2+at+b. In this case, a=-1 and b=-2, so we have t^2-t-2.

Step 2: Find two numbers that when multiplied give you the constant term (b) and when added give you the coefficient of the middle term (a). In this case, the numbers are -1 and 2, since (-1)(2)=-2 and -1+2=1.

Step 3: Rewrite the polynomial using these two numbers as coefficients for the middle term. t^2-t-2 can be rewritten as t^2+2t-t-2.

Step 4: Group the first two terms and the last two terms together. We get t(t+2)-1(t+2).

Step 5: Factor out the common term (t+2). We get (t+2)(t-1).

Therefore, the polynomial t^2-t-2 can be factored as (t+2)(t-1).

Now, let's look at the connection between the roots of this polynomial and the solutions of y"-y'-2y=0.

In the polynomial t^2-t-2, the roots are the values of t that make the polynomial equal to zero. In this case, the roots are t=-2 and t=1.

In the differential equation y"-y'-2y=0, the solutions are the values of y that make the equation true. In this case, the solutions are y=e^(-x) and y=e^(2x).

Notice that when we substitute t=-2 in the polynomial, we get (t+2)(t-1)=(-2+2)(-2-1)=0, which means that t=-2 is a root of the polynomial. Similarly, when we substitute t=1 in the polynomial, we get (t+2)(t-1)=(1+2)(1-1)=0, which means that t=1 is also a root of the polynomial.

This shows that the roots of the polynomial t^2-t-2 are directly related to the solutions of the differential equation y"-y'-2y=0. This is because both the polynomial and the differential equation have the
 

Related to How Are the Roots of t^2-t-2 Related to the Solutions of y''-y'-2y=0?

1. What does it mean to "factor" a polynomial?

Factoring a polynomial means finding the expressions or numbers that, when multiplied together, result in the original polynomial. It is essentially the reverse of distributing or multiplying polynomials.

2. How do I factor a polynomial with 3 terms?

To factor a polynomial with 3 terms, first check if there is a common factor among all terms. If there is, factor it out. Then, use the FOIL (First, Outer, Inner, Last) method or the grouping method to find the remaining factors. Finally, check your answer by distributing the factors and seeing if it results in the original polynomial.

3. What is the difference between factoring and simplifying a polynomial?

Factoring a polynomial involves breaking it down into smaller expressions or numbers, while simplifying a polynomial involves combining like terms and reducing the expression to its simplest form. Factoring can also be seen as the first step towards simplifying a polynomial.

4. Can I factor a polynomial with only two terms?

No, a polynomial must have at least three terms to be factored. If a polynomial has only two terms, it can be simplified, but not factored.

5. How do I know if I have factored a polynomial correctly?

You can check if you have factored a polynomial correctly by distributing the factors and seeing if it results in the original polynomial. You can also use the reverse of the FOIL method or the reverse of the grouping method to check your answer.

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