I am so stuck. I need help with this problem:
Between what integers are the real roots of g(x) = 3x^4 + 3x^3 – 2x^2 + x – 1 located?
Answer:
1. Count the number of possible real roots using Descartes’ rule of signs.
Since g(x) = 3x^4 + 3x^3 – 2x^2 + x – 1 changes sign 3 times, g has either 1 or 3 real positive zeros.
Since g(–x) = 3x^4 – 3x^3 – 2x^2 – x – 1 changes sign 1 time, g has exactly 1 negative real zero.
So I get this, but the next one I have no idea where they got the -2, -1, 0, and 1.
2. Find g(x) for integer values of x using substitution or synthetic division.
g(–2) = 3(–2)^4 + 3(–2)^3 – 2(–2)^2 + (–2) – 1 = 13
g(–1) = 3(–1)^4 + 3(–1)^3 – 2(–1)^2 + (–1) – 1 = –4
g(0) = 3(0)^4 + 3(0)^3 – 2(0)^2 + (0) – 1 = –1
g(1) = 3(1)^4 + 3(1)^3 – 2(1)^2 + (1) – 1 = 4
Any ideas?
>where they got the -2, -1, 0, and 1.
Those are examples values of argument. You could as well try plugging -3, -1.5, 0.33333 and 1.4.
Since our function g(x) = 3x^4 + 3x^3 – 2x^2 + x – 1 is of fourth degree and has small coefficients the most sensible values to check are those that are not too big. Of course plugging integers instead of fractions is easier here.
If we were to choose -100, -99, -98, -97 then all the values of the function would be of the same sign which doesn't help us all here.
>>149510
Oh okay thank you for help.
One last thing, what is the purpose of this? As in what do I do with 13, -4, -1, and 4?
>>149558
Pic related is some polynomial. Polynomials are continuous functions.
If for some argument "x1" the value of function is positive but for argument "x2" is negative, then the function must have a root between "x1" and "x2" - the blue X.
In your case for some consecutive arguments you have values that are:
g(-2) - positive
g(-1) - negative
g(0) - negative
g(1) - positive
From what I said earlier we can conclude that there exist roots between -2 and -1 AND between 0 and 1. So we know that our function has at least 2 roots. It may have more.
It's more related to Newton-Raphson method finding roots of functions which is actually used in real life computations.
If you still have problem visualizing it it would be best to plot the function in wolframalpha which I am too lazy to do.
