My Math Final Is Wednesday And I Have No Idea How To Do "Rational Root Therom" HELP PLEASE!!!!!!!!!!!!!!!!!!!

Question

Please Trying Explaining how to do it for this equation plz.

2x³-9x²-11x+8=0

Answer 1

Quoting from Wikipedia:

The Rational Root Theorem states a constraint on the roots to the polynomial equation

An*xⁿ + An-1* xⁿˉ¹ + ... + A1 *x + A0 = 0

with integer coefficients.

Let An be nonzero. Then each rational solution x can be written in the form x = p/q for p and q satisfying two properties:

p is an integer factor of the constant term A0, and
q is an integer factor of the leading coefficient An.

Thus, a list of possible rational roots of the equation can be derived using the formulae x = ± p/q.

So in the case of 2x³-9x²-11x+8=0, these are integer cooefficients and all non-zero so the "Rational Root Theorem" applies

An = 2 and A0 = 8

So

p is a member of {1,2,4,8} - the integer factors of A0=8; and
and q is a member of {1,2} - the integer factors of An=2

So the possible solutions are:

±1, ±1/2, ±2, ±4, ±8

As this is not a large set, you can just test them

+1 2 - 9 - 11 +8 = -10
-1 -2 - 9 +11 +8 = +8
+1/2, 2/8 - 9/4 - 11/2 + 8 = (1-9-22)/4 + 8 = 1/2
-1/2 -2/8 - 9/4 +11/2 + 8 = (-1-9+22)/4 + 8 = +11
+2 16 - 36 - 22 + 8 = -34
-2 -16 - 36 +22 + 8 = -22
+4 128-144-44 +8 = -52
-4 -128-144+44+8 = -220
+8 1024-576-88+8 = 368
-8 -1024-576+88+8 = -1504

So none of these are solutions and this means that the equation NO rational solution

Actually - and I will not put the working here as it has nothing to do with the Rational Root Theorem, the roots are:

(x+1.41041)(x-0.52680)(x-5.38361)

Now I am of course assuming you wrote the formula down right, because the following has at least one integer solution:

2x³-9x²+11x+8=0

=> (2x+1)(2x-5+i√7)(2x-5-i√7) = 0

Answer 2

If the polynomial has a rational root, the denominator will be a factor of the leading coefficient (the 2), and the numerator will be a factor of the constant (the 8). So possible rational roots are ±8/1, ±4/1, ±2/1, ±1/1, ±8/2, ± 4/2, ±2/2, and ±1/2. In this case some of those are duplicates. But also in this case, none of those work, so all your roots are irrational, but the graph shows 3 real ones.

Answer 3

The rational root theorem states that all possible rational roots of an equation in the form of an^3 + bn^2 + cn + d are any p/q, where q is all factors of a and p is all factors of d. These are only possible rational roots, and are not necessarily actual ones.

In this case, q can be +/- 1 or 2, and p can be +/- 1, 2, 4, or 8. Thus, p/q can be +/- 1/2, 1, 2, 4, or 8. When you find a root that divides in evenly, you can use long division or synthetic division to factor it out. For this equation, however, it can't be factored into rational factors. Using a graphing calculator, the solutions are (rounded to 4 places) 0.5268, -1.4104, and 5.3836.

Answer 4

The rational root theorem states that, for a cubic

f(x) = ax^3 + bx^2 + cx + d

The rational roots can be found by finding the factors of d
(let's call them d1, d2, d3, ...) divided by the factors of a(call them a1, a2, a3). Then, all of your possible rational roots would be:

d1/a1, d1/a2, d1/a3, ...
d2/a1, d2/a2, d2/a3, ....

And so forth

In our case, we have

2x^3 - 9x^2 - 11x + 8 = 0

The factors of 8 are (plus or minus) 1, 2, 4, 8
The factors of 2 are (plus or minus) 1, 2

Therefore, all of our possible rational roots are:
+/- 1,+/- 1/2,+/- 2,+/- 4

What we have to do is test each of those values one at time.
For f(x) = 2x^3 - 9x^2 - 11x + 8

Test f(-1) for 0.

f(-1) = 2(-1) - 9 + 11 + 8 = -2 - 9 + 11 + 8 = 8, which is non-zero.

f(1) = 2 - 9 - 11 + 8 = -18 + 8 = -10, which is non-zero.

f(-1/2) = 2[-1/2]^3 - 9[-1/2]^2 - 11[-1/2] + 8
= 2[-1/8] - 9[1/4] + 11/2 + 8 = -1/4 - 9/4 + 22/4 + 32/4
= [non-zero]

f(1/2) = 2(1/8) - 9(1/4) - 11(1/2) + 8 = 1/4 - 9/4 - 11/2 + 32/4 =
[non-zero]

Keeping trying those list of factors until you get one that equates to 0. Then, as soon as you find out r is a root, it follows that
(x - r) is a factor, and you do synthetic long division to find out the rest of the solutions.

Answer 5

rational root theorem: take the positve and negatives of thefactors of the constant / factors of leading corfficient.
so constant here is 8 and leading coefficient is 2 . factor of 8 is 2,4,1,8 and taking their positive and negative are +2, +4, +1,+8 and -2,-4,-1,-8. divide each of the eight numbers by the factors of the leading coefficient, 2. so 2,4,1,8,-2,-4,-1,-8,1/2,-1/2 are all possible factors of the problems. unless my calculations are wrong, this problems can't be factored evenly.

Answer 6

CAN U TELL ME MORE ABOUT YOUR CLASS AND SYLLABES . I HAVE ANSWER BUT IT IS OF ENGG. LEVEL