The function f(x) = 3x^4 + 2x^3 - 4x^2 + 5x - 6 has at least two rational roots. Use the rational root theorem to find those roots, then proceed to find all complex roots
The rational root theorem states that if a polynomial has a rational root p/q, where p and q are integers with no common factors other than 1, then p must divide the constant term 6 and q must divide the leading coefficient 3.
The constant term 6 = 2 * 3 can be factored into two positive numbers, which are the possible values for p: ±1, ±2, ±3, ±6. The leading coefficient 3 has only one positive factor, which is the possible value for q: ±1, ±3.
Now, we can test these possible rational roots by substituting them into the polynomial and checking if f(p/q) = 0.
When we substitute p = ±1, q = ±1, we get f(±1/±1) = 0, so x = ±1 is a rational root.
When we substitute p = ±2, q = ±1, we get f(±2/±1) = 0, so x = ±2 is also a rational root.
When we substitute p = ±3, q = ±1, we get f(±3/±1) = 0, so x = ±3 is a rational root as well.
Finally, when we substitute p = ±6, q = ±1, we get f(±6/±1) = 0, so x = ±6 is a rational root.
Hence, the rational roots are x = ±1, ±2, ±3, ±6.
To find the complex roots, we can use polynomial long division or synthetic division to divide f(x) by (x - r), where r is each of the rational roots we found.
Using synthetic division:
1 | 3 2 -4 5 -6
3 5 1 -1
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3 5 1 6 -7
So, (x - 1) is a factor of f(x).
Doing the same process with the remaining rational roots:
(x - 2) is also a factor:
2 | 3 5 1 6 -7
6 22 46 104
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3 11 23 52 97
(x + 2) is also a factor:
-2 | 3 11 23 52 97
-6 -10 -26 -52
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3 5 13 26 45
(x + 3) is also a factor:
-3 | 3 5 13 26 45
-9 -12 -3 0
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3 -4 1 23 45
(x - 3) is also a factor:
3 | 3 -4 1 23 45
9 15 48 213
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3 5 16 71 258
(x - 6) is also a factor:
6 | 3 5 16 71 258
18 138 858
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3 23 154 929 1116
Finally, the quotient we obtained is 3x^3 + 23x^2 + 154x + 929.
Since this new polynomial is still a cubic equation, we can find its complex roots using methods like factoring, the quadratic formula, or synthetic division. However, it is not guaranteed that all roots will be rational or even real.
To simplify the process, we can use the Rational Root Theorem again to potentially identify any rational roots of the cubic equation. In this case, we can observe that p must divide 929 and q must divide 3. However, after examining the factors of 929, the equation does not have any rational roots.
Therefore, the complex roots of the original function f(x) = 3x^4 + 2x^3 - 4x^2 + 5x - 6 are x = ±1, ±2, ±3, ±6, plus any additional complex roots obtained from the cubic equation.