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To determine the total number of roots of the polynomial function [tex]\( f(x) = 3x^6 + 2x^5 + x^4 - 2x^3 \)[/tex], we need to analyze and solve the polynomial equation [tex]\( 3x^6 + 2x^5 + x^4 - 2x^3 = 0 \)[/tex].
Here’s a detailed, step-by-step solution:
1. Identify and factor out the common term:
Notice that each term in the polynomial [tex]\( 3x^6 + 2x^5 + x^4 - 2x^3 \)[/tex] contains [tex]\( x^3 \)[/tex]. So, we can factor [tex]\( x^3 \)[/tex] out of the polynomial:
[tex]\[ f(x) = x^3 (3x^3 + 2x^2 + x - 2) \][/tex]
2. Solve for the simpler polynomial:
The polynomial is now split into [tex]\( x^3 = 0 \)[/tex] and [tex]\( 3x^3 + 2x^2 + x - 2 = 0 \)[/tex].
- For [tex]\( x^3 = 0 \)[/tex]:
[tex]\[ x = 0 \][/tex]
This gives us one root with a multiplicity of 3 because [tex]\( x^3 = 0 \)[/tex] represents the same root three times.
3. Solve for the more complicated polynomial:
Next, we solve the cubic polynomial [tex]\( 3x^3 + 2x^2 + x - 2 = 0 \)[/tex]. The exact roots for cubic polynomials can be obtained by applying methods like the Rational Root Theorem, synthetic division, or using more sophisticated algebraic techniques.
4. Roots calculation:
After solving the cubic polynomial [tex]\( 3x^3 + 2x^2 + x - 2 = 0 \)[/tex], we obtain three distinct (possibly complex) roots:
[tex]\[ \text{Roots are: } -\frac{2}{9} + (-\frac{1}{2} - \frac{\sqrt{3}i}{2}) \left(\frac{262}{729} + \frac{\sqrt{849}}{81}\right)^{1/3} - \frac{5}{81(-\frac{1}{2} - \frac{\sqrt{3}i}{2}) \left(\frac{262}{729} + \frac{\sqrt{849}}{81}\right)^{1/3}}, \\ -\frac{2}{9} - \frac{5}{81(-\frac{1}{2} + \frac{\sqrt{3}i}{2}) \left(\frac{262}{729} + \frac{\sqrt{849}}{81}\right)^{1/3}} + (-\frac{1}{2} + \frac{\sqrt{3}i}{2}) \left(\frac{262}{729} + \frac{\sqrt{849}}{81}\right)^{1/3}, \\ -\frac{2}{9} - \frac{5}{81\left(\frac{262}{729} + \frac{\sqrt{849}}{81}\right)^{1/3}} + \left(\frac{262}{729} + \frac{\sqrt{849}}{81}\right)^{1/3} \][/tex]
5. Total number of roots:
Combining the multiplicity of the real root [tex]\( x = 0 \)[/tex] and the three distinct roots from the cubic polynomial, we get a total of 4 roots.
Therefore, the total number of roots of the polynomial function [tex]\( f(x) = 3x^6 + 2x^5 + x^4 - 2x^3 \)[/tex] is [tex]\( \boxed{4} \)[/tex].
Here’s a detailed, step-by-step solution:
1. Identify and factor out the common term:
Notice that each term in the polynomial [tex]\( 3x^6 + 2x^5 + x^4 - 2x^3 \)[/tex] contains [tex]\( x^3 \)[/tex]. So, we can factor [tex]\( x^3 \)[/tex] out of the polynomial:
[tex]\[ f(x) = x^3 (3x^3 + 2x^2 + x - 2) \][/tex]
2. Solve for the simpler polynomial:
The polynomial is now split into [tex]\( x^3 = 0 \)[/tex] and [tex]\( 3x^3 + 2x^2 + x - 2 = 0 \)[/tex].
- For [tex]\( x^3 = 0 \)[/tex]:
[tex]\[ x = 0 \][/tex]
This gives us one root with a multiplicity of 3 because [tex]\( x^3 = 0 \)[/tex] represents the same root three times.
3. Solve for the more complicated polynomial:
Next, we solve the cubic polynomial [tex]\( 3x^3 + 2x^2 + x - 2 = 0 \)[/tex]. The exact roots for cubic polynomials can be obtained by applying methods like the Rational Root Theorem, synthetic division, or using more sophisticated algebraic techniques.
4. Roots calculation:
After solving the cubic polynomial [tex]\( 3x^3 + 2x^2 + x - 2 = 0 \)[/tex], we obtain three distinct (possibly complex) roots:
[tex]\[ \text{Roots are: } -\frac{2}{9} + (-\frac{1}{2} - \frac{\sqrt{3}i}{2}) \left(\frac{262}{729} + \frac{\sqrt{849}}{81}\right)^{1/3} - \frac{5}{81(-\frac{1}{2} - \frac{\sqrt{3}i}{2}) \left(\frac{262}{729} + \frac{\sqrt{849}}{81}\right)^{1/3}}, \\ -\frac{2}{9} - \frac{5}{81(-\frac{1}{2} + \frac{\sqrt{3}i}{2}) \left(\frac{262}{729} + \frac{\sqrt{849}}{81}\right)^{1/3}} + (-\frac{1}{2} + \frac{\sqrt{3}i}{2}) \left(\frac{262}{729} + \frac{\sqrt{849}}{81}\right)^{1/3}, \\ -\frac{2}{9} - \frac{5}{81\left(\frac{262}{729} + \frac{\sqrt{849}}{81}\right)^{1/3}} + \left(\frac{262}{729} + \frac{\sqrt{849}}{81}\right)^{1/3} \][/tex]
5. Total number of roots:
Combining the multiplicity of the real root [tex]\( x = 0 \)[/tex] and the three distinct roots from the cubic polynomial, we get a total of 4 roots.
Therefore, the total number of roots of the polynomial function [tex]\( f(x) = 3x^6 + 2x^5 + x^4 - 2x^3 \)[/tex] is [tex]\( \boxed{4} \)[/tex].
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