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Answer :
To solve the polynomial equation [tex]\( f(x) = x^5 - 9x^4 + 29x^3 - 45x^2 + 54x - 54 \)[/tex], we need to find the roots of the polynomial. Here’s a step-by-step explanation:
1. Understand the Polynomial: The polynomial is a quintic (degree 5), so it can have up to five roots. These roots can be real or complex numbers.
2. Finding Real Roots: In this polynomial, we have found that one of the real roots is [tex]\( x = 3 \)[/tex].
3. Checking for Complex Roots: The polynomial can also have complex roots, which usually come in conjugate pairs if the coefficients are real numbers. This polynomial has complex roots, which are [tex]\( -\sqrt{2}i \)[/tex] and [tex]\( \sqrt{2}i \)[/tex].
4. Summary of Roots:
- Real Root: [tex]\( x = 3 \)[/tex]
- Complex Roots: [tex]\( -\sqrt{2}i, \sqrt{2}i \)[/tex]
These roots fully satisfy the equation given the polynomial, meaning that if you substitute these values back into the original polynomial, it should equate to zero. This concludes our solution to the polynomial equation.
1. Understand the Polynomial: The polynomial is a quintic (degree 5), so it can have up to five roots. These roots can be real or complex numbers.
2. Finding Real Roots: In this polynomial, we have found that one of the real roots is [tex]\( x = 3 \)[/tex].
3. Checking for Complex Roots: The polynomial can also have complex roots, which usually come in conjugate pairs if the coefficients are real numbers. This polynomial has complex roots, which are [tex]\( -\sqrt{2}i \)[/tex] and [tex]\( \sqrt{2}i \)[/tex].
4. Summary of Roots:
- Real Root: [tex]\( x = 3 \)[/tex]
- Complex Roots: [tex]\( -\sqrt{2}i, \sqrt{2}i \)[/tex]
These roots fully satisfy the equation given the polynomial, meaning that if you substitute these values back into the original polynomial, it should equate to zero. This concludes our solution to the polynomial equation.
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