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Answer :
To find all the zeros of the polynomial [tex]\( f(x) = x^5 - 7x^4 + 20x^3 - 68x^2 + 99x - 45 \)[/tex], we use the Linear Factors Theorem. This polynomial is of degree 5, which means it has 5 roots counting multiplicities.
The zeros of this polynomial are:
1. [tex]\( x = 1 \)[/tex]
2. [tex]\( x = 5 \)[/tex]
3. [tex]\( x = -3i \)[/tex]
4. [tex]\( x = 3i \)[/tex]
These solutions include both real and complex numbers. The complex roots [tex]\( -3i \)[/tex] and [tex]\( 3i \)[/tex] are conjugates, which often happens when finding roots of polynomials with real coefficients. The Linear Factors Theorem tells us that these roots correspond to factors of the polynomial, so we could express [tex]\( f(x) \)[/tex] as a product of factors corresponding to these zeros.
Here’s a breakdown of each root:
- Real roots: [tex]\( x = 1 \)[/tex] and [tex]\( x = 5 \)[/tex]
- Complex roots: [tex]\( x = -3i \)[/tex] and [tex]\( x = 3i \)[/tex]
For comprehensive understanding, remember that when dealing with polynomials, especially of higher degree, complex solutions often appear in conjugate pairs. This means if you have a root [tex]\( a + bi \)[/tex] (where [tex]\( i \)[/tex] is the imaginary unit), you'll generally also have the root [tex]\( a - bi \)[/tex].
This should give you a complete set of zeros for the given polynomial.
The zeros of this polynomial are:
1. [tex]\( x = 1 \)[/tex]
2. [tex]\( x = 5 \)[/tex]
3. [tex]\( x = -3i \)[/tex]
4. [tex]\( x = 3i \)[/tex]
These solutions include both real and complex numbers. The complex roots [tex]\( -3i \)[/tex] and [tex]\( 3i \)[/tex] are conjugates, which often happens when finding roots of polynomials with real coefficients. The Linear Factors Theorem tells us that these roots correspond to factors of the polynomial, so we could express [tex]\( f(x) \)[/tex] as a product of factors corresponding to these zeros.
Here’s a breakdown of each root:
- Real roots: [tex]\( x = 1 \)[/tex] and [tex]\( x = 5 \)[/tex]
- Complex roots: [tex]\( x = -3i \)[/tex] and [tex]\( x = 3i \)[/tex]
For comprehensive understanding, remember that when dealing with polynomials, especially of higher degree, complex solutions often appear in conjugate pairs. This means if you have a root [tex]\( a + bi \)[/tex] (where [tex]\( i \)[/tex] is the imaginary unit), you'll generally also have the root [tex]\( a - bi \)[/tex].
This should give you a complete set of zeros for the given polynomial.
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