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Answer :
To determine the possible number of imaginary zeros of the polynomial [tex]\( f(x) = x^5 - 5x^4 - 9x^3 + 45x^2 + 18x - 90 \)[/tex], we need to follow these steps:
1. Identify the Degree of the Polynomial:
The degree of the polynomial is determined by the highest power of [tex]\( x \)[/tex] in the expression. In this case, the highest power is 5, so the degree of the polynomial is 5.
2. Understand the Concept of Imaginary Zeros:
Imaginary zeros refer to complex roots that are not real numbers. For polynomials with real coefficients, these complex roots occur in conjugate pairs. This means if [tex]\( a + bi \)[/tex] is a root (where [tex]\( b \neq 0 \)[/tex]), then [tex]\( a - bi \)[/tex] must also be a root.
3. Determine Possible Numbers of Imaginary Zeros:
Since imaginary zeros must occur in conjugate pairs, the number of these zeros must be even. Given the polynomial with a degree of 5, the highest number of zeros (real and imaginary combined) is 5. This leads to the following possibilities:
- All zeros could be real, therefore 0 imaginary zeros.
- 2 zeros could be imaginary, the rest being real.
- 4 zeros could be imaginary, and 1 zero must be real since the total number of zeros is 5.
From these observations, the possible number of imaginary zeros for the polynomial are:
- 0
- 2
- 4
These are the valid possibilities for the number of imaginary zeros in this polynomial.
1. Identify the Degree of the Polynomial:
The degree of the polynomial is determined by the highest power of [tex]\( x \)[/tex] in the expression. In this case, the highest power is 5, so the degree of the polynomial is 5.
2. Understand the Concept of Imaginary Zeros:
Imaginary zeros refer to complex roots that are not real numbers. For polynomials with real coefficients, these complex roots occur in conjugate pairs. This means if [tex]\( a + bi \)[/tex] is a root (where [tex]\( b \neq 0 \)[/tex]), then [tex]\( a - bi \)[/tex] must also be a root.
3. Determine Possible Numbers of Imaginary Zeros:
Since imaginary zeros must occur in conjugate pairs, the number of these zeros must be even. Given the polynomial with a degree of 5, the highest number of zeros (real and imaginary combined) is 5. This leads to the following possibilities:
- All zeros could be real, therefore 0 imaginary zeros.
- 2 zeros could be imaginary, the rest being real.
- 4 zeros could be imaginary, and 1 zero must be real since the total number of zeros is 5.
From these observations, the possible number of imaginary zeros for the polynomial are:
- 0
- 2
- 4
These are the valid possibilities for the number of imaginary zeros in this polynomial.
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