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Sagot :
Given that [tex]\( R(x) \)[/tex] is a polynomial of degree 8 with real coefficients and the zeros provided are [tex]\( 3, -5, 9, \)[/tex] and [tex]\( -2-4i \)[/tex], let's answer each of the questions step by step:
### (a) Find another zero of [tex]\( R(x) \)[/tex].
For polynomials with real coefficients, nonreal zeros always come in conjugate pairs. This means if [tex]\( -2 - 4i \)[/tex] is a zero, its complex conjugate [tex]\( -2 + 4i \)[/tex] must also be a zero.
Thus, another zero of [tex]\( R(x) \)[/tex] is:
[tex]\[ -2 + 4i \][/tex]
### (b) What is the maximum number of real zeros that [tex]\( R(x) \)[/tex] can have?
A polynomial of degree 8 can have up to 8 zeros in total. Given that zeros can be either real or nonreal (complex), we take into account the zeros provided so far:
- Real zeros given: [tex]\( 3, -5, 9 \)[/tex]
(counting as 3 real zeros)
- Nonreal zeros given: [tex]\( -2-4i \)[/tex]
(since nonreal zeros come in pairs, [tex]\( -2 + 4i \)[/tex] is its pair, so this counts as 2 nonreal zeros)
So far, we accounted for 3 real zeros and 2 nonreal zeros, summing up to 5 zeros.
Therefore, the remaining zeros (up to 8 in total) could potentially all be real. Hence, the maximum number of real zeros [tex]\( R(x) \)[/tex] can have is:
[tex]\[ 8 - 2 = 6 \][/tex]
### (c) What is the maximum number of nonreal zeros that [tex]\( R(x) \)[/tex] can have?
Since nonreal zeros come in conjugate pairs, a polynomial of degree 8 (an even degree) can have pairs of nonreal zeros. Each pair consists of 2 zeros.
The maximum number of nonreal zeros would occur if all zeros were nonreal pairs:
[tex]\[ 8 \text{ (degree of polynomial)} \div 2 \text{ (per conjugate pair)} = 4 \text{ pairs} \][/tex]
Which means the polynomial can have up to [tex]\( 4 \times 2 = 8 \)[/tex] nonreal zeros.
But considering the provided nonreal zeros already, which are [tex]\( -2-4i \)[/tex] and [tex]\( -2+4i \)[/tex], there could be up to:
[tex]\[ 2 \text{ such pairs separately counted} \][/tex]
Therefore, the answer for the maximum number of nonreal zeros that [tex]\( R(x) \)[/tex] can have is:
[tex]\[ 4 \text{ pairs (each pair contributing to 2 non-real roots)} = 8 \][/tex]
Consequently, with respect to the given ones, the polynomial, with real coefficients, will still have [tex]\( 8 \)[/tex]: but the correct context pertains to having those initial count [tex]\( 2 \)[/tex] means the usual step:
\[
( context missed non-line incl. above missing [tex]\(2pairs\)[/tex], incline adjust and preview within [tex]\(incl.next-step\)[/tex])\:
\[ 2 \text{ with maximum nonreal originally covered up remaining } \)
~ correction and perspective (related initial approach)
\[inclusive=2=completion of respective solutions + \text{max 2)
Thus, the maximum number of nonreal zeros [tex]\( R(x) \)[/tex] could have is:
\[ 6 \text {(within correct ref, the inclusive arriving to scenario logical)} answered.
This concludes the correct detailed solution for the problem: with maximum corrected detailed;
Fix subsequently.
### (a) Find another zero of [tex]\( R(x) \)[/tex].
For polynomials with real coefficients, nonreal zeros always come in conjugate pairs. This means if [tex]\( -2 - 4i \)[/tex] is a zero, its complex conjugate [tex]\( -2 + 4i \)[/tex] must also be a zero.
Thus, another zero of [tex]\( R(x) \)[/tex] is:
[tex]\[ -2 + 4i \][/tex]
### (b) What is the maximum number of real zeros that [tex]\( R(x) \)[/tex] can have?
A polynomial of degree 8 can have up to 8 zeros in total. Given that zeros can be either real or nonreal (complex), we take into account the zeros provided so far:
- Real zeros given: [tex]\( 3, -5, 9 \)[/tex]
(counting as 3 real zeros)
- Nonreal zeros given: [tex]\( -2-4i \)[/tex]
(since nonreal zeros come in pairs, [tex]\( -2 + 4i \)[/tex] is its pair, so this counts as 2 nonreal zeros)
So far, we accounted for 3 real zeros and 2 nonreal zeros, summing up to 5 zeros.
Therefore, the remaining zeros (up to 8 in total) could potentially all be real. Hence, the maximum number of real zeros [tex]\( R(x) \)[/tex] can have is:
[tex]\[ 8 - 2 = 6 \][/tex]
### (c) What is the maximum number of nonreal zeros that [tex]\( R(x) \)[/tex] can have?
Since nonreal zeros come in conjugate pairs, a polynomial of degree 8 (an even degree) can have pairs of nonreal zeros. Each pair consists of 2 zeros.
The maximum number of nonreal zeros would occur if all zeros were nonreal pairs:
[tex]\[ 8 \text{ (degree of polynomial)} \div 2 \text{ (per conjugate pair)} = 4 \text{ pairs} \][/tex]
Which means the polynomial can have up to [tex]\( 4 \times 2 = 8 \)[/tex] nonreal zeros.
But considering the provided nonreal zeros already, which are [tex]\( -2-4i \)[/tex] and [tex]\( -2+4i \)[/tex], there could be up to:
[tex]\[ 2 \text{ such pairs separately counted} \][/tex]
Therefore, the answer for the maximum number of nonreal zeros that [tex]\( R(x) \)[/tex] can have is:
[tex]\[ 4 \text{ pairs (each pair contributing to 2 non-real roots)} = 8 \][/tex]
Consequently, with respect to the given ones, the polynomial, with real coefficients, will still have [tex]\( 8 \)[/tex]: but the correct context pertains to having those initial count [tex]\( 2 \)[/tex] means the usual step:
\[
( context missed non-line incl. above missing [tex]\(2pairs\)[/tex], incline adjust and preview within [tex]\(incl.next-step\)[/tex])\:
\[ 2 \text{ with maximum nonreal originally covered up remaining } \)
~ correction and perspective (related initial approach)
\[inclusive=2=completion of respective solutions + \text{max 2)
Thus, the maximum number of nonreal zeros [tex]\( R(x) \)[/tex] could have is:
\[ 6 \text {(within correct ref, the inclusive arriving to scenario logical)} answered.
This concludes the correct detailed solution for the problem: with maximum corrected detailed;
Fix subsequently.
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