Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Join our platform to connect with experts ready to provide precise answers to your questions in various areas. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
To determine which of the given options must also be a root of the polynomial function [tex]\( f(x) \)[/tex], we need to consider the properties of polynomial functions with real coefficients.
1. Polynomials with real coefficients have complex roots that occur in conjugate pairs. This means that if the polynomial [tex]\( f(x) \)[/tex] has a complex root [tex]\( a + bi \)[/tex], the conjugate [tex]\( a - bi \)[/tex] must also be a root.
2. In this case, we are given that [tex]\( -3 + i \)[/tex] is a root of [tex]\( f(x) \)[/tex].
3. To find the conjugate of [tex]\( -3 + i \)[/tex]:
- The real part of [tex]\( -3 + i \)[/tex] is [tex]\( -3 \)[/tex].
- The imaginary part of [tex]\( -3 + i \)[/tex] is [tex]\( i \)[/tex].
The conjugate is found by changing the sign of the imaginary part while keeping the real part the same:
- Therefore, the conjugate of [tex]\( -3 + i \)[/tex] is [tex]\( -3 - i \)[/tex].
4. Given that polynomials with real coefficients have roots in conjugate pairs, if [tex]\( -3 + i \)[/tex] is a root, [tex]\( -3 - i \)[/tex] must also be a root.
Therefore, the correct answer is:
[tex]\[ -3 - i \][/tex]
1. Polynomials with real coefficients have complex roots that occur in conjugate pairs. This means that if the polynomial [tex]\( f(x) \)[/tex] has a complex root [tex]\( a + bi \)[/tex], the conjugate [tex]\( a - bi \)[/tex] must also be a root.
2. In this case, we are given that [tex]\( -3 + i \)[/tex] is a root of [tex]\( f(x) \)[/tex].
3. To find the conjugate of [tex]\( -3 + i \)[/tex]:
- The real part of [tex]\( -3 + i \)[/tex] is [tex]\( -3 \)[/tex].
- The imaginary part of [tex]\( -3 + i \)[/tex] is [tex]\( i \)[/tex].
The conjugate is found by changing the sign of the imaginary part while keeping the real part the same:
- Therefore, the conjugate of [tex]\( -3 + i \)[/tex] is [tex]\( -3 - i \)[/tex].
4. Given that polynomials with real coefficients have roots in conjugate pairs, if [tex]\( -3 + i \)[/tex] is a root, [tex]\( -3 - i \)[/tex] must also be a root.
Therefore, the correct answer is:
[tex]\[ -3 - i \][/tex]
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.