Discover a wealth of knowledge at Westonci.ca, where experts provide answers to your most pressing questions. Discover comprehensive answers to your questions from knowledgeable professionals on our user-friendly platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
To analyze the polynomial [tex]\( y = x^4 + 4x^3 + 5x^2 + 4x + 4 \)[/tex], we will examine each statement carefully.
1. The function is of degree 10:
- The degree of a polynomial is the highest power of the variable [tex]\( x \)[/tex] in the polynomial.
- In [tex]\( y = x^4 + 4x^3 + 5x^2 + 4x + 4 \)[/tex], the highest power of [tex]\( x \)[/tex] is 4.
- Therefore, the degree of this polynomial is 4, not 10.
- This statement is false.
2. The function has at least one zero in the set of complex numbers:
- According to the Fundamental Theorem of Algebra, every non-constant polynomial has at least one complex root.
- Since our polynomial is of degree 4 (which is non-constant), it must have at least one complex root.
- This statement is true.
3. The function has a zero with a multiplicity of 5:
- The multiplicity of a zero is the number of times that zero appears as a root of the polynomial.
- Since the polynomial is of degree 4, the maximum possible multiplicity for any zero would be 4.
- Therefore, it is impossible for this polynomial to have a zero with a multiplicity of 5.
- This statement is false.
4. The function cannot be graphed:
- A polynomial function can always be graphed because it is a continuous and smooth function.
- Therefore, this statement is false.
In summary, the only true statement about the polynomial [tex]\( y = x^4 + 4x^3 + 5x^2 + 4x + 4 \)[/tex] is:
The function has at least one zero in the set of complex numbers.
1. The function is of degree 10:
- The degree of a polynomial is the highest power of the variable [tex]\( x \)[/tex] in the polynomial.
- In [tex]\( y = x^4 + 4x^3 + 5x^2 + 4x + 4 \)[/tex], the highest power of [tex]\( x \)[/tex] is 4.
- Therefore, the degree of this polynomial is 4, not 10.
- This statement is false.
2. The function has at least one zero in the set of complex numbers:
- According to the Fundamental Theorem of Algebra, every non-constant polynomial has at least one complex root.
- Since our polynomial is of degree 4 (which is non-constant), it must have at least one complex root.
- This statement is true.
3. The function has a zero with a multiplicity of 5:
- The multiplicity of a zero is the number of times that zero appears as a root of the polynomial.
- Since the polynomial is of degree 4, the maximum possible multiplicity for any zero would be 4.
- Therefore, it is impossible for this polynomial to have a zero with a multiplicity of 5.
- This statement is false.
4. The function cannot be graphed:
- A polynomial function can always be graphed because it is a continuous and smooth function.
- Therefore, this statement is false.
In summary, the only true statement about the polynomial [tex]\( y = x^4 + 4x^3 + 5x^2 + 4x + 4 \)[/tex] is:
The function has at least one zero in the set of complex numbers.
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.