Answer :
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.
