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Answer :
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We are given a polynomial [tex]\(k(x) = 45x^5 - 114x^4 + 26x^3 + 44x^2 + x - 2\)[/tex].
### Determining the Degree of the Polynomial
The degree of a polynomial is the highest power of the variable [tex]\(x\)[/tex] that appears in the polynomial with a non-zero coefficient. In this polynomial, the term with the highest power of [tex]\(x\)[/tex] is [tex]\(45x^5\)[/tex]. Therefore, the degree of [tex]\(k(x)\)[/tex] is:
- 5
### Determining the Leading Coefficient
The leading coefficient is the coefficient of the term with the highest degree. In our polynomial, the leading term is [tex]\(45x^5\)[/tex], so the leading coefficient is:
- 45
### End Behavior of the Polynomial
The end behavior of a polynomial describes how the values of the polynomial behave as [tex]\(x\)[/tex] approaches positive or negative infinity. This behavior is determined by the degree and the leading coefficient of the polynomial.
1. Right Hand End Behavior:
- Since the degree (5) is odd and the leading coefficient (45) is positive, as [tex]\(x\)[/tex] approaches infinity ([tex]\(x \rightarrow \infty\)[/tex]), the value of the polynomial [tex]\(k(x)\)[/tex] also approaches infinity ([tex]\(k(x) \rightarrow \infty\)[/tex]).
- Therefore, the right-hand end behavior is: As [tex]\(x \rightarrow \infty\)[/tex], [tex]\(k(x) \rightarrow \infty\)[/tex].
2. Left Hand End Behavior:
- Again, because the degree is odd, and the leading coefficient is positive, as [tex]\(x\)[/tex] approaches negative infinity ([tex]\(x \rightarrow -\infty\)[/tex]), the value of the polynomial [tex]\(k(x)\)[/tex] approaches negative infinity ([tex]\(k(x) \rightarrow -\infty\)[/tex]).
- So, the left-hand end behavior is: As [tex]\(x \rightarrow -\infty\)[/tex], [tex]\(k(x) \rightarrow -\infty\)[/tex].
These steps provide a comprehensive understanding of the characteristics of the polynomial based on its degree and leading coefficient.
We are given a polynomial [tex]\(k(x) = 45x^5 - 114x^4 + 26x^3 + 44x^2 + x - 2\)[/tex].
### Determining the Degree of the Polynomial
The degree of a polynomial is the highest power of the variable [tex]\(x\)[/tex] that appears in the polynomial with a non-zero coefficient. In this polynomial, the term with the highest power of [tex]\(x\)[/tex] is [tex]\(45x^5\)[/tex]. Therefore, the degree of [tex]\(k(x)\)[/tex] is:
- 5
### Determining the Leading Coefficient
The leading coefficient is the coefficient of the term with the highest degree. In our polynomial, the leading term is [tex]\(45x^5\)[/tex], so the leading coefficient is:
- 45
### End Behavior of the Polynomial
The end behavior of a polynomial describes how the values of the polynomial behave as [tex]\(x\)[/tex] approaches positive or negative infinity. This behavior is determined by the degree and the leading coefficient of the polynomial.
1. Right Hand End Behavior:
- Since the degree (5) is odd and the leading coefficient (45) is positive, as [tex]\(x\)[/tex] approaches infinity ([tex]\(x \rightarrow \infty\)[/tex]), the value of the polynomial [tex]\(k(x)\)[/tex] also approaches infinity ([tex]\(k(x) \rightarrow \infty\)[/tex]).
- Therefore, the right-hand end behavior is: As [tex]\(x \rightarrow \infty\)[/tex], [tex]\(k(x) \rightarrow \infty\)[/tex].
2. Left Hand End Behavior:
- Again, because the degree is odd, and the leading coefficient is positive, as [tex]\(x\)[/tex] approaches negative infinity ([tex]\(x \rightarrow -\infty\)[/tex]), the value of the polynomial [tex]\(k(x)\)[/tex] approaches negative infinity ([tex]\(k(x) \rightarrow -\infty\)[/tex]).
- So, the left-hand end behavior is: As [tex]\(x \rightarrow -\infty\)[/tex], [tex]\(k(x) \rightarrow -\infty\)[/tex].
These steps provide a comprehensive understanding of the characteristics of the polynomial based on its degree and leading coefficient.
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