Questions: State the degree and end behavior of f(x)=-x+4x^3-x^5+1. Explain or show your reasoning.

State the degree and end behavior of f(x)=-x+4x^3-x^5+1. Explain or show your reasoning.
Transcript text: 3. State the degree and end behavior of $f(x)=-x+4 x^{3}-x^{5}+1$. Explain or show your reasoning.
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Solution

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Solution Steps

Step 1: Identify the Degree of the Polynomial

The degree of a polynomial is the highest power of \( x \) in the expression. For \( f(x) = -x + 4x^{3} - x^{5} + 1 \), the highest power of \( x \) is \( 5 \). Therefore, the degree of the polynomial is \( 5 \).

Step 2: Determine the Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree. In \( f(x) = -x + 4x^{3} - x^{5} + 1 \), the term with the highest degree is \( -x^{5} \), so the leading coefficient is \( -1 \).

Step 3: Analyze the End Behavior

The end behavior of a polynomial is determined by its degree and leading coefficient:

  • If the degree is odd and the leading coefficient is positive, the graph falls to the left and rises to the right.
  • If the degree is odd and the leading coefficient is negative, the graph rises to the left and falls to the right.

Since the degree is \( 5 \) (odd) and the leading coefficient is \( -1 \) (negative), the graph of \( f(x) \) rises to the left and falls to the right.

Final Answer

The degree of the polynomial is \( 5 \) and the end behavior is that the graph rises to the left and falls to the right.

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