To determine the end behavior of the polynomial [tex]\( f(x) = -2x^4 + 3x^3 + x + 5 \)[/tex], we need to analyze the term with the highest degree, which is known as the leading term. The leading term will dictate the polynomial's end behavior because it grows faster than the other terms as [tex]\( x \)[/tex] becomes very large or very small.
1. Identify the leading term:
The term with the highest degree in the polynomial [tex]\( f(x) = -2x^4 + 3x^3 + x + 5 \)[/tex] is [tex]\( -2x^4 \)[/tex].
2. Determine the coefficient and the sign of the leading term:
The coefficient of the leading term [tex]\( -2x^4 \)[/tex] is [tex]\(-2\)[/tex], which is negative.
3. Analyze the end behavior based on the leading term:
- When [tex]\( x \to \infty \)[/tex] (as [tex]\( x \)[/tex] approaches positive infinity):
- The leading term [tex]\( -2x^4 \)[/tex] will dominate. Since the exponent is even ([tex]\(4\)[/tex]) and the coefficient is negative ([tex]\(-2\)[/tex]), [tex]\( -2x^4 \)[/tex] will tend toward [tex]\( -\infty \)[/tex]. Therefore, [tex]\( f(x) \to -\infty \)[/tex] as [tex]\( x \to \infty \)[/tex].
- When [tex]\( x \to -\infty \)[/tex] (as [tex]\( x \approaches negative infinity):
- Again, the leading term \( -2x^4 \)[/tex] will dominate. With an even exponent and a negative coefficient, [tex]\( -2x^4 \)[/tex] will still tend to [tex]\( -\infty \)[/tex]. Thus, [tex]\( f(x) \to -\infty \)[/tex] as [tex]\( x \to -\infty \)[/tex].
In summary, the end behavior of the polynomial [tex]\( f(x) = -2x^4 + 3x^3 + x + 5 \)[/tex] is as follows:
- [tex]\( f(x) \to -\infty \)[/tex] as [tex]\( x \to \infty \)[/tex]
- [tex]\( f(x) \to -\infty \)[/tex] as [tex]\( x \to -\infty \)[/tex]
