Determine the end behavior of P(x) = -4x^5 + 3x^4 - x + 7 as x → ∞ and as x → -∞.

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Multiple Choice

Determine the end behavior of P(x) = -4x^5 + 3x^4 - x + 7 as x → ∞ and as x → -∞.

Explanation:
End behavior of polynomials is determined by the leading term, because that term dominates as x becomes very large in magnitude. Here the leading term is -4x^5, an odd-degree term with a negative coefficient. For very large positive x, x^5 is positive, so -4x^5 is negative and P(x) tends to -∞. For very large negative x, x^5 is negative, and -4x^5 becomes positive, so P(x) tends to ∞. This means the right end goes toward -∞ while the left end goes toward ∞. The other described possibilities would require a different combination of degree parity and leading sign.

End behavior of polynomials is determined by the leading term, because that term dominates as x becomes very large in magnitude. Here the leading term is -4x^5, an odd-degree term with a negative coefficient. For very large positive x, x^5 is positive, so -4x^5 is negative and P(x) tends to -∞. For very large negative x, x^5 is negative, and -4x^5 becomes positive, so P(x) tends to ∞. This means the right end goes toward -∞ while the left end goes toward ∞. The other described possibilities would require a different combination of degree parity and leading sign.

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