Algebra Question of the Day

1. $p(x)$ grows faster long-term because it has two terms.
2. $q(x)$ grows faster long-term and will eventually exceed $p(x)$. (correct answer)
3. They grow at the same rate because both increase as $x$ increases.
4. $q(x)$ can never exceed $p(x)$ because $1.5$ is less than $2$.

Explanation: This question tests your understanding of a fundamental principle in mathematics: exponential functions eventually grow faster than any polynomial function—even very high-degree polynomials—when we look at large enough x-values. The growth hierarchy for large x-values is: exponential > polynomial > linear. Even a slow exponential like (1.01)^x will eventually exceed a fast polynomial like x^100 if we go far enough to the right. This happens because exponential growth is multiplicative (multiply by the same factor repeatedly), which compounds much faster than polynomial growth, which is additive-based (even if accelerating). Comparing p(x) = x² + 10x (quadratic) and q(x) = 1.5^x (exponential): At x = 5, we have p(5) = 25 + 50 = 75 and q(5) = 1.5^5 ≈ 7.59. At x = 10, we have p(10) = 100 + 100 = 200 and q(10) = 1.5^10 ≈ 57.67. At x = 15, we have p(15) = 225 + 150 = 375 and q(15) = 1.5^15 ≈ 437.89. The exponential has caught up! At x = 20, we have p(20) = 400 + 200 = 600 and q(20) = 1.5^20 ≈ 3325.26. The exponential dominates! Choice B correctly states that q(x) grows faster long-term and will eventually exceed p(x), recognizing that the exponential function 1.5^x must eventually dominate the quadratic polynomial x² + 10x. Choice C incorrectly reasons that because 1.5 < 2, the exponential can never exceed the polynomial—but this confuses the base with growth rate. Any exponential with base > 1 eventually exceeds any polynomial! The growth hierarchy you MUST remember: Exponential > Any Polynomial > Linear (for large x). Within polynomials: higher degree > lower degree. Within exponentials: larger base > smaller base. This hierarchy is a fundamental property of these function types—exponential growth is multiplicative and compounds, beating any additive pattern no matter how fast. When comparing functions long-term, just identify their types and apply this hierarchy!