Given two polynomials of the same degree, how could you tell wich one grows faster doing calculations and by just graphing?
To determine which polynomial grows faster, you can compare their coefficients and exponents. Here are the steps to determine it both mathematically and graphically:
Mathematical Analysis:
1. Consider two polynomials of the same degree written in standard form, such as:
P(x) = a_n * x^n + a_{n-1} * x^{n-1} + ... + a_1 * x
2. Compare the leading term, which is the term with the highest degree exponent (n) and its coefficient (a_n).
- If the coefficient of the leading term (a_n) of polynomial P(x) is greater than the coefficient of polynomial Q(x), then P(x) grows faster than Q(x).
- If the coefficients of the leading terms are equal, compare the coefficients of the next highest degree terms until a difference is found.
Graphical Approach:
1. Plot the graphs of the two polynomials on the same coordinate system.
2. Observe the behavior of the graphs as x approaches positive or negative infinity.
- If the graph of P(x) is steeper, it represents faster growth compared to the graph of Q(x).
- If the graphs are parallel, the polynomials have the same rate of growth.
- If the graph of Q(x) is steeper, it represents faster growth compared to the graph of P(x).
It is essential to remember that these methods work when comparing two polynomials of the same degree. If the degrees differ, the polynomial with the higher degree will eventually grow faster.