Advanced Algebra
Polynomials
Get to know the polynomial family: power functions and how their graphs grow, even and odd powers, shifting and stretching graphs, evaluating and graphing polynomials, behaviour far from and near zero, roots, adding and multiplying polynomials, and factoring to find roots.
THE SYLLABUS
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A First Look At Polynomials
A polynomial is built by adding and subtracting powers of x, like x⁴ − 9x²; its graph can be read for points on it, the places it crosses the x-axis, where it is negative, and where it rises or falls.
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Power Functions
Power functions such as x, x², x³ and x¹⁰ are the building blocks of polynomials; every power is 0 at 0, even powers send −1 to 1 and odd powers keep −1 at −1, and x⁰ is defined to be 1 everywhere.
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Comparing Powers
The exponent n of xⁿ is its degree; for x > 1 a higher degree gives a larger value, for a negative input even powers are positive and odd powers negative, and power functions of higher degree grow in size more quickly.
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The Dominance of Powers
For positive x the ratio of x^(n+1) to xⁿ is x itself, so a higher power is k times the next lower one exactly at x = k; the higher power dominates the lower one for large x and overtakes any multiple of it eventually.
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Between −1 and 1
For inputs between −1 and 1, raising to a higher power moves the output closer to 0, so higher powers flatten toward zero there; outside that band they grow faster, which decides where one power is bigger than another.
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Level Check
Fifteen questions with no hints on power functions: where a shifted power is negative, where a higher power beats a multiple of a lower one, plotted values of power functions, missing crossings, chains of powers of fractions, and a look back at powers of ten and evaluating powers.
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