Chapter 13
Polynomials, Exponentials, and Logarithms
Fold a sheet of paper in half, then in half again. After ten folds it is thicker than a book. After fifty, if you could manage them, it would reach past the moon. Nothing you have written this year behaves that way. The expressions you have built, powers of a letter added together, climb at a pace you can picture. Folding does not. What changes there is the count of folds, sitting up where the exponent goes. Growth like that runs away from you, and working back to the count is a question of its own.
What You'll Explore
5 lessons.
Some lessons are marked Advanced because they go beyond core Algebra I. Skip them or explore them if you are curious.
- Adding and Subtracting Polynomials
Two expressions built from powers of x get added together, and only some of their pieces collapse into one. Which pieces are allowed to merge, and what changes when the second expression is subtracted instead, are the questions here, along with what qualifies as a polynomial in the first place.
- Multiplying Polynomials
Multiplying two polynomials pairs every piece of one with every piece of the other, and it is not obvious how many of those pairings survive once they combine, or what happens to the powers along the way. Here you will find out whether this needs a rule of its own, or whether a property you already trust does all of it.
- Exponential Functions
Watch a quantity that doubles every day against one built from ordinary powers, and the doubling quantity eventually pulls ahead for good. That difference is not really about speed, and this lesson tracks it down to one feature of how the rule is written, the same feature that decides whether a quantity climbs or fades.
- Compound Interest
Deposit the same amount into two accounts advertising the same yearly rate, one paying out once a year and the other every month. After ten years, are the balances equal? This lesson follows what happens when earned interest is left to sit with the deposit, and asks whether paying more often can help forever.
- Introduction to Logarithms Advanced. This lesson goes beyond core Algebra I. You can skip it.
Some missing exponents are easy to find, since doubling reaches four, then eight, then sixteen. Ten never appears in that list, yet the smooth curve of that doubling passes through ten exactly once. This lesson introduces the tool built for that gap, and the questions it finally makes answerable.