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# Quadratics; Inequalities and Absolute Values; Long-division, Remainders

factoring
A+ S 101-003: High school mathematics from a more advanced point of view

1. We’ll continue with quadratic equations today : the connection between finding
roots, graphs, and factoring.

2. We may see up to two presentations today.

3. As time permits: Why does factoring a function f(x) help find roots of f (x)?

4. Division stuff : numbers and polynomials

Quick review

Problem 1. Let q(x) = x2. How can you find the minimum value of f(x)? Can you
do it without graphing, or even visualization? (That is, argue you have the correct min
using number facts .)

Problem 2. Let f(x) = (x − 2)2 + 5. How can you find the minimum value of f(x)?
Can you do it without graphing, or even visualization?

Problem 3. Let h(x) = |x − 4| + 5. How can you find the minimum value of f(x)?
Can you do it without graphing, or even visualization?

Problem 4. Make up real-world maximization (or minimization) problem modeled

Discussion problem 5. What’s a root of a quadratic function? What connection
does a root a quadratic have with its graph? Can you explain that connection?

Problem 6. Find the roots of
1. f(x) = x2 − 3x + 2
2. g(x) = −x2 + 4x + 1

Problem 7. What connection does the factorization of a quadratic have to with its
roots as a function? For example, let q(x) = x2−3x+2. Factor q(x). Determine its roots.

Problem 8. Suppose we have factored a quadratic q(x) = (x − c)(x − d). What are
the roots of q(x)? Explain. How can you be sure that we haven’t missed any roots? Before
you explain, work on the next problem.

Problem 9. Let’s work in Z12. Let f(x) = (x − 2)(x − 4) Are the roots of f(x) just
2 and 4?

Problem 10. What property of real numbers (not possessed by Z12) guarantees that
factoring completely determines the root set of a quadratic?

Absolute values and inequalities too: because they are so important in high school math

Let’s recall the 50 cent definition of absolute value: |x| = x, if x ≥ 0; otherwise,
|x| = −x (when x < 0).

Problem 11. Explain how to find solutions to |x| = 2?

To |x − 1| = 2?
To |2x − 1| − 2 = 0?

Problem 12. Now you want to find solutions to |x| > 1. How many solutions are
there? Can you provide a graphical description of the solution set ?

Problem 13. Solve |2x − 1| ≥ 3.

Problem 14. Connect absolute values and the idea of distance on the number line.

Problem 15. How would you solve |x2 − 4| ≥ 0?

Loose ends: should we go here?

Please look this over this week. I think some discussion of long-division, divisibility
(in numbers and polynomials ) might be interesting (and useful ultimately to you).

The famous Division Algorithm (so obvious it’s “hard to see”.. but having so many
uses).

Let a, b be integers, with b ≠ 0. Then there exist unique q (the quotient) and r (the
remainder) , with r satisfying 0 ≤ r < b, such that a = bq + r.

Let a = 1001 and b = 21. Find q and r guaranteed by the Division Algorithm.

Unless you’re good at mental arithmetic , you might do a long-division.

One consequence of the uniqueness aspect of the Division Algorithm: an integer is
either even or odd and it can’t be both! (Of course everyone knows that anyway.. so of
course it’s a consequence of a detestable theorem)

For that matter, why does the long-division algorithm (on integers) work anyway?
Simple explanation ? Ever try to find one? (I didn’t.. but now I’m sorry I didn’t– because
I have to teach the validity of that algorithm to pre -service elementary teachers, which
makes me feel sort of phony but relieved to find out it’s explainable in simple terms .)

Theorem. Let p(x) be a polynomial of degree n. Then r is a root of p(x) if and only
if p(x) = (x − r)q(x), where q(x) is a polynomial of degree n − 1.

What does this theorem say about the number of roots of a polynomial of degree n?

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