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

# 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) = x^{2}. 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

by a quadratic function.

**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) = x^{2} − 3x + 2

2. g(x) = −x^{2} + 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) = x^{2}−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 Z_{12}. 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 Z_{12})
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 |x^{2} − 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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