Joe Bernat
Wednesday December 1st
We began the class with a discussion about the use of writing mathematics in secondary schools. We
discussed the benefits, such as better understanding of the material, as well as the downfalls, such as
making sure the assignment is clear and that all students know what the teacher is asking. We also
discussed the fact that the school district should be in agreement with the use of writing and that it is
difficult for one teacher to enforce the technique if others do not.
The discussion moved to the density of the points around the circle through angle rotation. The question
that was asked was if an angle of 1.05 radians was continuously rotated around the unit circle, how
dense would the points become around the circle? How dense would the points become around the
point (1,0)? It was conjectured that they would become as dense as the real numbers. The discussion
led further to the statement that the points around the circle would be as dense as the rationals would
be in the set of real numbers. Professor Berger and a student discussed the function
𝑓𝑛 (𝑥) = 𝑠𝑖𝑛2 (𝑛𝑥)
The student said they saw this in another class, and this student suggested looking at the integral of the
limit of this function.
2𝜋
∫ lim 𝑖𝑛𝑓(𝑠𝑖𝑛2 (𝑛𝑥))
𝑛→∞
0
Next the discussion moved to completing the square. The example given was
(1) 0 = 𝑥 2 + 8𝑥 − 9
and was compared to
(2) 𝑦 = 𝑥 2 + 8𝑥 − 9.
At this time the class discussed a relationship between these two equations.
The first relationship looked at the values of x in (1) which solved the equation, and related to the xintercepts in the graph of (2). These solutions in fact had the same values for x, but in equation (2) gave
an ordered pair, (𝑥, 𝑦), where the y-value was 0 and the x-value was the same as in (1).
Next we looked at the names of these x-values. In equation (1), the solution gave the x-values that
solved the equation or roots. In equation (2), the solution gave the zeros of the function or the graph
(aka the x-intercepts).
In both cases, the procedure for solving for x is to factor by completing the square of the quadratic
polynomial, or by using the quadratic formula. This concept led to the next part of the discussion.
The steps for completing the square of a general quadratic polynomial with real coefficients 𝑎𝑥 2 +
𝑏𝑥 + 𝑐 = 0 were given by a student as follows
(1)
(2)
(3)
(4)
(5)
If 𝑎 ≠ 1, divide by the value of a.
Subtract c from both sides of the equation.
Divide 𝑏 by 2. Square this value and add this to both sides.
Factor the left side into its binomial roots.
Subtract the sum of the right side in order to bring this to the left side.
Professor Berger hinted at the fact that students might be able to understand the method for
completing the square. She recommended that students learn this instead of memorizing the quadratic
formula. In her experience, she found that students were not pushed to understand the concept of
completing the square and instead were taught the quadratic formula with no mathematical
understanding. Her recommendation is to teach students how to complete the square instead of just
memorizing the formula. Someone said that completing the square was messy and that it is easier to
plug into the quadratic formula. Another use for completing the square was the ability to put the
equation of a parabola into vertex form. This form makes it easy for the students to identify the graph of
the parabola.
Example: write the above example in vertex form.
𝑥 2 + 8𝑥 + 16 = 9 + 16
(𝑥 + 4)2 = 25
(𝑥 + 4)2 − 25 = 0
Thus the equation in vertex form is:
𝑦 = (𝑥 + 4)2 − 25
And the roots of the equation are:
(𝑥 + 4)2 = 25
𝑥 + 4 = ±5
𝑥 = −4 ± 5
𝑥 = {−9,1}
In class we proved the quadratic formula by completing the square on an arbitrary quadratic equation
as follows:
𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0
𝑏
𝑐
𝑥2 + 𝑥 + = 0
𝑎
𝑎
𝑏
𝑏2
−𝑐
𝑏2
𝑥2 + 𝑥 +
=
+
𝑎
(2𝑎)2
𝑎
(2𝑎)2
(𝑥 +
𝑥+
𝑏 2 −𝑐 𝑏 2
) =
+ 2
2𝑎
𝑎
4𝑎
𝑏
𝑏 2 − 4𝑎𝑐
= ±√
2𝑎
4𝑎2
𝑥=
−𝑏 √𝑏 2 − 4𝑎𝑐
±
2𝑎
2𝑎
𝑥=
−𝑏 ± √𝑏 2 − 4𝑎𝑐
2𝑎
𝑐
𝑎
𝑏
𝑎
A student stated that the product of the roots is ; and that the sum of the roots is .
The above examples show the uses and importance of completing the square. Along the same lines, the
graph of a parabola can be determined by looking at the vertex form of the equation. In vertex form, the
value inside the parenthesis determines the distance between the vertex and the y-axis, and the
constant added to the outside determines the distance between the vertex and the x-axis.
Example:
𝑦 = (𝑥 + 5)2 + 7
This graph opens up and the vertex is in the second quadrant. More specifically, the vertex is at the
point (−5,7).
Next we began a discussion about the discriminant of a quadratic equation. This is the part under the
square root of the quadratic formula: 𝑏 2 − 4𝑎𝑐. This tells us important information regarding the roots
of the quadratic polynomial. It tells us whether the roots are: real or non-real, rational or irrational, and
whether there are one or two roots.
If the discriminant is negative, there are two non-real roots.
If the discriminant is zero, there is one real root which is repeated twice.
If the discriminant is positive, then there are two real unequal roots. Further, if the discriminant is a
perfect square, the roots are rational.
After the discussion of the discriminant for quadratics was held, Professor Berger asked if anyone knew
the discriminant for cubic polynomials, or more generally any nth degree polynomial. Can we make an
equivalent formula for cubic polynomials as the quadratic formula is for a quadratic polynomial?
It is important to remember that we have real valued coefficients for the polynomials in question. So
with this idea in mind, look at the possible roots of a cubic polynomial:
(1)
(2)
(3)
(4)
2 real roots, 1 non-real root
1 real root, 2 non-real roots
3 real roots
3 non-real roots
Looking at each case above, we concluded that we had to eliminate the 1st and 4th case, because the
polynomial in this instance would have non-real coefficients. Thus we have two options, (2) and (3). In
case two, the graph of the equation crossed the x-axis only once, and in case three, the graph crosses in
three places.
In case (1) and (4), the roots of the equation would be 𝑥 = {𝑎1 , 𝑎2 , 𝑎3 + 𝑏𝑖} and 𝑥 = {𝑎1 + 𝑏1 𝑖, 𝑎2 +
𝑏2 𝑖, 𝑎3 + 𝑏3 𝑖} respectively. In either case, the polynomial must have real valued coefficients, thus
𝑎𝑗 , 𝑏𝑗 ∈ ℝ, 𝑗 ∈ ℕ, and the above cases result in non-real coefficients because of the odd number of nonreal roots. Next we showed that case (2) and (3), however, must be solutions to a real valued polynomial
of degree three.
Case (3): The set of real numbers is closed under addition and multiplication, thus this is trivial and must
give a real valued polynomial.
Case (2) must have roots as follows: 𝑥 = {𝑎1 , 𝑎2 + 𝑏𝑖, 𝑎2 − 𝑏𝑖}. Note that both non-real roots must be
complex conjugates. Thus the polynomial can be represented as:
(𝑥 − 𝑎1 )(𝑥 − (𝑎2 + 𝑏𝑖))(𝑥 − (𝑎2 − 𝑏𝑖)) = (𝑥 − 𝑎1 )(𝑥 2 − 2𝑎2 𝑥 + 𝑎22 + 𝑏 2 )
= 𝑥 3 − 2𝑎2 𝑥 2 + 𝑎22 𝑥 + 𝑏 2 𝑥 − 𝑎1 𝑥 2 + 2𝑎2 𝑎1 𝑥 − 𝑎1 𝑎22 − 𝑎1 𝑏 2
= 𝑥 3 + (−2𝑎2 − 𝑎1 )𝑥 2 + (𝑏 2 + 𝑎22 + 2𝑎2 𝑎1 )𝑥 − 𝑎1 𝑎22 − 𝑎1 𝑏 2
Thus the coefficients are real valued.
The next part of the discussion was about the coefficients of a general cubic equation where
𝑎, 𝑏, 𝑐, 𝑑 ∈ ℝ:
𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑 = 0
𝑡𝑜 𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑦, 𝑙𝑒𝑡 𝑎 = 1
In a quadratic polynomial, the coefficients were determined by the sum or product of the roots. Is the
same true for cubics? Yes. Let 𝛼 represent arbitrary roots of the polynomial.
𝑑 = −𝛼1 𝛼2 𝛼3
𝑐 = 𝛼1 𝛼2 + 𝛼2 𝛼3 + 𝛼1 𝛼3
𝑏 = −(𝛼1 +𝛼2 +𝛼3 )
We determined that the discriminant of a quadratic polynomial can be found by squaring the difference
of the two roots. If the result is negative, the roots are non-real, if the result is non-negative, the roots
are real. ((𝑎 + 𝑏𝑖) − (𝑎 − 𝑏𝑖))((𝑎 + 𝑏𝑖) − (𝑎 − 𝑏𝑖))
Can an equivalent statement be said about cubic polynomials?
The professor gave us the formula for the discriminant as follows:
(𝛼1 −𝛼2 )2 (𝛼1 − 𝛼3 )2 (𝛼2 − 𝛼3 )2
In general, the discriminant for a monic (The leading coefficient is equal to 1) degree 𝑛 polynomial is
defined to be:
2
∏(𝛼𝑖 − 𝛼𝑗 ) ,
𝑖<𝑗
Where the roots of the polynomial are {𝛼1 … 𝛼𝑛 }.
Next we looked at the family of quadratics with the form below,
𝑦 = 𝑥 2 + 2𝑥 + 𝑐
We studied the different possible roots for this family by substituting different values in for 𝑐. This gave
us different functions to look at. We analyzed the following in class:
𝑓(𝑥) = 𝑥 2 + 2𝑥 + 1
𝑔(𝑥) = 𝑥 2 + 2𝑥 − 3
ℎ(𝑥) = 𝑥 2 + 2𝑥 + 5
𝑗(𝑥) = 𝑥 2 + 2𝑥 + 3
We allowed 𝑐 to be 1 and noticed that there was only one root, or x-intercept at the point (−1,0). Next
we let 𝑐 = −3, and looked at the roots again. This translated the vertex down, in turn giving two
distinct roots, or zeroes. The zeroes occurred at the points (−3,0) and (1,0). Professor Berger
questioned what the relationship is for the distance between the zeroes or roots and the translation of
the vertex. We noticed that the further down the vertex was moved, the greater the distance between
the x-intercepts. Should this direct relationship exist if the vertex is moved up? The class looked at the
translation of the vertex when 𝑐 = 5 and also when 𝑐 = 3. In both of these cases, there were two nonreal roots. For the equation ℎ(𝑥), the roots were {−1 + 2𝑖, −1 − 2𝑖} and for 𝑗(𝑥), the roots were {−1 +
√2𝑖, −1 − √2𝑖}. We next examined the roots and noticed an equivalent relationship as before: the
further the vertex was moved from the original location, the greater the distance between the roots.
After noting this, we stated that the distance between the roots was in a direct correspondence with the
distance the vertex was moved for both the real case and the non-real case. Also we noticed that the
roots for this family all lied along one of two lines. For the real case, the roots were on the line 𝑦 = 0.
And for the non-real case, the roots were on the line 𝑥 = −1. In general we noticed that fixing 𝑎 = 1,
(monic polynomial) and fixing 𝑏 to a number, the roots follow a specific pattern:
Real roots are on the line 𝑦 = 0.
𝑏
Non-real roots are on the line 𝑥 = − 2𝑎.
Further inquiry:
In class, it was discussed that quadratic and cubic polynomials must have specific types and amounts of
roots (real and non-real). Is this true for n-degree polynomials? Can you list the possible roots for degree
four and degree five polynomials?