Fun with polynomials!

[Scott1328]

That's a long proof, there is a much shorter one any sequence of iconsecutive natural numbers of length P will contain at least one multiple of every natural number less than or equal P. A fact Calli cited in his proof.

Let w, x, y, z be four consecutive natural numbers greater than 0.

Since one of those numbers must a multiple of 4, let that number be
z=4*k

Since one of these numbers must be a multiple of 3, let that number be
y=3*j

Since one of these numbers must be a multiple of and not a multiple of 4, let that number be
x=2*i

Therefore
w*x*y*z = w * 2*i * 3j * 4*k = 24 * w * i * j * k

QED

[Brian Peacock]

[Xamonas Chegwé]

Scott beat me to it.

[Seth]

Here is my solution,, with full working.

The polynomial equation to solve is a4+4x3+11x2+14x+15=0. Since we are told that this has complex solutions c1, c2, c3 and c4, such that c1=c2*, and c3=c4*, where z* denotes the complex conjugate of z, this means that the quartic equation must be factorisable into two quadratic factors Q1(x) and Q2(x), such that the solutions of Q1(x)=0 are c1 and c2, and the solutions of Q2(x)=0 are c3 and c4.

Therefore, we write the quartic in the form (x2+ax+b)(x2+cx+d)=0, and multiply out, giving:

x4+(a+c)x3+(b+d+ac)x2+(ad+bc)x+bd=0

Equating coefficients, we have:

[1] a+c=4
[2] b+d+ac=11
[3] ad+bc=14
[4] bd=15

Taking [4] first, 15 can be factorised as either 3×5 or 15×1. Let us choose b=3, d=5, and see if this choice leads to a consistent solution.

Substituting these values into [3] yields 5a+3c=14. Multiplying [1] by 3 throughout, and solving the resulting simultaneous equations, we have:

[5] 5a+3c=14
[6] 3a+3c=12

Subtracting [6] from [5] gives 2a=2, therefore a=1. Substituting this result into [1] gives c=3.

These values form a consistent solution. We therefore have a=1, b=3, c=3, d=5, yielding:

[7] Q1(x) = x2+x+3
[8] Q2(x) = x2+3x+5

and the quartic factorises as:

(x2+x+3)(x2+3x+5)=0

From the quadratic formula, the roots of the equation Q1(x)=0 are:

[-1±(1-12)]/2

which gives as our first roots:

[9] c1 = (-½+11½i)
[10] c2 = (-½-11½i)

Likewise, the roots of the equation Q2(x)=0 are:

[-3±(9-20)]/2

which gives as our second roots:

[11] c3 = (-3/2+11½i)
[12] c4 = (-3/2-11½i)

EDIT: Bugger, someone got there first. But without providing full working. I'm sure JimC will see fit to award bonus points for full working being provided.

WTF are you people talking about?

[Xamonas Chegwé]

Seth can't do maths. Explains his views on economics.

[Brian Peacock]

[mistermack]

For fuck's sake.

Do you people actually know the meaning of the word fun?

Hint. It's not '' fucked up numbers ''.

[JimC]

Fun is giving a maths test!

(for me, at least...)

[Seth]

Seth can't do maths. Explains his views on economics.

Economics ain't about polynomials, it's about money in, money out and politics. Addition and subtraction I can do.

If you try to make economics about polynomials, you're probably a politician trying to pull the wool over the eyes of the people who pay your salary.

[Xamonas Chegwé]

Seth can't do maths. Explains his views on economics.

Economics ain't about polynomials, it's about money in, money out and politics. Addition and subtraction I can do.

If you try to make economics about polynomials, you're probably a politician trying to pull the wool over the eyes of the people who pay your salary.

Economics is about far more complex mathematics than merely polynomials. How do you think supermarkets decide their pricing structures, stock-levels, staffing levels, etc? How are oil prices set? What about interest rates? Adding and subtracting?

Like I said, you don't understand any of this stuff. But, as with any ignoramus, your opinion is just as valid as anyone else's because FREEDUM, DAMMIT!

[Brian Peacock]

Is there any common conjunction between polynomials and Iswasawa theory?

[Svartalf]

Is there any common conjunction between polynomials and Iswasawa theory?

uh, whatcha talking aboot?

[Seth]

Seth can't do maths. Explains his views on economics.

Economics ain't about polynomials, it's about money in, money out and politics. Addition and subtraction I can do.

If you try to make economics about polynomials, you're probably a politician trying to pull the wool over the eyes of the people who pay your salary.

Economics is about far more complex mathematics than merely polynomials. How do you think supermarkets decide their pricing structures, stock-levels, staffing levels, etc? How are oil prices set? What about interest rates? Adding and subtracting?

Like I said, you don't understand any of this stuff. But, as with any ignoramus, your opinion is just as valid as anyone else's because FREEDUM, DAMMIT!

Yup, adding and subtracting. Employee X costs us A, so we need to generate revenue Y so we can make profit Z. Addition and subtraction.

[Xamonas Chegwé]

Seth can't do maths. Explains his views on economics.

Economics ain't about polynomials, it's about money in, money out and politics. Addition and subtraction I can do.

If you try to make economics about polynomials, you're probably a politician trying to pull the wool over the eyes of the people who pay your salary.

Economics is about far more complex mathematics than merely polynomials. How do you think supermarkets decide their pricing structures, stock-levels, staffing levels, etc? How are oil prices set? What about interest rates? Adding and subtracting?

Like I said, you don't understand any of this stuff. But, as with any ignoramus, your opinion is just as valid as anyone else's because FREEDUM, DAMMIT!

Yup, adding and subtracting. Employee X costs us A, so we need to generate revenue Y so we can make profit Z. Addition and subtraction.

Yes, Seth. It's all just adding and subtracting. Nobody needs higher any Maths to work out compound interest, optimisation or dynamic analysis except for corrupt politicians.

[Brian Peacock]

Is there any common conjunction between polynomials and Iwasawa theory?

uh, whatcha talking aboot?

I was hoping to illuminate for Seth's benefit the concordance between expressions consisting of indeterminates and coefficients and expressions of projective algebraic variety - specifically, theorems concerning addition and subtraction.