Formula for Root Product

In solving this problem, I found a formula for a product (1+x1)(1+x2)(1+x3)(1+x4)...(1+xd)(1+x_{1})(1+x_{2})(1+x_{3})(1+x_{4})...(1+x_{d}) up until the last root of a polynomial of degree d. I challenge Brilliant members to find this formula, with proof.

Note by Tristan Shin
5 years, 3 months ago

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it would be kind enough if u tell us......

Max B - 5 years, 3 months ago

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There is a pattern.....for example...............

( a + 1 ) ( b + 1 ) ( C + 1 ) = 1 + a + b + c + a b + b c + c a + a b c .

Hope you want this!

Satvik Golechha - 5 years, 3 months ago

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Yes, this is the pattern, but it can be simplified even further. Think about the certain terms in your example.

Tristan Shin - 5 years, 3 months ago

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Well, Tristan, I can't get it......please post the answer.....

Satvik Golechha - 5 years, 3 months ago

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Let the polynomial be f(x) f(x) with degree n n and roots r1,r2,,rn1,rn r_1, r_2, \dots, r_{n-1}, r_{n}

Suppose f(x)=a0xn+a1xn1++an1x+an=a0(xr1)(xr2)(xrn1)(xrn) f(x) = a_0x^n + a_1x^{n-1} + \dots + a_{n-1}x + a_n = a_0(x-r_1)(x-r_2)\dots(x-r_{n-1})(x-r_{n})

Then f(1)=a0(1)n+a1(1)n1++an1(1)+an=a0(1r1)(1r2)(1rn1)(1rn) f(-1) = a_0(-1)^n + a_1(-1)^{n-1} + \dots + a_{n-1}(-1) + a_n = a_0(-1-r_1)(-1-r_2)\dots(-1-r_{n-1})(-1-r_n)

f(1)=a0(1)(1+r1)(1)(1+r2)(1)(1+rn1)(1)(1+rn) f(-1) = a_0(-1)(1+r_1)(-1)(1+r_2)\dots(-1)(1+r_{n-1})(-1)(1+r_n)

f(1)a0=(1)n(1+r1)(1+r2)(1+rn1)(1+rn) \dfrac{f(-1)}{a_0} = (-1)^n(1+ r_1)(1 + r_2)\dots(1+ r_{n-1})(1+ r_n)

(1+r1)(1+r2)(1+rn1)(1+rn)={f(1)a0:n1(mod2)f(1)a0:n0(mod2) (1+ r_1)(1 + r_2)\dots(1+ r_{n-1})(1+ r_n) = \left\{ \begin{array}{lr} \frac{-f(-1)}{a_0} & : n \equiv 1 \pmod2 \\ \frac{f(-1)}{a_0} & : n \equiv 0 \pmod2 \end{array} \right.

Siddhartha Srivastava - 5 years, 3 months ago

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The solution is almost complete. The third time you state f(1)f\left(-1\right), this can be simplified after in the next step. In addition, what if there is a coefficient of the xnx^{n} term? Consider these and make an edit.

Tristan Shin - 5 years, 3 months ago

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I don't understand what you meant in the first line. I've edited for the case when there is a coefficient of the leading term.

Siddhartha Srivastava - 5 years, 3 months ago

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@Siddhartha Srivastava Your solution is just one thing away. Instead of using modular arithmetic in your final formula, what can you do with the sign instead?

Tristan Shin - 5 years, 3 months ago

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@Tristan Shin Are you implying (1)n(-1)^n?

Daniel Liu - 5 years, 3 months ago

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I'll let some more people try, then I'll post this answer sometime next week(April 14 to 18).

Tristan Shin - 5 years, 3 months ago

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It seems that Siddhartha Srivastava has come very close to solving this problem. Congratulations!

Tristan Shin - 5 years, 3 months ago

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