[math-fun] polynomial puzzle
consider polynomials with the property that the line passing through two inflection points on the graph is tangent to the graph at a third point. there are polynomials of degree 5 or more with this property. i can find one of degree 6 with integer coefficients. not hard at all. but i cannot seem to find one with degree 5 with integer coefficients. can you? erich friedman
How about p(x) = x^5 - 126x^4 + 5040x^3 - 62720x^2 = x^2 (x-56) (x^2-70x+1120) and the second derivative is p''(x) = 4 (5x-28) (x^2-70x+1120) so the roots of x^2-70x+1120 namely 35+sqrt(105) and 35-sqrt(105) are inflection points on the x-axis, and it is tangent to y=p(x) because of the x^2 term. On Sat, May 9, 2015 at 10:30 AM, Erich Friedman <erichfriedman68@gmail.com> wrote:
consider polynomials with the property that the line passing through two inflection points on the graph is tangent to the graph at a third point.
there are polynomials of degree 5 or more with this property.
i can find one of degree 6 with integer coefficients. not hard at all.
but i cannot seem to find one with degree 5 with integer coefficients. can you?
erich friedman _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
very nice. what was your method james? i tried something similar, finding a function function with inflection points on the x-axis, but couldn't solve the resulting equations. erich On May 9, 2015, at 4:57 PM, James Buddenhagen wrote:
How about p(x) = x^5 - 126x^4 + 5040x^3 - 62720x^2 = x^2 (x-56) (x^2-70x+1120) and the second derivative is p''(x) = 4 (5x-28) (x^2-70x+1120) so the roots of x^2-70x+1120 namely 35+sqrt(105) and 35-sqrt(105) are inflection points on the x-axis, and it is tangent to y=p(x) because of the x^2 term.
On Sat, May 9, 2015 at 10:30 AM, Erich Friedman <erichfriedman68@gmail.com> wrote:
consider polynomials with the property that the line passing through two inflection points on the graph is tangent to the graph at a third point.
there are polynomials of degree 5 or more with this property.
i can find one of degree 6 with integer coefficients. not hard at all.
but i cannot seem to find one with degree 5 with integer coefficients. can you?
erich friedman _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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p(x) = x^2 (x-c) (28x^2-35xc+10c^2) gives p''(x) = 2(10x-c) (28x^2-35xc+10c^2) and so works for any non-zero integer c. Taking c=1 gives a simpler example than that of my previous message. I looked at functions of the form p(x) = x^2 (x-a) (x-b) (x-c) and solved {p''(a)=0, p''(b)=0} for {a,b} with c as parameter. On Sat, May 9, 2015 at 4:16 PM, Erich Friedman <erichfriedman68@gmail.com> wrote:
very nice. what was your method james?
i tried something similar, finding a function function with inflection points on the x-axis, but couldn't solve the resulting equations.
erich
On May 9, 2015, at 4:57 PM, James Buddenhagen wrote:
How about p(x) = x^5 - 126x^4 + 5040x^3 - 62720x^2 = x^2 (x-56) (x^2-70x+1120) and the second derivative is p''(x) = 4 (5x-28) (x^2-70x+1120) so the roots of x^2-70x+1120 namely 35+sqrt(105) and 35-sqrt(105) are inflection points on the x-axis, and it is tangent to y=p(x) because of the x^2 term.
On Sat, May 9, 2015 at 10:30 AM, Erich Friedman < erichfriedman68@gmail.com> wrote:
consider polynomials with the property that the line passing through two inflection points on the graph is tangent to the graph at a third point.
there are polynomials of degree 5 or more with this property.
i can find one of degree 6 with integer coefficients. not hard at all.
but i cannot seem to find one with degree 5 with integer coefficients. can you?
erich friedman _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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participants (2)
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James Buddenhagen