Polynomials from Cambridge (1 Viewer)

[.ben]

1. The equation xⁿ+px-q=0 has a double root. Show that (p/n)ⁿ+(q/(n-1))^n-1=0

2. The equation x³+3px²+3px+r=0 where p² does not equal q, has a double root. Show that (pq-r)²=4(p²-q)(q²-pr).

3. The equation x³+3x+2=0 has roots a, b, c. Fin the equation with roots a+1/a, b+1/b, c+1/c.

4. When P(x)=x⁴+ax²+bx is divided by x²+1, the remainder is x+2. Find the values of a and b.

 

[Mountain.Dew]

1. The equation xⁿ+px-q=0 has a double root. Show that (p/n)ⁿ+(q/(n-1))^n-1=0

2. The equation x³+3px²+3px+r=0 where p² does not equal q, has a double root. Show that (pq-r)²=4(p²-q)(q²-pr).

3. The equation x³+3x+2=0 has roots a, b, c. Fin the equation with roots a+1/a, b+1/b, c+1/c.

4. When P(x)=x⁴+ax²+bx is divided by x²+1, the remainder is x+2. Find the values of a and b.

1) find f'(x) = 0. substitute the value for x into the original equation and do some smart algebraic manipulation to get ur answer.
2) same thing
3) if all else fails, just simply get coefficients of sum of single roots, double roots and product of roots of new equation and ur done.
4) realise that u substitute x = i and x = -i. you have two equations. solve simultaenously to get a and b.

 

[.ben]

1) find f'(x) = 0. substitute the value for x into the original equation and do some smart algebraic manipulation to get ur answer.
2) same thing
3) if all else fails, just simply get coefficients of sum of single roots, double roots and product of roots of new equation and ur done.
4) realise that u substitute x = i and x = -i. you have two equations. solve simultaenously to get a and b.

could you expand on the first two please, cos i tried and it didn't come off. thanks

 

[haboozin]

1. The equation xn+px-q=0 has a double root. Show that (p/n)n+(q/(n-1))n-1=0

2. The equation x3+3px2+3px+r=0 where p2 does not equal q, has a double root. Show that (pq-r)2=4(p2-q)(q2-pr).


hi,

do you know the definition?
that if p(x) has a double root then P'(x) has that root aswell.
u should be able to prove it aswell... it has come up in hsc exams a few times.

 

[Mountain.Dew]

could you expand on the first two please, cos i tried and it didn't come off. thanks

for questions 1 and 2, solve the equation f'(x) = 0. find the value for x, then substitute THAT value back into the original f(x).