polynomials (1 Viewer)

[MC Squidge]

1. The polynomial ax^(n+1) + bx^n + 1 is divisible by (x - 1)^2. Show that a = n, and b = -(1 + n).

2. Let x = a be a root of the quartic polynomial P(x) = x^4 + Ax^3 + Bx^2 + Ax + 1 where (2 + B)^2 NOT EQUALS 4A^2.
i. Show that a cannot be 0, 1, or -1.
ii. Show that x = 1/a
is a root.
iii. Deduce that if a is a multiple root, then its multiplicity is 2 and 4B = 8 + A^2.

 

[Affinity]

1. The polynomial ax^(n+1) + bx^n + 1 is divisible by (x - 1)^2. Show that a = n, and b = -(1 + n).

2. Let x = a be a root of the quartic polynomial P(x) = x^4 + Ax^3 + Bx^2 + Ax + 1 where (2 + B)^2 NOT EQUALS 4A^2.
i. Show that a cannot be 0, 1, or -1.
ii. Show that x = 1/a
is a root.
iii. Deduce that if a is a multiple root, then its multiplicity is 2 and 4B = 8 + A^2.

1. substitute 1 into the polynomial and it's derivative, you get

a + b + 1 = 0, (n+1)a + nb = 0, etc.

2.i. 0 obviously doesn't work.. and if you test 1 or -1, the given condition will rule them out.
ii.substitute 1/a into the polynomial then multiply through by a^4
iii. Will leave you this one.. one thing to think about is how to rule out the case of a,a,a,1/a. part i will be useful

 

[gurmies]

For x^4 + Ax^3 + Bx^2 + Ax + 1=0

x^2 + Ax+B+A/x+1/x^2 = 0

(x+1/x)^2 - 2 + A(x+1/x)+B=0

(x+1/x)^2 + A(x+1/x) + B - 2 = 0

Also shows that 1/x is a root