Math ( Polynomial )
posted by Jack
This time three questions - 1. If (x^2 - 1 ) is a factor of polynomial ax^4 + bx^3 + cx^2 + dx + e, show that a + c + e = b + d = 0.
2. Let R1 and R2 be the remainders when polynomials x^3 + 2x^2 - 5ax - 7 and x^ 3 + ax^2 - 12 x + 6 are divided by ( x + 1 ) and ( x - 2 ) respectively. If 2R1 + R2 = 6, find a.
3. If alpha and beta are the zeros of the polynomial ax^2 + bx + c then evaluateA. (alpha)^2 / beta + (beta)^2 / alpha
B. alpha^2 .beta + alpha.beta^2
C. 1/(alpha)^4 + 1/(beta)^4.
Please work the complete solutions.
I will do the first one, the second is similar
if (x^2 -1) is a factor, then (x+1) and (x-1) are both factors
f(1) = 0 and f(-1) = 0
f(x) = ax^4 + bx^3 + cx^2 + dx + c
f(1) = a+b+c+d + e = 0
f(-1) = a - b + c -d + e = 0
add them: ---> 2a + 2c + 2e = 0
or a + c + e = 0
2b + 2d = 0
b + d + 0
so a+c+e = b+d = 0
use remaider theorem .