Question

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.

Answer 1

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
or
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
subtract them:
2b + 2d = 0
b + d + 0

so a+c+e = b+d = 0

Answer 2

x^3+2x^2-5ax -7/x+1
R1= 5a-6
x^3+ax^2-12x+6/x-2
R2=4a-10
2R1+R2=6
2(5a-6)+4a-10=6
a=2
use remaider theorem .