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Question 6
For which values of a and b, the zeroes of q(x)=x3+2x2+a are also the zeroes of the polynomial p(x)=x5x44x3+3x2+3x+b? which zeroes of p (x) are not the zeroes of q(x)?

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Solution

Given that the zeroes of q(x)=x3+2x2+a are also the zeroes of the polynomial p(x)=x5x44x3+3x2+3x+b i.e., q(x) is a factor of p(x). then, we use a division algorithm
x3+2x2+ax23x+2x5x44x3+3x2+3x+b
x5+2x4+ax23x44x3+(3a)x2+3x+b
3x46x33ax2x3+(3a)x2+(3+3a)x+b
2x3+4x2+2a(1+a)x2+(3+3a)x+(b2a)

if (x3+2x2+a) is a factor of (x5x44x3+3x2+3x+b), then remainder should be zero.........(Factor theorem)
i.e., (1+a)x2+(3+3a)x+(b2a)=0
=0.x2+0.x+0
On comparing the coefficient of x, we get
a+1=0
a=1
and b2a=0
b=2a
b=2(1)=2
For a = -1 and b = -2, the zeroes of q(x) are also the zeroes of the polynomial p(x)
1(x)=x3+2x21
and p(x)=x55x44x3+3x2+3x2
now, dividend = divisor × quotient + remainder
p(x)=9x3+2x21)(x23x+2)+0
=(x3+2x21)(x22xx+2)
(x3+2x21)(x2)(x1)
Hence, the zeroes of p(x) are 1 and 2 which are not the zeroes of q (x).

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