# For Which Values And Zeroes Are Also Zeroes Polynomial 4×3 3×2 Which Zeroes Are Not Zeroes

For which values of a and b, the zeroes of q(x) = x3 + 2×2 + a are also the zeroes of the polynomial … zeroes of p(x) are not the…

Given that the zeroes of q(x) = x3 + 2x+ a are also the zeroes of the polynomial p(x) = x– x4 – 4x3+ 3xz + 3x + b i.e., q(x) is a factor of p(x). Then, we use a division algorithm.

If (x3 + 2x2 + a) is a factor of (x5 – x4 – 4x3 + 3x2 + 3x + b), then remainder should be zero.
i.e.,– (1 + a) x2 + (3 + 3a) x + (b – 2a) = 0
= 0.x2 + 0.x+0
On comparing the coefficient of x, we get
a + 1 = 0
⇒a = -1
and b – 2a = 0
⇒b =2a
b = 2(-1) = -2                                                   [va = -1]
For a = -1 and b = – 2, the zeroes ofq(x) are also the zeroes of the polynomial p(x).
∴                                              q(x) = x3 + 2x2 -1
and                                           p(x) = x5 – x– 4x3 + 3x2 + 3x – 2
Now,                  Divident = divisor x quotient + remainder
p(x) = (x3 +2x2 -1)(x2 -3x + 2)+ 0
= (x3 + 2x2 -1){x-2x – x + 2}
= (x3 + 2x2 – 1) (x – 2) (x – 1)
Hence, the zeroes of p(x) are land 2 which are not the zeroes of q(x).