1. If f(x)=x7+x5+x4+x3+x+1 and g(x)=x3+x+1, find f(x) – g(x).
a) x7+x5+x4+x3
b) x6+x4+x2+x
c) x4+x2+x+1
d) x7+x5+x4
Answer
Answer: d [Reason:] Perform Modular subtraction.
2. 5/3 mod 7 =
a) 2
b) 3
c) 4
d) 5
Answer
Answer: c [Reason:] 5/3 mod 7 = (5×3-1) mod 7 = (5×5) mod 7 = 4.
3. The polynomial x4+1 can be represented as –
a) (x+1)(x3+x2+1)
b) (x+1)(x3+x2+x)
c) (x)(x2+x+1)
d) None of the mentioned
Answer
Answer: d [Reason:] (x4+1) = (x+1)(x3+x2+x+1).
4. -5 mod -3 =
a) 3
b) 2
c) 1
d) 5
Answer
Answer: c [Reason:] -5 mod -3 = -2 mod -3 = 1 mod -3.
5. Multiply the polynomials P1 = x5 +x2+ x) by P2 = (x7 + x4 +x3+x2 + x) in GF(28) with irreducible polynomial (x8 + x4 + x3 + x + 1). The result is
a) x4+ x3+ x+1
b) x5+ x3+x2+x+1
c) x5+ x4+ x3+x+1
d) x5+ x3+x2+x
Answer
Answer: b [Reason:] On performing polynomial multiplication we get with respect to modulus (x8 + x4 + x3 + x + 1) we get x5+ x3+x2+x+1.
6. Multiply 00100110 by 10011110 in GF(2^8) with modulus 100011011.The result is
a) 00101111
b) 00101100
c) 01110011
d) 11101111
Answer
Answer: a [Reason:] On performing polynomial multiplication with respect to modulus 100011011 we get 00101111.
7.Find the inverse of (x7+x+1) modulo (x8 + x4 + x3+ x + 1).
a) x7+x
b) x6+x3
c) x7
d) x5+1
Answer
Answer: c [Reason:] Finding the inverse with respect to (x8 + x4 + x3+ x + 1) we get x7 as the inverse.
8. 7x = 6 mod 5. Then the value of x is
a) 2
b) 3
c) 4
d) 5
Answer
Answer: b [Reason:] 7 x 3 mod 5 = 6 mod 5 = 1.
State whether the following few statement are true or false over a field.
9. The product of monic polynomials is monic.
a) True
b) False
c) Can’t Say
d) None of the mentioned
Answer
Answer: a [Reason:] This is always true over a field.
10. The product of polynomials of degrees m and n has a degree m+n+1.
a) True
b) False
c) Can’t Say
d) None of the mentioned
Answer
Answer: b [Reason:] The product of polynomials of degrees m and n has a degree m+n.
11. The sum of polynomials of degrees m and n has degree max[m,n].
a) True
b) False
c) Can’t Say
d) None of the mentioned
Answer
Answer: c [Reason:] True when m is not equal to n; in that case the highest degree coefficient is of degree max[m,n]. But false in general when m = n, because the highest-degree coefficients might cancel (be additive inverses).
12. (7x + 2)-(x2 + 5) in Z_10 =
a) 9x2 + 7x + 7
b) 9x2+ 6x + 10
c) 8x2 + 7x + 6
d) None of the mentioned
Answer
Answer: a [Reason:](7x + 2) – (x2 + 5) in Z_10 = 9x2 + 7x +7. We can find this via basic polynomial arithmetic in Z_10.
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