& Give an example of a polynomial (x) € Z[] and a prime p such that f(x) is reducible in Qx] but 7(x) is irreducible in Z,JK]. Does this contradict
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
#8 ignore the last part of the question.
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(a) -x* +x +x+x+2
(c) 3x + 2 - 7x + 2x
(b) x* + 4x + x- x
(d) 2x - 5x+ 3x² + 4x – 6
(e) 2x + 7x + 5x² + 7x + 3
(f) 6x* – 31x + 25x2 + 33x + 7
2. Show that Vp is irrational for every positive prime integer p. [Hint: What are
the roots of x2 - p? Do you prefer this proof to the one in Exercises 30 and 31
of Section 1.37]
3. If a monic polynomial with integer coefficients has a root in Q, show that this
root must be an integer.
4. Show that each polynomial is irreducible in Q[x], as in Example 3.
(a) x* + 2x' +x + 1
(b) x* - 2x2 + 8x + 1
5. Use Eisenstein's Criterion to show that each polynomial is irreducible in Q[x]:
(a) x – 4x + 22
(b) 10 – 15x + 25x² – 7x
(c) 5x" – 6x* + 12x + 36x – 6
6. Show that there are infinitely many integers k such that + 12x – 21x + k
is irreducible in Q[x].
7. Show that each polynomial f(x) is irreducible in Q[x] by finding a prime p
such that f(x) is irreducible in Z,[x]
(a) 7x + 6x? + 4x + 6
(Ъ) 9х + 4x3—3х + 7
8. Give an example of a polynomial f(x) E Z[x] and a prime p such that f(x)
is reducible in Q[x] but ƒ(x) is irreducible in Z[x]. Does this contradict
Theorem 4.25?
9. Give an example of a polynomial in Z[x] that is irreducible in Q[x] but factors
when reduced mod 2, 3, 4, and 5.
10. If a monic polynomial with integer coefficients factors in Z[x] as a product of
polynomials of degrees m and n, prove that it can be factored as a product of
monic polynomials of degrees m and n in Z[x].
B. 11. Prove that 30x – 91 (where n e Z, n>1) has no roots in Q.
12. Let Fbe a field and f(x) E F[×]. If c e Fand f(x +c) is irreducible in F[x],
prove that f(x) is irreducible in F[x]. [Hint: Prove the contrapositive.]
13. Prove that f(x) = x + 4x + 1 is irreducible in Q[x] by using Eisenstein's
Criterion to show that f(x + 1) is irreducible and applying Exercise 12.
14. Prove that f(x) = x* + x +²+x+lis irreducible in O[x]. [Hint: Use the
hint for Exercise 21 with p = 5.]
15. Let f(x) = a,t + an-1x1 +• ·•+ a,x + a, be a polynomial with integer
coefficients. If p is a prime such that p|a1, p|az, ... ,P|a, but p } m, and
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