AlgebraMath 294A: Problem Solving Seminar1 PolynomialsOne of the most important things to know when solving a problem involving polynomials is how to factor. Thereare a large number of identities which c an be useful when factoring, but the most basic ones arean− bn= (a − b)(an−1+ an−2b + · · · + abn−2+ bn−1),an+ bn= (a + b)(an−1− an−2b + · · · − abn−2+ bn−1), n odd.Similar to division and factoring in the integers, we may apply the Division algorithm to polynomials.Division Algorithm. Let f(x) and g(x) be either polynomials with coefficients in R, C, or Q, or monic poly-nomials over Z. Then there exist unique polynomials (of the same type) q(x) and r(x), such thatf(x) = q(x)g(x) + r(x),where deg(r(x)) < deg(g(x)), and g(x) divides f(x) precisely when r(x) is the zero polynomial.Like in the integers, the division algorithm can be used to find the greatest common divisor of two polynomials.Example 1. Let x1and x2be the roots of the equationx2− (a + d)x + (ad − bc) = 0.Show that x31and x32are the roots of the equationy2− (a3+ d3+ 3abc + 3bcd)y + (ad − bc)3= 0.Example 2. Prove that the fraction (n3+ 2n)/(n4+ 3n2+ 1) is irreducible for every natural number n.Example 3. Let N be the number which consists of 91 consecutive 1’s in base ten expansion. Prove that N iscomposite.Problem 1. Show that n4− 20n2+ 4 is composite for any integer n.Problem 2. Determine all solutions in the real numbers x, y, z, w of the systemx + y + z = w, 1/x + 1/y + 1/z = 1/w.Problem 3. For what n is the polynomial 1+x2+x4+· · ·+x2n−2divisible by the polynomial 1+x+x2+· · ·+xn−1?Problem 4. Consider all lines which meet the graph ofy = 2x4+ 7x3+ 3x − 5in four distinct points, say (xi, yi), i = 1, 2, 3, 4. Show thatx1+ x2+ x3+ x44is independent of the line, and find its value.Problem 5. Prove that there are no prime numbers in the infinite sequence of integers10001, 100010001, 1000100010001, . . . .Problem 6. Given numbers x, y, z such thatx + y + z = 3, x2+ y2+ z2= 5, x3+ y3+ z3= 7,find the value of x4+ y4+ z4.Problem 7. If n > 1, show that (x + 1)n− xn− 1 = 0 has a multiple root if and only if n − 1 is divisibleby 6.Problem 8. Let P (x) be the following polynomial, with real coefficients:P (x) = anxn+ an−1xn−1+ · · · + a3x3+ x2+ x + 1,where n ≥ 2. Show that the equation P (x) = 0 cannot have all real roots.2 GroupsLet S be a set. A binary operation on S is a function from S × S to S. For example, addition is a binary op erationon Z. A binary operation ∗ on S is associative if r ∗ (s ∗ t) = (r ∗ s) ∗ t for all r, s, t ∈ S. A group is a nonempty setG with an associative binary operation ∗ such that(i) G contains an identity element, e ∈ G, which has the property e ∗ g = g ∗ e = g for every g ∈ G.(ii) G contains inverses of elements. That is, for every g ∈ G, there is an element h ∈ G such thatg ∗ h = h ∗ g = e,where e is the identity element of G.So, for example, Z with addition is a group. Note that the operation is not required to be commutative. Anexample of a group with a non-commutative operation is the group of two-by-two matrices over R with multiplication.In an arbitrary group, the binary operation is often denoted by juxtaposing two elements (as in multiplication), sothat ab means the operation p e rformed on a and b.The theory of groups is a huge subject, so we restrict ourselves to only a few facts and notions. Firstly, theidentity element and inverses in groups are unique (prove this as an exercise ). Secondly, groups have the left andright cancellation property. That is, for all a, b, c ∈ G, ab = ac implies b = c, and ab = cb implies a = c. A subgroupof a group G is a subset of G which forms a group under the same operations as H. For example, the set of evenintegers is a subgroup of Z under addition.Example 4. Let a and b be two elements in a group such that aba = ba2b, a3= e, and b2n−1= e for some positiveinteger n. Prove that b = e, where e is the identity element.Problem 9. Let G be a set with an associative binary operation such that for all a, b ∈ G, a2b = b = ba2.Show that G is a group and the operation is commutative.Problem 10. Let A b e a subset of a finite group G such that A contains more that half of the elements ofG. Prove that each element of G is the product of two elements of A.Problem 11. Prove that no group is a union of two proper subgroups (that is, subgroups which are not thegroup itself).Problem 12. Let S be a nonempty set with an associative binary operation with the left and right cancella-tion properties. Assume that for every a ∈ S, the set {an| n = 1, 2, 3, . . .} is finite. Must S be a