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Algebra
Math 294A: Problem Solving Seminar
1 Polynomials
One of the most important things to know when solving a problem involving polynomials is how to factor. There are a large number of identities which can be useful when factoring, but the most basic ones are
an^ − bn^ = (a − b)(an−^1 + an−^2 b + · · · + abn−^2 + bn−^1 ),
an^ + bn^ = (a + b)(an−^1 − an−^2 b + · · · − 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 that
f (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 x 1 and x 2 be the roots of the equation
x^2 − (a + d)x + (ad − bc) = 0.
Show that x^31 and x^32 are the roots of the equation
y^2 − (a^3 + d^3 + 3abc + 3bcd)y + (ad − bc)^3 = 0.
Example 2. Prove that the fraction (n^3 + 2n)/(n^4 + 3n^2 + 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 is composite.
Problem 1. Show that n^4 − 20 n^2 + 4 is composite for any integer n.
Problem 2. Determine all solutions in the real numbers x, y, z, w of the system
x + y + z = w, 1 /x + 1/y + 1/z = 1/w.
Problem 3. For what n is the polynomial 1+x^2 +x^4 +· · ·+x^2 n−^2 divisible by the polynomial 1+x+x^2 +· · ·+xn−^1?
Problem 4. Consider all lines which meet the graph of
y = 2x^4 + 7x^3 + 3x − 5
in four distinct points, say (xi, yi), i = 1, 2 , 3 , 4. Show that
x 1 + x 2 + x 3 + x 4 4
is independent of the line, and find its value.
Problem 5. Prove that there are no prime numbers in the infinite sequence of integers
10001 , 100010001 , 1000100010001 ,....
Problem 6. Given numbers x, y, z such that
x + y + z = 3, x^2 + y^2 + z^2 = 5, x^3 + y^3 + z^3 = 7,
find the value of x^4 + y^4 + z^4.
Problem 7. If n > 1, show that (x + 1)n^ − xn^ − 1 = 0 has a multiple root if and only if n − 1 is divisible by 6.
Problem 8. Let P (x) be the following polynomial, with real coefficients:
P (x) = anxn^ + an− 1 xn−^1 + · · · + a 3 x^3 + x^2 + x + 1,
where n ≥ 2. Show that the equation P (x) = 0 cannot have all real roots.
2 Groups
Let S be a set. A binary operation on S is a function from S × S to S. For example, addition is a binary operation on 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 set G 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 that
g ∗ 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. An example of a group with a non-commutative operation is the group of two-by-two matrices over R with multiplication.
