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Chapter 4: Problem 11

Give the greatest common divisor and least common multiple of the following pairs of polynomials with complex coefficients: (a) \((t-2)^{3}(t-3)^{4}(t-i)\) and \((t-1)(t-2)(t-3)^{3}\) (b) \(\left(t^{2}+1\right)\left(t^{2}-1\right)\) and \((t+i)^{3}\left(t^{3}-1\right)\)

Short Answer

Expert verified
(a) GCD: \((t-2)(t-3)^3\), LCM: \((t-1)(t-2)^3(t-3)^4(t-i)\) (b) GCD: \((t-1)\), LCM: \((t^2 + 1)(t+1)(t-1)(t+i)^3 (t^2 + t + 1)\)

Step by step solution

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(a) Finding the GCD of \((t-2)^{3}(t-3)^{4}(t-i)\) and \((t-1)(t-2)(t-3)^{3}\)#:tag_content# To find the GCD of these polynomials, we first need to identify the common factors in both polynomials. In this case, we have the common factors \((t-2)\) and \((t-3)\). Now, we need to determine the smallest power of these common factors in both polynomials. For \((t-2)\) the smallest power is 1, and for \((t-3)\) it's 3. Hence, the GCD of the given polynomials is: \( GCD\left( (t-2)^{3}(t-3)^{4}(t-i),(t-1)(t-2)(t-3)^{3} \right) = (t-2)(t-3)^3 \)

(a) Finding the LCM of \((t-2)^{3}(t-3)^{4}(t-i)\) and \((t-1)(t-2)(t-3)^{3}\)#:tag_content# To find the LCM, we can use the formula: \(LCM(A, B) = \frac{A \cdot B}{GCD(A, B)}\) For our given polynomials, substituting the values we get: \(LCM\left((t-2)^{3}(t-3)^{4}(t-i), (t-1)(t-2)(t-3)^{3}\right) = \frac{(t-2)^{3}(t-3)^{4}(t-i)(t-1)(t-2)(t-3)^{3}}{(t-2)(t-3)^3}\) Simplifying: \(LCM\left((t-2)^{3}(t-3)^{4}(t-i), (t-1)(t-2)(t-3)^{3}\right) = (t-1)(t-2)^{3}(t-3)^{4}(t-i)\)
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(b) Finding the GCD of \(\left(t^{2}+1\right)\left(t^{2}-1\right)\) and \((t+i)^{3}\left(t^{3}-1\right)\)#:tag_content# Let's rewrite the given polynomials in their factorized form: 1. \(\left(t^{2}+1\right)\left(t^{2}-1\right) = (t^2 + 1)(t+1)(t-1)\) 2. \((t+i)^{3}\left(t^{3}-1\right) = (t+i)^3 \cdot (t-1)(t^2 + t + 1)\) Now, we can identify the common factor between both polynomials, which is \((t-1)\). Both polynomials have this factor with the same power (i.e., power 1). Therefore, the GCD is simply: \( GCD\left( \left(t^{2}+1\right)\left(t^{2}-1\right),(t+i)^{3}\left(t^{3}-1\right) \right) = (t-1) \)

(b) Finding the LCM of \(\left(t^{2}+1\right)\left(t^{2}-1\right)\) and \((t+i)^{3}\left(t^{3}-1\right)\)#:tag_content# Using the formula for LCM, we get: \(LCM\left(\left(t^{2}+1\right)\left(t^{2}-1\right), (t+i)^{3}\left(t^{3}-1\right)\right) = \frac{(t^2 + 1)(t+1)(t-1)(t+i)^3 \cdot (t-1)(t^2 + t + 1)}{(t-1)}\) Simplifying: \(LCM\left(\left(t^{2}+1\right)\left(t^{2}-1\right), (t+i)^{3}\left(t^{3}-1\right)\right) = (t^2 + 1)(t+1)(t-1)(t+i)^3 (t^2 + t + 1)\) And that's the final answer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Least Common Multiple of Polynomials
Understanding the least common multiple (LCM) of polynomials can be quite similar to finding LCM for integers, albeit with a few twists. It's important when working with polynomial expressions, especially when adding, subtracting, or finding equivalencies.

The LCM of two or more polynomials is the smallest polynomial that is divisible by each of the polynomials without a remainder. A step to this process, as seen in the given exercise, involves factoring polynomials to their irreducible factors and then multiplying each factor the greatest number of times it occurs in any of the polynomials. This concept ensures a solid understanding of how polynomials behave in division, aiding students in grasping the idea that finding the LCM is akin to overlaying the polynomials and choosing the most extensive 'coverage' of factors.
Polynomials with Complex Coefficients
When dealing with polynomials with complex coefficients, the fundamental principles of algebra still apply. However, the inclusion of complex numbers, such as 'i', the imaginary unit where \(i^2 = -1\), adds an additional layer of complexity.

Factoring such polynomials may reveal not just real roots (solutions) but also complex ones. As seen in the solutions provided, the presence of \(t-i\) introduces the need to consider complex conjugates and their properties during factorization and when finding the GCD or LCM. It is beneficial for students to realize that polynomials with complex coefficients are managed with the same operations as those with real coefficients; the solutions just may not be graphable on a conventional two-dimensional plane.
Factorization of Polynomials
Factorization is the process of breaking down a polynomial into the product of its simplest parts鈥攊ts factors. It's akin to breaking a multiplication problem into a division problem with smaller, more manageable numbers. For example, if we had the polynomial \(t^2 - 4\), it can be factorized into \((t + 2)\) and \((t - 2)\), since \((t + 2)\times(t - 2) = t^2 - 4\).

This process is profoundly useful when finding the GCD or LCM of polynomials because it simplifies the expressions and highlights common factors between them. In the step-by-step solution, polynomials are factorized to their irreducible factors, illustrating this process. It's important to highlight that sometimes factorization can introduce complex factors, especially in polynomials with complex coefficients, and that every polynomial expression can be factorized into a product of irreducible factors over the complex numbers, according to the Fundamental Theorem of Algebra.

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Most popular questions from this chapter

Show that \(t^{n}-1\) is divisible by \(t-1\).

(a) Let \(K\) be a subfield of a field \(E\), and \(\alpha \in E\). Let \(J\) be the set of all polynomials \(f(t)\) in \(K[t]\) such that \(f(\alpha)=0\). Show that \(J\) is an ideal. If \(J\) is not the zero ideal, show that the monic generator of \(J\) is irreducible. (b) Conversely, let \(p(t)\) be irreducible in \(K[t]\) and let \(\alpha\) be a root. Show that the ideal of polynomials \(f(t)\) in \(K[t]\) such that \(f(x)=0\) is the ideal generated by \(p(t)\).

Let \(W_{n}\) be the set of primitive \(n\) -th roots of unity in \(\mathbf{C}^{*}\). Define the \(n\) -th cyclotomic polynomial to be $$ \Phi_{n}(t)=\prod_{\zeta \in w_{v}}(t-\zeta) . $$ (a) Prove that \(t^{n}-1=\prod_{d \mid n} \Phi_{d} d t\). (c) Let \(p\) be a prime number and let \(k\) be a positive integer. Prove $$ \Phi_{p}(t)=\Phi_{p}\left(t^{p^{k+2}}\right) \quad \text { and } \quad \Phi_{p}(t)=t^{p-1}+\cdots+1 . $$ (d) Compute explicitly \(\Phi_{n}(t)\) for \(n \leqq 10 .\)

Show that \(t^{4}+4\) can be factored as a product of polynomials of degree 2 with integer coefficients. [Hint: try \(\left.t^{2} \pm 2 t+2 .\right]\)

Let \(R=g / f\) be a rational function with deg \(g<\) deg \(f\). Let $$ \frac{g}{f}=\frac{h_{1}}{p_{1}^{l_{1}}}+\cdots+\frac{h_{n}}{P_{a}^{l_{n}}} $$ be its partial fraction decomposition. L.et \(d_{x}-\operatorname{deg} p_{v}\). Show that the coefficicnts of \(h_{1} \ldots \ldots, h_{n}\) are the solutions of a system of linear equations, such that Ihe number of variables is equal to the number of equations, namely $$ \operatorname{deg} f=i_{1} d_{1}+\cdots+i_{n} d_{m^{-}} $$
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