AnswersDriveIn linear algebra, the minimal polynomial μA of an n × n matrix A over a field F is the monic polynomial P over F of least degree such that P(A) = 0. Any other polynomial Q with Q(A) = 0 is a (polynomial) multiple of μA. λ is a root of the characteristic polynomial χA of A, λ is an eigenvalue of matrix A.
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What does it mean for a polynomial to be Monic?
In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form.
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What is the generator polynomial?
Polynomial code. From Wikipedia, the free encyclopedia. In coding theory, a polynomial code is a type of linear code whose set of valid code words consists of those polynomials (usually of some fixed length) that are divisible by a given fixed polynomial (of shorter length, called the generator polynomial).
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What does it mean for a polynomial to be primitive?
A primitive polynomial is a polynomial that generates all elements of an extension field from a base field. Primitive polynomials are also irreducible polynomials. For any prime or prime power and any positive integer , there exists a primitive polynomial of degree over GF( ). There are.
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What does it mean for a polynomial to be irreducible?
In mathematics, an irreducible polynomial is, roughly speaking, a non-constant polynomial that cannot be factored into the product of two non-constant polynomials. On the other hand, with several indeterminates, there are absolutely irreducible polynomials of any degree, such as for any positive integer n.
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What is an irreducible element?
Irreducible Element. An element of a ring which is nonzero, not a unit, and whose only divisors are the trivial ones (i.e., the units and the products , where is a unit). Equivalently, an element is irreducible if the only possible decompositions of into the product of two factors are of the form.
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What does it mean when a polynomial is prime?
A polynomial with integer coefficients that cannot be factored into polynomials of lower degree , also with integer coefficients, is called an irreducible or prime polynomial . Example 1: x 2 + x + 1.
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How do you know when a Trinomial is prime?
Therefore, it is impossible to write the trinomial as a product of two binomials. This trinomial cannot be factored. Similarly to prime numbers, which do not have any factors other than 1 and themselves, the trinomials that cannot be factored are called prime trinomials.
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What does it mean for a polynomial to be factored completely?
Factoring polynomials involves breaking up a polynomial into simpler terms (the factors) such that when the terms are multiplied together they equal the original polynomial. Factoring helps solve complex equations so they are easier to work with. Finding the greatest common factor. Grouping like terms.
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Why is it important to know factoring?
Setting an equation equal to zero and factoring is an invaluable technique. It does more than just factor for no reason as it's presented in an Algebra 1 class. Factoring can be used to solved polynomial inequalities, quadratic equations, simplify expressions for them to be easier to work with etc.
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How do you factor a quadratic expression?
Solution
- First, factorise the quadratic equation x2- 9x + 20 = 0.
- Find two numbers which add up to 9 and multiply to give 20. These numbers are 4 and 5.
- Now find the value x so that when these brackets are multiplied together the answer is 0.
- You can check these answers by substitutuing 4 and 5 in to the equation:
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What is the box method?
A fairly new method, or algorithm, called the box method is being used to multiply two binomials together. When a trinomial of the form ax2 + bx + c can be factored into the product of two binomials, the format of the factorization is (dx + e)(fx + g) where d x f = a and e x g = c.
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What is the definition of a quadratic sequence?
A quadratic sequence is a sequence of numbers in which the second differences between each consecutive term differ by the same amount, called a common second difference.
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What is a quadratic pattern?
Recognizing a Quadratic Pattern. A sequence of numbers has a quadratic pattern when its sequence of second differences is constant. Here is an example.
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What is the Fibonacci sequence?
The Fibonacci sequence is a series of numbers where a number is found by adding up the two numbers before it. Starting with 0 and 1, the sequence goes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so forth. Written as a rule, the expression is xn = xn-1 + xn-2.
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What is the golden ratio spiral?
In geometry, a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio. That is, a golden spiral gets wider (or further from its origin) by a factor of φ for every quarter turn it makes.
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What is special about the golden ratio?
The Golden ratio is a special number found by dividing a line into two parts so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part. It is often symbolized using phi, after the 21st letter of the Greek alphabet. Phi is usually rounded off to 1.618.