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Can you find a third degree polynomial equation with rational coefficients that has the given numbers as roots 3i and 7?
Wiki User
∙ 12y ago
Yes, easily.
Even though the question did not ask what the polynomial was, only if I could find it, here is how you would find the polynomial:
Since the coefficients are rational, the complex (or imaginary) roots must form a conjugate pair. That is to say, the two complex roots are + 3i and -3i. The third root is 7. So the polynomial, in factorised form, is
(x - 3i)(x + 3i)(x - 7) = (x2 + 9)(x - 7) = x3 - 7x2 + 9x - 63
Wiki User
∙ 12y ago
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What does transcendental mean in mathematics?
An algebraic number is one which is a root of a polynomial equation with rational coefficients. All rational numbers are algebraic numbers. Irrational numbers such as square roots, cube roots, surds etc are algebraic but there are others that are not. A transcendental number is such a number: an irrational number that is not an algebraic number. pi and e (the base of the exponential function) are both transcendental.
What is a rational function?
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational numbers.In the case of one variable, , a function is called a rational function if and only if it can be written in the formwhere and are polynomial functions in and is not the zero polynomial. The domain of is the set of all points for which the denominator is not zero, where one assumes that the fraction is written in its lower degree terms, that is, and have several factors of the positive degree.Every polynomial function is a rational function with . A function that cannot be written in this form (for example, ) is not a rational function (but the adjective "irrational" is not generally used for functions, but only for numbers).An expression of the form is called a rational expression. The need not be a variable. In abstract algebra the is called an indeterminate.A rational equation is an equation in which two rational expressions are set equal to each other. These expressions obey the same rules as fractions. The equations can be solved by cross-multiplying. Division by zero is undefined, so that a solution causing formal division by zero is rejected.
What statement must be true of an equation before you can use the quadratic formula to find the solutions?
The quadratic formula can be used to find the solutions of a quadratic equation - not a linear or cubic, or non-polynomial equation. The quadratic formula will always provide the solutions to a quadratic equation - whether the solutions are rational, real or complex numbers.
What are the numbers called in an equation?
The number and variables when next to each other in an equation are called a term. The variable or letters are coefficients.4x + 3x = 28.4x and 3x are terms and the variable "x" is the coefficient.
How many rational numbers are there between two consecutive rational numbers?
There are no consecutive rational numbers. Between any two rational numbers there are an infinity of rational numbers.
What is algebraic numbers?
An algebraic number is one that is a root to a non-zero polynomial, in one variable, whose coefficients are rational numbers.Equivalently, if the polynomial is multiplied by the LCM of the coefficients, the coefficients of the polynomial will all be integers.
What are the Basic concepts of rational function?
Thee basic concept is that an rational function is one polynomial divided by another polynomial. The coefficients of these polynomials need not be rational numbers.
Are all coefficients whole numbers in linear functions?
No. The equation 3/2 x + 2/3 y - 7 = 42 is a linear equation. But the coefficients of x and y are both rational numbers, not whole numbers.
What does transcendental mean in mathematics?
An algebraic number is one which is a root of a polynomial equation with rational coefficients. All rational numbers are algebraic numbers. Irrational numbers such as square roots, cube roots, surds etc are algebraic but there are others that are not. A transcendental number is such a number: an irrational number that is not an algebraic number. pi and e (the base of the exponential function) are both transcendental.
What does it mean if a number is transcendental?
It means that it is not algebraic... that is, it is not the root of a polynomial expression with rational coefficients.There are infinitely more transcendental numbers than algebraic numbers, but only a few are of any practical importance... most prominently pi and e, the base of the natural logarithms.All (real) transcendental numbers are irrational, but not all irrational numbers are transcendental. For example, the square root of two is irrational, but it is not transcendental as it is a root of the equation x2 - 2 = 0, a polynomial expression with rational coefficients.
How do I write an equation in standard form with integer coefficients?
The answer will depend on the form of the equation. Whether it is an equation in one or more variables, whether it is linear or polynomial, there are different standard forms for exponential equations.
What does polynomial with rational coefficients mean?
Let's define this question one word at a time. A polynomial is an equation with the variable x raised to whole number powers other than 0. This may include 2x + 3, or x2 - 8x + 16, or even x5 - 4x3 + 9. Coefficients are the numbers multiplied by the x term in question. The term 6x3 has a coefficient of 6, the term -x/2 has a coefficient of -1/2 and the term x2 has a coefficient of 1. Rational numbers are those which can be written as a ratio, or a fraction. This means its decimal notation will either have a finite amount of digits, like 0.625 (5/8), or a repeating series of decimals, e.g. 2.16666... or 13/6. Rational numbers can only be formed with addition, subtraction, multiplication and division - this means it excludes functions like taking the square root, the sine, or the log of a number. In summary, a polynomial with rational coefficients is an expression with multiple terms, such as ax2 + bx + c, where the coefficients 'a' and 'b' (and typically 'c' as well, as it is the coefficient of x0 which is 1 by definition, and is therefore being multiplied by 1) are rational numbers. This can extend to mean a polynomial of any degree, be it linear (x), cubic (x3), quartic (x4) or anything higher - so long as the coefficients of all the x terms are rational.
What is irreducible equation?
An irreducible equation is an irreducible polynomial which is equal to zero. A polynomial is irreducible over a particular type of number if it cannot be factorised into the products of two or more lower degree polynomials with coefficients of that type of number. For example, the equation x2 + 1 =0 is irreducible over the real numbers; there are no lower order polynomials, containing only real coefficients, which could be multiplied together to give this equation.
What rule describes a pattern that will have even numbers?
It can be any polynomial rule with integer coefficients in which there are an even number of odd coefficients.
what is irreducible?
An irreducible equation is an irreducible polynomial which is equal to zero. A polynomial is irreducible over a particular type of number if it cannot be factorised into the products of two or more lower degree polynomials with coefficients of that type of number. For example, the equation x2 + 1 =0 is irreducible over the real numbers; there are no lower order polynomials, containing only real coefficients, which could be multiplied together to give this equation.
What is the definition of cofficient of polynomial?
The coefficients of polynomials are the numbers in front of the variable expressions. Ex: In the polynomial: 3x^5 + 12x^2 - 45x + 134 the numerical coefficients are: 3,12,& -45
How would you differentiate rational algebraic expressions from those which are not?
The coefficients in a rational expression would be rational numbers.
