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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.
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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.
Which polynomial has rational coefficients a leading leading coefficient of 1 and the zeros at 2-3i and 4?
There cannot be such a polynomial. If a polynomial has rational coefficients, then any complex roots must come in conjugate pairs. In this case the conjugate for 2-3i is not a root. Consequently, either (a) the function is not a polynomial, or (b) it does not have rational coefficients, or (c) 2 - 3i is not a root (nor any other complex number), or (d) there are other roots that have not been mentioned. In the last case, the polynomial could have any number of additional (unlisted) roots and is therefore indeterminate.
Pi the math term is transcendental what does that mean?
A transcendental number is one which is not algebraic. An algebraic number is one which is a root of a non-zero polynomial with rational coefficients.
How do you write a polynomial function with rational coefficients in standard form with given zeros of -1 -1 1?
The polynomial is (x + 1)*(x + 1)*(x - 1) = x3 + x2 - x - 1
How is a rational function the ratio of two polynomial functions?
That's the definition of a "rational function". You simply divide a polynomial by another polynomial. The result is called a "rational function".
