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Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form. 2 -4 and 1 plus 3i?
To write a polynomial function with real coefficients given the zeros 2, -4, and (1 + 3i), we must also include the conjugate of the complex zero, which is (1 - 3i). The polynomial can be expressed as (f(x) = (x - 2)(x + 4)(x - (1 + 3i))(x - (1 - 3i))). Simplifying the complex roots, we have ((x - (1 + 3i))(x - (1 - 3i)) = (x - 1)^2 + 9). Thus, the polynomial in standard form is:
[ f(x) = (x - 2)(x + 4)((x - 1)^2 + 9). ]
Expanding this gives the polynomial (f(x) = (x - 2)(x + 4)(x^2 - 2x + 10)), which can be further simplified to the standard form.
What else can I help you with?
Is x over 3 a polynomial?
No, ( \frac{x}{3} ) is not considered a polynomial because it is expressed as a rational expression. However, if you rewrite it as ( \frac{1}{3}x ), it is a polynomial of degree 1, since it can be expressed in the standard form ( ax^n ), where ( a = \frac{1}{3} ) and ( n = 1 ). Polynomials can include constants, whole number coefficients, and non-negative integer exponents only.
Is monomials are polynomials?
Yes, monomials are a specific type of polynomial. A monomial is a polynomial that consists of only one term, which can include variables raised to non-negative integer exponents and coefficients. In contrast, a polynomial can have multiple terms, such as binomials (two terms) or trinomials (three terms). Therefore, all monomials are polynomials, but not all polynomials are monomials.
How alike the polynomial and non polynomial?
"Non-polynomial" can mean just about anything... How alike it is with the polynomial depends on what specifically you choose to include.
What is the polinomical- real life examples?
A polynomial is a mathematical expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents. Real-life examples of polynomials include calculating the area of a rectangular garden (length times width), modeling the trajectory of a projectile, or determining profit based on the number of products sold, where profit can be expressed as a polynomial function of the quantity sold. These expressions help in various fields, such as physics, economics, and engineering, to represent relationships and predict outcomes.
What is polynomial spline?
A polynomial spline is a piecewise-defined polynomial function used to create smooth curves that pass through or near a set of data points. It is constructed by connecting multiple polynomial segments, each defined on an interval, ensuring that these segments are continuous and have continuous derivatives up to a specified order at their junctions (knots). This flexibility allows polynomial splines to effectively model complex shapes and relationships in data. Common types include linear, quadratic, and cubic splines, with cubic splines being particularly popular for their smoothness and simplicity.
Write a polynomial degree with real coefficients whose zeros include 13 plus i and 2 Write the polynomial degree in standard form?
To have a zero at 2, you need to include a factor (x - 2). To have a zero at 13 + i, you need a factor (x - (13 + i)). To have real coefficients, for each non-real factor you need to include its complex conjugate, so in this case, (x + 13 + i).Thus, you have the factors: (x - 2)(x - (13 + i))(x + 13 + i) You can multiply the factors together to get the polynomial into standard form. I suggest you start with the complex conjugates, that makes it easier.
Is x over 3 a polynomial?
No, ( \frac{x}{3} ) is not considered a polynomial because it is expressed as a rational expression. However, if you rewrite it as ( \frac{1}{3}x ), it is a polynomial of degree 1, since it can be expressed in the standard form ( ax^n ), where ( a = \frac{1}{3} ) and ( n = 1 ). Polynomials can include constants, whole number coefficients, and non-negative integer exponents only.
What is the different of polynomial and non-polynomial?
Briefly: A polynomial consists only of powers of the variables - ie the variables multiplied by themselves or one another. A non polynomial can include any other function such as trigonometric, exponential, logarithmic etc.
How can I utilize the Wolfram Polynomial Calculator to solve complex polynomial equations efficiently?
To efficiently solve complex polynomial equations using the Wolfram Polynomial Calculator, input the polynomial equation you want to solve into the calculator. Make sure to include all coefficients and variables. The calculator will then provide you with the solution, including real and complex roots, if applicable. You can also adjust the settings to customize the output format and precision of the results.
Is monomials are polynomials?
Yes, monomials are a specific type of polynomial. A monomial is a polynomial that consists of only one term, which can include variables raised to non-negative integer exponents and coefficients. In contrast, a polynomial can have multiple terms, such as binomials (two terms) or trinomials (three terms). Therefore, all monomials are polynomials, but not all polynomials are monomials.
What are the alike of polynomial and non-polynomial?
That depends a lot on what you choose to include in "non-polynomial" - it can be just about anything. If you are referring to functions, what they have in common is anything that defines a function - mainly, the fact that for every value of an independent variable, a unique value is defined for the independent variable.
How alike the polynomial and non polynomial?
"Non-polynomial" can mean just about anything... How alike it is with the polynomial depends on what specifically you choose to include.
What is the polinomical- real life examples?
A polynomial is a mathematical expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents. Real-life examples of polynomials include calculating the area of a rectangular garden (length times width), modeling the trajectory of a projectile, or determining profit based on the number of products sold, where profit can be expressed as a polynomial function of the quantity sold. These expressions help in various fields, such as physics, economics, and engineering, to represent relationships and predict outcomes.
How do you draw graph for Fourier series?
To draw a graph for a Fourier series, first, calculate the Fourier coefficients by integrating the function over one period. Then, construct the Fourier series by summing the sine and cosine terms using these coefficients. Plot the resulting function over the desired interval, ensuring to include enough terms in the series to capture the function's behavior accurately. Finally, compare the Fourier series graph against the original function to visualize the approximation.
What are the two kinds of coefficients?
In terms of mathematics, a coefficient plays the role of a multiplicative factor in a series or an expression. The two different kinds of coefficients include numbers and letters.
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 polynomial spline?
A polynomial spline is a piecewise-defined polynomial function used to create smooth curves that pass through or near a set of data points. It is constructed by connecting multiple polynomial segments, each defined on an interval, ensuring that these segments are continuous and have continuous derivatives up to a specified order at their junctions (knots). This flexibility allows polynomial splines to effectively model complex shapes and relationships in data. Common types include linear, quadratic, and cubic splines, with cubic splines being particularly popular for their smoothness and simplicity.
