# Explain how to write a polynomial in standard form

Basically, the procedure is carried out like long division of real numbers. The procedure is explained in the textbook if you're not familiar with it. One key point about division, and this works for real numbers as well as for polynomial division, needs to be pointed out. When you divide the dividend by the divisor, you get a quotient and a remainder. Polynomials of small degree have been given specific names. A polynomial of degree zero is a constant polynomial or simply a constant. Polynomials of degree one, two or three are respectively linear polynomials, quadratic polynomials and cubic polynomials.

For higher degrees the specific names are not commonly used, although quartic polynomial for degree four and quintic polynomial for degree five are sometimes used.

The names for the degrees may be applied to the polynomial or to its terms. The polynomial 0, which may be considered to have no terms at all, is called the zero polynomial. Unlike other constant polynomials, its degree is not zero. The zero polynomial is also unique in that it is the only polynomial having an infinite number of roots.

In the case of polynomials in more than one indeterminate, a polynomial is called homogeneous of degree n if all its non-zero terms have degree n. The zero polynomial is homogeneous, and, as homogeneous polynomial, its degree is undefined.

For more details, see homogeneous polynomial. The commutative law of addition can be used to rearrange terms into any preferred order.

In polynomials with one indeterminate, the terms are usually ordered according to degree, either in "descending powers of x", with the term of largest degree first, or in "ascending powers of x". The polynomial in the example above is written in descending powers of x. The first term has coefficient 3, indeterminate x, and exponent 2.

The third term is a constant. Because the degree of a non-zero polynomial is the largest degree of any one term, this polynomial has degree two.

It may happen that this makes the coefficient 0. The term "quadrinomial" is occasionally used for a four-term polynomial. A real polynomial is a polynomial with real coefficients.

The argument of the polynomial is not necessarily so restricted, for instance the s-plane variable in Laplace transforms. A real polynomial function is a function from the reals to the reals that is defined by a real polynomial.

Similarly, an integer polynomial is a polynomial with integer coefficients, and a complex polynomial is a polynomial with complex coefficients. A polynomial in one indeterminate is called a univariate polynomial, a polynomial in more than one indeterminate is called a multivariate polynomial.

A polynomial with two indeterminates is called a bivariate polynomial. These notions refer more to the kind of polynomials one is generally working with than to individual polynomials; for instance when working with univariate polynomials one does not exclude constant polynomials which may result, for instance, from the subtraction of non-constant polynomialsalthough strictly speaking constant polynomials do not contain any indeterminates at all.

It is possible to further classify multivariate polynomials as bivariate, trivariate, and so on, according to the maximum number of indeterminates allowed. Again, so that the set of objects under consideration be closed under subtraction, a study of trivariate polynomials usually allows bivariate polynomials, and so on.

It is common, also, to say simply "polynomials in x, y, and z", listing the indeterminates allowed. The evaluation of a polynomial consists of substituting a numerical value to each indeterminate and carrying out the indicated multiplications and additions. For polynomials in one indeterminate, the evaluation is usually more efficient lower number of arithmetic operations to perform using Horner's method:Explanation.

We begin by attempting to find any rational roots using the Rational Root Theorem, which states that the possible rational roots are the positive or negative versions of the possible fractional combinations formed by placing a factor of the constant term in the numerator and a factor of the leading coefficient in the denominator.

A polynomial is considered prime if it cannot be factored into the standard linear form of (x+a)((x+b). A given expression is a polynomial if it has more than one term. An example of a polynomial that can be factored would be x 2 +4x+4. When writing a polynomial in standard form, you always want to start from the highest degree (exponent), so let's simplify our expression before doing so + 6 = 4; This number will go first as it has the highest degree.

Up next will be '-3x', as it has a degree of 1. After this, we have 6, having a degree of /5(3). Write three different quadratic functions in factored form that have an axis of symmetry of x = 2 but have different x -intercepts.

Use words and/or numbers to show how you determined your answer. Sketch and write an equation of a polynomial that has the following characteristics: crosses the 𝑥𝑥-axis only at −2 and 4, touches the 𝑥𝑥-axis at 0 and 2, and is above the 𝑥𝑥-axis between 2 and 4.

The simplest possible basis is the monomial basis: \$\{1,x,x^2,x^3,\ldots,x^n\}\$. Recall the definition of a basis.

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The key property is that some linear combination of .

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