# HornerForm

HornerForm[poly]

puts the polynomial poly in Horner form.

HornerForm[poly,vars]

puts poly in Horner form with respect to the variable or variable list vars.

HornerForm[poly1/poly2]

puts the rational function poly1/poly2 in Horner form by nesting poly1 and poly2.

HornerForm[poly1/poly2,vars1,vars2]

puts poly1/poly2 in Horner form using the variables or variable lists vars1 and vars2 for poly1 and poly2, respectively.

# Details

• When variables are not specified, HornerForm puts the polynomial or rational function into Horner form with respect to the variables identified using Variables.

# Examples

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## Basic Examples(3)

Horner form of a polynomial in x:

Put a polynomial into Horner form with respect to a given variable:

Horner form of a rational function:

## Scope(2)

Order a bivariate polynomial with respect to x then y:

With respect to y then x:

Construct a rational function in two variables:

Convert the function to Horner form:

Convert to Horner form using y before x in the numerator:

## Generalizations & Extensions(1)

An expression with rational exponents:

## Applications(1)

Improve speed and stability for numeric evaluation of large polynomials:

## Properties & Relations(2)

Horner form of a rational function is the ratio of Horner forms:

Obtained as a ratio of Horner forms:

HornerForm recursively factors out powers of variables:

Collect groups based on powers of variables:

Factor gives the factored form:

## Possible Issues(1)

Exponents must be integers or rationals:

## Neat Examples(1)

Wolfram Research (2007), HornerForm, Wolfram Language function, https://reference.wolfram.com/language/ref/HornerForm.html.

#### Text

Wolfram Research (2007), HornerForm, Wolfram Language function, https://reference.wolfram.com/language/ref/HornerForm.html.

#### CMS

Wolfram Language. 2007. "HornerForm." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HornerForm.html.

#### APA

Wolfram Language. (2007). HornerForm. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HornerForm.html

#### BibTeX

@misc{reference.wolfram_2022_hornerform, author="Wolfram Research", title="{HornerForm}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/HornerForm.html}", note=[Accessed: 29-May-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_hornerform, organization={Wolfram Research}, title={HornerForm}, year={2007}, url={https://reference.wolfram.com/language/ref/HornerForm.html}, note=[Accessed: 29-May-2023 ]}