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_2025_hornerform, author="Wolfram Research", title="{HornerForm}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/HornerForm.html}", note=[Accessed: 18-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_hornerform, organization={Wolfram Research}, title={HornerForm}, year={2007}, url={https://reference.wolfram.com/language/ref/HornerForm.html}, note=[Accessed: 18-April-2025 ]}