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HornerForm
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HornerForm

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HornerForm[poly]

puts the polynomial poly in Horner form.

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HornerForm[poly,vars]

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

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HornerForm[poly1/poly2]

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

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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)Summary of the most common use cases

Horner form of a polynomial in x:

In[1]:=1
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https://wolfram.com/xid/0d6ioyodqy-b9goi5
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0d6ioyodqy-d481m2
Out[1]=1

Horner form of a rational function:

In[1]:=1
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https://wolfram.com/xid/0d6ioyodqy-cxck9f
Out[1]=1

Scope  (2)Survey of the scope of standard use cases

Order a bivariate polynomial with respect to x then y:

In[1]:=1
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https://wolfram.com/xid/0d6ioyodqy-zwe94
Out[1]=1

With respect to y then x:

In[2]:=2
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https://wolfram.com/xid/0d6ioyodqy-b6umnd
Out[2]=2

Construct a rational function in two variables:

In[1]:=1
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https://wolfram.com/xid/0d6ioyodqy-k3poyw
Out[1]=1

Convert the function to Horner form:

In[2]:=2
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https://wolfram.com/xid/0d6ioyodqy-d9r08b
Out[2]=2

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

In[3]:=3
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https://wolfram.com/xid/0d6ioyodqy-frpo5r
Out[3]=3

Generalizations & Extensions  (1)Generalized and extended use cases

An expression with rational exponents:

In[1]:=1
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https://wolfram.com/xid/0d6ioyodqy-tkk0u
Out[1]=1

Applications  (1)Sample problems that can be solved with this function

Improve speed and stability for numeric evaluation of large polynomials:

In[1]:=1
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https://wolfram.com/xid/0d6ioyodqy-i3gz2n
In[2]:=2
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https://wolfram.com/xid/0d6ioyodqy-lcgi9
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0d6ioyodqy-bu5myu
Out[3]=3

Properties & Relations  (2)Properties of the function, and connections to other functions

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

In[1]:=1
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https://wolfram.com/xid/0d6ioyodqy-bjv7lx
In[2]:=2
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https://wolfram.com/xid/0d6ioyodqy-tbivh
Out[2]=2

Obtained as a ratio of Horner forms:

In[3]:=3
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https://wolfram.com/xid/0d6ioyodqy-fgo0jr
Out[3]=3

HornerForm recursively factors out powers of variables:

In[1]:=1
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https://wolfram.com/xid/0d6ioyodqy-pj0w92
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0d6ioyodqy-d1xpci
Out[2]=2

Collect groups based on powers of variables:

In[3]:=3
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https://wolfram.com/xid/0d6ioyodqy-lkgig6
Out[3]=3

Factor gives the factored form:

In[4]:=4
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https://wolfram.com/xid/0d6ioyodqy-d8fls2
Out[4]=4

Possible Issues  (1)Common pitfalls and unexpected behavior

Exponents must be integers or rationals:

In[1]:=1
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https://wolfram.com/xid/0d6ioyodqy-cgbvpm
Out[1]=1

Neat Examples  (1)Surprising or curious use cases

In[1]:=1
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https://wolfram.com/xid/0d6ioyodqy-dky9xv
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0d6ioyodqy-c8qip7
Out[2]=2
Wolfram Research (2007), HornerForm, Wolfram Language function, https://reference.wolfram.com/language/ref/HornerForm.html.
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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.

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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.

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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

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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: 29-March-2025 ]}

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@misc{reference.wolfram_2025_hornerform, author="Wolfram Research", title="{HornerForm}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/HornerForm.html}", note=[Accessed: 29-March-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: 29-March-2025 ]}

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@online{reference.wolfram_2025_hornerform, organization={Wolfram Research}, title={HornerForm}, year={2007}, url={https://reference.wolfram.com/language/ref/HornerForm.html}, note=[Accessed: 29-March-2025 ]}