將直線（即一次多項式）推廣到 
次多項式 多項式
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(1)
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殘差由下式給出
![R^2=sum_(i=1)^n[y_i-(a_0+a_1x_i+...+a_kx_i^k)]^2.](/images/equations/LeastSquaresFittingPolynomial/NumberedEquation2.svg) |
(2)
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偏導數（同樣省略上標）是
 |  | ![-2sum_(i=1)^(n)[y-(a_0+a_1x+...+a_kx^k)]=0](/images/equations/LeastSquaresFittingPolynomial/Inline4.svg) |
(3)
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 |  | ![-2sum_(i=1)^(n)[y-(a_0+a_1x+...+a_kx^k)]x=0](/images/equations/LeastSquaresFittingPolynomial/Inline7.svg) |
(4)
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 |  | ![-2sum_(i=1)^(n)[y-(a_0+a_1x+...+a_kx^k)]x^k=0.](/images/equations/LeastSquaresFittingPolynomial/Inline10.svg) |
(5)
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這些匯出以下方程
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(6)
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(7)
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(8)
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或者，以矩陣形式表示
![[n sum_(i=1)^(n)x_i ... sum_(i=1)^(n)x_i^k; sum_(i=1)^(n)x_i sum_(i=1)^(n)x_i^2 ... sum_(i=1)^(n)x_i^(k+1); | | ... |; sum_(i=1)^(n)x_i^k sum_(i=1)^(n)x_i^(k+1) ... sum_(i=1)^(n)x_i^(2k)][a_0; a_1; |; a_k]=[sum_(i=1)^(n)y_i; sum_(i=1)^(n)x_iy_i; |; sum_(i=1)^(n)x_i^ky_i].](/images/equations/LeastSquaresFittingPolynomial/NumberedEquation3.svg) |
(9)
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這是一個範德蒙矩陣。我們也可以透過以下方式獲得最小二乘擬合的矩陣
![[1 x_1 ... x_1^k; 1 x_2 ... x_2^k; | | ... |; 1 x_n ... x_n^k][a_0; a_1; |; a_k]=[y_1; y_2; |; y_n].](/images/equations/LeastSquaresFittingPolynomial/NumberedEquation4.svg) |
(10)
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![[1 1 ... 1; x_1 x_2 ... x_n; | | ... |; x_1^k x_2^k ... x_n^k][1 x_1 ... x_1^k; 1 x_2 ... x_2^k; | | ... |; 1 x_n ... x_n^k][a_0; a_1; |; a_k]
=[1 1 ... 1; x_1 x_2 ... x_n; | | ... |; x_1^k x_2^k ... x_n^k][y_1; y_2; |; y_n],](/images/equations/LeastSquaresFittingPolynomial/NumberedEquation5.svg) |
(11)
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因此
![[n sum_(i=1)^(n)x_i ... sum_(i=1)^(n)x_i^k; sum_(i=1)^(n)x_i sum_(i=1)^(n)x_i^2 ... sum_(i=1)^(n)x_i^(k+1); | | ... |; sum_(i=1)^(n)x_i^k sum_(i=1)^(n)x_i^(k+1) ... sum_(i=1)^(n)x_i^(2k)][a_0; a_1; |; a_k]=[sum_(i=1)^(n)y_i; sum_(i=1)^(n)x_iy_i; |; sum_(i=1)^(n)x_i^ky_i].](/images/equations/LeastSquaresFittingPolynomial/NumberedEquation6.svg) |
(12)
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與之前一樣，給定 
個點 
並用多項式係數 
, ..., 
擬合，得到
![[y_1; y_2; |; y_n]=[1 x_1 x_1^2 ... x_1^k; 1 x_2 x_2^2 ... x_2^k; | | | ... |; 1 x_n x_n^2 ... x_n^k][a_0; a_1; |; a_k],](/images/equations/LeastSquaresFittingPolynomial/NumberedEquation7.svg) |
(13)
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在矩陣符號中，多項式擬合的方程由下式給出
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(14)
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這可以透過左乘轉置 
來求解，
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(15)
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這個矩陣方程可以數值求解，或者如果形式良好，可以直接求逆，以得到解向量
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(16)
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在上述方程中設定 
可以重現線性解。
