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和立方.
和立方是數學公式的一種,它屬於因式分解、乘法公式及恆等式,被普遍使用。和立方是指一個數項,加上另一個數項後,總和的立方:
![{\displaystyle (a+b)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3}\,\!=a^{3}+b^{3}+3ab(a+b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d317a513fee415c2acf26f09580f939f8e1d7c4c)
和立方可直接計算驗證:
![{\displaystyle (a+b)^{3}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5edbcc2f1e6fd8526a5947a110d56a51953e25c)
![{\displaystyle =(a+b)(a+b)(a+b)\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0b93cf5489d77e6ac29c9842ff09b24aec2b0a7)
![{\displaystyle =a(a+b)(a+b)+b(a+b)(a+b)\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39b85f817000c8988864bc7d15fd36a67036e724)
![{\displaystyle =(a^{2}+ab)(a+b)+(ab+b^{2})(a+b)\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b980d06e266d95c0acb85ed934c58ba8cfb0148)
![{\displaystyle =a(a^{2}+ab)+b(a^{2}+ab)+a(ab+b^{2})+b(ab+b^{2})\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/143875c698d036aeed13cb2845dd2fcf6221b22f)
![{\displaystyle =a^{3}+a^{2}b+a^{2}b+ab^{2}+a^{2}b+ab^{2}+ab^{2}+b^{3}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0efead6ef31b260075e41eb70273046682ddab69)
![{\displaystyle =a^{3}+3a^{2}b+3ab^{2}+b^{3}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89056766942618a695768f3136540e6750669585)
以上計算方式便可證明:
和立方亦可運用和平方驗證,首先要知道和平方的公式是:
![{\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3e790659d23b83235c75b9954ba869c53dd1eb6)
然後,利用和平方計算出和立方:
![{\displaystyle (a+b)^{3}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5edbcc2f1e6fd8526a5947a110d56a51953e25c)
![{\displaystyle =(a+b)^{2}(a+b)\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d21aa062d4a9344f67626ff9a73ba1824fcd299)
![{\displaystyle =(a^{2}+2ab+b^{2})(a+b)\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c4a17ea74f697484b3ea553ce26629c4789f06e)
![{\displaystyle =a(a^{2}+2ab+b^{2})+b(a^{2}+2ab+b^{2})\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f77021e6dceed994c64871113c596fc851952225)
![{\displaystyle =a^{3}+2a^{2}b+ab^{2}+a^{2}b+2ab^{2}+b^{3}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29a1bdd97232ab26d73fd0ea74e4fa22c13aa511)
![{\displaystyle =a^{3}+3a^{2}b+3ab^{2}+b^{3}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89056766942618a695768f3136540e6750669585)
以上計算方式便可證明:
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