Задание 438
Преобразуйте выражение в многочлен:
- \({\largeа)}\ (5-a)(3-a)-(a-4)^2;\)
- \({\largeб)}\ (x+3)^2+3(x-2)^2;\)
- \({\largeв)}\ 3(2-m)^2+2(2-m)^2;\)
- \({\largeг)}\ 5(2p-3)^2+2(5-2p)^2;\)
- \({\largeд)}\ 4(3-5a)^2-5(a-3)(2a-3);\)
- \({\largeе)}\ (a+1)^2+2(a+1)-3(a-1)(a+1);\)
- \({\largeж)}\ 3-2(5-x)(x-5)-2(5+x)^2;\)
- \({\largeз)}\ (x-y-z)(x-y-z)-(x-y)^2;\)
- \({\largeи)}\ (x+y+z)(x-y-z)-(x+y-z)(x-y+z);\)
- \({\largeк)}\ (x+y-z)(x-y+z)-(x+y+z)(x-y-z).\)
Источник заимствования: Алгебра. 7 класс. Учебник для общеобразовательных организаций / С.М. Никольский, М.К. Потапов, Н.Н. Решетников, А.В. Шевкин – Просвещение, 2013. – 116 c. ISBN 978-5-09-027739-6
А+АА-
Решение:
- \({\largeа)}\ (5-a)(3-a)-(a-4)^2=15-5a-3a+a^2-(a^2-8a+16)=\) \(15\) \(-\) \(8a\) \(+\) \(a^2\) \(-\) \(a^2\) \(+\) \(8a\) \(-\) \(16\) \(={-}1\)
- \({\largeб)}\ (x+3)^2+3(x-2)^2=x^2+6x+9+3(x^2-4x+4)=\) \(x^2\) \(+\) \(6x\) \(+\) \(9\) \(+\) \(3x^2\) \(-\) \(12x\) \(+\) \(12\) \(=4x^2-6x+21\)
\(\ \largeв)\) Преобразуем выражение в многочлен первым способом:
- \(3(2-m)^2+2(2-m)^2=(2-m)^2\cdot(3+2)=(4-4m+m^2)\cdot5=5m^2-20m+20\)
Преобразуем выражение в многочлен вторым способом:
- \(3(2-m)^2+2(2-m)^2=3(4-4m+m^2)+2(4-4m+m^2)=\) \(12\) \(-\) \(12m\) \(+\) \(3m^2\) \(+\) \(8\) \(-\) \(8m\) \(+\) \(2m^2\) \(=5m^2-20m+20\)
- \({\largeг)}\ 5(2p-3)^2+2(5-2p)^2=5(4p^2-12p+9)+2(25-20p+4p^2)=\) \(20p^2\) \(-\) \(60p\) \(+\) \(45\) \(+\) \(50\) \(-\) \(40p\) \(+\) \(8p^2\) \(=28p^2-100p+95\)
- \({\largeд)}\ 4(3-5a)^2-5(a-3)(2a-3)=4(9-30a+25a^2)-5(2a^2-3a-6a+9)=36-120a+100a^2-5(2a^2-9a+9)=\) \(36\) \(-\) \(120a\) \(+\) \(100a^2\) \(-\) \(10a^2\) \(+\) \(45a\) \(-\) \(45\) \(=90a^2-75a-9\)
\(\ \largeе)\) Преобразуем выражение в многочлен первым способом:
- \((a+1)^2+2(a+1)-3(a-1)(a+1)=(a+1)(a+1+2-3a+3)=(a+1)(6-2a)=\) \(6a\) \(-\ 2a^2+6\ -\) \(2a\) \(={-}2a^2+4a+6\)
Преобразуем выражение в многочлен вторым способом:
- \((a+1)^2+2(a+1)-3(a-1)(a+1)=a^2+2a+1+2a+2-3(a^2-1)=\) \(a^2\) \(+\) \(2a\) \(+\) \(1\) \(+\) \(2a\) \(+\) \(2\) \(-\) \(3a^2\) \(+\) \(3\) \(={-}2a^2+4a+6\)
- \({\largeж)}\ 3-2(5-x)(x-5)-2(5+x)^2=3+2(x-5)(x-5)-2(25+10x+x^2)=3+2(x-5)^2-50-20x-2x^2=3+2(x^2-10x+25)-50-20x-2x^2=\) \(3\) \(+\) \(2x^2\) \(-\) \(20x\) \(+\) \(50\) \(-\) \(50\) \(-\) \(20x\) \(-\) \(2x^2\) \(=3-40x\)
- \({\largeз)}\ (x-y-z)(x-y-z)-(x-y)^2=(x-y-z)^2-(x^2-2xy+y^2)=((x-y)-z)^2-x^2+2xy-y^2=(x-y)^2-2(x-y)z+z^2-x^2+2xy-y^2=\) \(x^2\) \(-\) \(2xy\) \(+\) \(y^2\) \(-\ 2xz+2yz+z^2\ -\) \(x^2\) \(+\) \(2xy\) \(-\) \(y^2\) \(=z^2-2xz+2yz\)
- \({\largeи)}\ (x+y+z)(x-y-z)-(x+y-z)(x-y+z)=(x+(y+z))(x-(y+z))-(x+(y-z))(x-(y-z))=x^2-(y+z)^2-(x^2-(y-z)^2)=x^2-(y+z)^2-x^2+(y-z)^2=x^2-(y^2+2yz+z^2)-x^2+y^2-2yz+z^2=\) \(x^2\) \(-\) \(y^2\) \(-\) \(2yz\) \(-\) \(z^2\) \(-\) \(x^2\) \(+\) \(y^2\) \(-\) \(2yz\) \(+\) \(z^2\) \(={-}4yz\)
- \({\largeк)}\ (x+y-z)(x-y+z)-(x+y+z)(x-y-z)=(x+(y-z))(x-(y-z))-(x+(y+z))(x-(y+z))=x^2-(y-z)^2-(x^2-(y+z)^2)=x^2-(y-z)^2-x^2+(y+z)^2=x^2-(y^2-2yz+z^2)-x^2+y^2+2yz+z^2=\) \(x^2\) \(-\) \(y^2\) \(+\) \(2yz\) \(-\) \(z^2\) \(-\) \(x^2\) \(+\) \(y^2\) \(+\) \(2yz\) \(+\) \(z^2\) \(=4yz\)
