Question

Разложите на множители выражение 1) a^2+b^2+2a-2b-2ab 2) a^6-8 3) (x-5)^2+(x-5)(x+5) 4) -0.027x^6+y^3/64 5) 2a^2-5a+2/ab-2b-3a+6

Answer 1

1) a^2+b^2+2a-2b-2ab=(a^2-2ab+b^2)+(2a-2b)=(a-b)^2+2(a-b)=(a-b)(a-b+2)

2) a^6-8=(a^2)^3-2^3=(a^2-2)(a^4+2a^2+4)

3) (x-5)^2+(x-5)(x+5)=(x-5)(x-5+x+5)=2x(x-5)

4) -0,027x^6+y^3/64=(-0,3)x^2)^3+(y/4)^3=(-0,3x^2+y/4)(0,09x^4-(-0,3x^2y/4)+y^2/16)=

=(0,25y-0,3x^2)(0,09x^4+0,075x^2y+0,0625y^2)

5) (2a^2-5a+2)/(ab-2b-3a+6)=(2a^2-a-4a+2)/(b(a-2)-3(a-2))=

=(a(2a-1)-2(2a-1))/((a-2)(b-3))=((2a-1)(a-2))/((a-2)(b-3))=(2a-1)/(b-3)

Answer 2

1) $a^{2}+b^{2}+2a-2b-2ab=(a^{2}-2ab+b^{2})+(2a-2b)=(a-b)^{2}+2(a-b)=(a-b)(a+2-b)$

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2) $a^{6}-8=(a^{2})^{3}-2^{3}=(a^{2}-2)(a^{4}+2a^{2}+4)$

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3) $(x-5)^{2}+(x-5)(x+5)=(x-5)(x-5)+(x-5)(x+5)=(x-5)(x-5+x+5)=(x-5)((x+x)+(5-5))=2x(x-5)$

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4) $-0,027x^{6}+\frac{y^{3}}{64}=-\frac{27}{1000}x^{6}+\frac{y^{3}}{64}=((-\frac{3}{10})x^{2})^{3}+(\frac{y}{4})^{3}=$

$(-\frac{3}{10}x^{2}+\frac{y}{4})(\frac{9}{100}x^{4}-(-\frac{3}{10}x^{2}\cdot\frac{y}{4})+\frac{y^{2}}{16})=$

$(-\frac{3}{10}x^{2}+\frac{1}{4}y)(\frac{9}{100}x^{4}+\frac{3}{40}x^{2}y+\frac{y^{2}}{16})= (\frac{1}{4}y-\frac{3}{10}x^{2})(\frac{9}{100}x^{4}+\frac{3}{40}x^{2}y+\frac{y^{2}}{16})$

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5) $\frac{2a^{2}-5a+2}{ab-2b-3a+6}=\frac{2a^{2}+(-a-4a)+2}{(ab-2b)+(-3a+6)}=\frac{(2a^{2}-a)+(-4a+2)}{(ab-2b)-(3a-6)}=\frac{(2a^{2}-a)-(4a-2)}{b(a-2)-3(a-2)}=\frac{a(2a-1)-2(2a-1)}{(a-2)(b-3)}=$

$\frac{(2a-1)(a-2)}{(a-2)(b-3)}=\frac{2a-1}{b-3}$