Ôn tập phép nhân và phép chia đa thức

Lời giải:

\(x^4+y^4=x^4+2.x^2y^2+y^2-2(xy)^2\)

\(=(x^2+y^2)^2-2(xy)^2=[(x+y)^2-2xy]^2-2(xy)^2\)

\(=(a^2-2b)^2-2b^2\)

\(=a^4-4a^2b+2b^2\)

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\(x^5+y^5=(x^4+y^4)(x+y)-x^4y-xy^4\)

\(=(x^4+y^4)(x+y)-xy(x^3+y^3)=(x^4+y^4)(x+y)-xy[(x+y)^3-3xy(x+y)]\)

\(=(a^4-4a^2b+2b^2)a-b(a^3-3ab)\)

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

Câu 1:

\(a(b^3-c^3)+b(c^3-a^3)+c(a^3-b^3)\)

\(=a(b^3-c^3)-b(a^3-c^3)+c(a^3-b^3)\)

\(=a(b^3-c^3)-b[(b^3-c^3)+(a^3-b^3)]+c(a^3-b^3)\)

\(=(a-b)(b^3-c^3)-(b-c)(a^3-b^3)\)

\(=(a-b)(b-c)(b^2+bc+c^2)-(b-c)(a-b)(a^2+ab+b^2)\)

\(=(a-b)(b-c)(b^2+bc+c^2-a^2-ab-b^2)\)

\(=(a-b)(b-c)(c^2+bc-a^2-ab)\)

\(=(a-b)(b-c)[(c^2-a^2)+b(c-a)]\)

\(=(a-b)(b-c)(c-a)(c+a+b)\)

Câu 2:

\(a^3(b^2-c^2)+b^3(c^2-a^2)+c^3(a^2-b^2)\)

\(=a^3(b^2-c^2)-b^3(a^2-c^2)+c^3(a^2-b^2)\)

\(=a^3(b^2-c^2)-b^3[(b^2-c^2)+(a^2-b^2)]+c^3(a^2-b^2)\)

\(=(a^3-b^3)(b^2-c^2)-(b^3-c^3)(a^2-b^2)\)

\(=(a-b)(a^2+ab+b^2)(b-c)(b+c)-(b-c)(b^2+bc+c^2)(a-b)(a+b)\)

\(=(a-b)(b-c)[(b+c)(a^2+ab+b^2)-(a+b)(b^2+bc+c^2)]\)

\(=(a-b)(b-c)(a^2b+a^2c-ac^2-bc^2)\)

\(=(a-b)(b-c)[b(a^2-c^2)+ac(a-c)]\)

\(=(a-b)(b-c)[b(a-c)(a+c)+ac(a-c)]\)

\(=(a-b)(b-c)(a-c)(ab+bc+ac)\)