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d
\(x^4+2x^3-4x-4\)
\(=\left(x^4+2x^3+x^2\right)-\left(x^2+4x+4\right)\)
\(=\left(x^2+x\right)^2-\left(x+2\right)^2\)
\(=\left(x^2-2\right)\left(x^2+2x+2\right)\)
d: \(=\left(x^2-2\right)\left(x^2+2\right)+2x\left(x^2-2\right)\)
\(=\left(x^2-2\right)\left(x^2+2x+2\right)\)
e: \(=27x^3-9x^2-18x^2+6x+12x-4\)
\(=\left(3x-1\right)\left(x^2-6x+4\right)\)
Bài giải:
a) x3 + = x3 + ()3 = (x + )(x2 – x . + ()2)
=(x + )(x2 – x + )
b) (a + b)3 – (a - b)3
= [(a + b) – (a – b)][(a + b)2 + (a + b) . (a – b) + (a – b)2]
= (a + b – a + b)(a2 + 2ab + b2 + a2 – b2 + a2 – 2ab + b2)
= 2b . (3a3 + b2)
c) (a + b)3 + (a – b)3 = [(a + b) + (a – b)][(a + b)2 – (a + b)(a – b) + (a – b)2]
= (a + b + a – b)(a2 + 2ab + b2 – a2 +b2 + a2 – 2ab + b2]
= 2a . (a2 + 3b2)
d) 8x3 + 12x2y + 6xy2 + y3 = (2x)3 + 3 . (2x)2 . y +3 . 2x . y + y3 = (2x + y)3
e) - x3 + 9x2 – 27x + 27 = 27 – 27x + 9x2 – x3 = 33 – 3 . 32 . x + 3 . 3 . x2 – x3 = (3 – x)3
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a: x^3-7x-6
=x^3-x-6x-6
=x(x-1)(x+1)-6(x+1)
=(x+1)(x^2-x-6)
=(x-3)(x+2)(x+1)
b: =2x^3+x^2-2x^2-x+6x+3
=x^2(2x+1)-x(2x+1)+3(2x+1)
=(2x+1)(x^2-x+3)
c: =2x^3-3x^2-2x^2+3x+2x-3
=x^2(2x-3)-x(2x-3)+(2x-3)
=(2x-3)(x^2-x+1)
d: =2x^3+x^2+2x^2+x+2x+1
=(2x+1)(x^2+x+1)
e: =3x^3+x^2-3x^2-x+6x+2
=(3x+1)(x^2-x+2)
f: =27x^3-9x^2-18x^2+6x+12x-4
=(3x-1)(9x^2-6x+4)
a) \(x^3-7x-6\)
\(=x^3-x-6x-6\)
\(=\left(x^3-x\right)-\left(6x+6\right)\)
\(=x\left(x^2-1\right)-6\left(x+1\right)\)
\(=x\left(x+1\right)\left(x-1\right)-6\left(x+1\right)\)
\(=\left(x+1\right)\left(x^2-x-6\right)\)
b) \(2x^3-x^2+5x+3\)
\(=2x^3+x^2-2x^2-x+6x+3\)
\(=\left(2x^3+x^2\right)-\left(2x^2+x\right)+\left(6x+3\right)\)
\(=x^2\left(2x+1\right)-x\left(2x+1\right)+3\left(2x+1\right)\)
\(=\left(x^2-x+3\right)\left(2x+1\right)\)
c) \(2x^3-5x^2+5x+1\)
\(=2x^3-3x^2-2x^2+3x+2x-3\)
\(=\left(2x^3-3x^2\right)-\left(2x^2-3x\right)+\left(2x-3\right)\)
\(=x^2\left(2x-3\right)-x\left(2x-3\right)+\left(2x-3\right)\)
\(=\left(x^2-x+1\right)\left(2x-3\right)\)
d) \(2x^3+3x^2+3x+1\)
\(=2x^3+x^2+2x^2+x+2x+1\)
\(=\left(2x^3+x^2\right)+\left(2x^2+x\right)+\left(2x+1\right)\)
\(=x^2\left(2x+1\right)+x\left(2x+1\right)+\left(2x+1\right)\)
\(=\left(2x+1\right)\left(x^2+x+1\right)\)
e) \(3x^3-2x^2+5x+2\)
\(=3x^3+x^2-3x^2-x+6x+2\)
\(=\left(3x^3+x^2\right)-\left(3x^2+x\right)+\left(6x+2\right)\)
\(=x^2\left(3x+1\right)-x\left(3x+1\right)+2\left(3x+1\right)\)
\(=\left(3x-1\right)\left(x^2-x+2\right)\)
f) \(27x^3-27x^2+18x-4\)
\(=27x^3-9x^2-18x^2+6x+12x-4\)
\(=\left(27x^3-9x^2\right)-\left(18x^2-6x\right)+\left(12x-4\right)\)
\(=9x^2\left(3x-1\right)-6x\left(3x-1\right)+4\left(3x-1\right)\)
\(=\left(3x-1\right)\left(9x^2-6x+4\right)\)
1) \(\left(a-b\right)\left(c-a\right)\left(c-b\right)\left(c+b+a\right)\)
\(a.\) \(ax^2-a^2x-x+a\)
\(=\left(ax^2-a^2x\right)-\left(x-a\right)\)
\(=ax\left(x-a\right)-\left(x-a\right)\)
\(=\left(ax-1\right)\left(x-a\right)\)
\(b.\) \(18x^3-12x^2+2x\)
\(=2x\left(9x^2-6x+1\right)\)
\(=2x\left(3x-1\right)^2\)
\(c.\) \(x^3-5x^2-4x+20\)
\(=\left(x^3-5x^2\right)-\left(4x-20\right)\)
\(=x^2\left(x-5\right)-4\left(x-5\right)\)
\(=\left(x^2-4\right)\left(x-5\right)\)
\(=\left(x-2\right)\left(x+2\right)\left(x-5\right)\)
\(d.\) \(\left(x+7\right)\left(x+15\right)+15\)
\(=x^2+15x+7x+105+15\)
\(=x^2+22x+120\)
\(=\left(x+10\right)\left(x+12\right)\)
Phân tích đa thức sau thành nhân tử :
\(a^3\left(b-c\right)+b^3\left(c-a\right)+c^3\left(a-b\right)\)
\(a^3\left(b-c\right)+b^3\left(c-a\right)+c^3\left(a-b\right)=a^3\left(b-c\right)+b^3c-b^3a+c^3a-c^3b\\ \)
\(\Rightarrow\)\(a^3\left(b-c\right)+bc\left(b^2-c^2\right)-a\left(b^3-c^3\right)\)
\(\Rightarrow\)\(a^3\left(b-c\right)+bc\left(b-c\right)\left(b+c\right)-a\left(b-c\right)\left(b^2+bc+c^2\right)\)
\(\Rightarrow\)\(\left(b-c\right)\left(a^3+bc\left(b+c\right)-a\left(b^2+bc+c^2\right)\right)\)
\(\Rightarrow\)\(\left(b-c\right)\left(a^3+b^2c+bc^2-ab^2-abc-ac^2\right)\)
\(\Rightarrow\)\(\left(b-c\right)\left(bc\left(c-a\right)+b^2\left(c-a\right)-a\left(c^2-a^2\right)\right)\)
\(\Rightarrow\)\(\left(b-c\right)\left(c-a\right)\left(bc+b^2-a\left(c+a\right)\right)\)
\(\Rightarrow\)\(\left(b-c\right)\left(c-a\right)\left(bc+b^2-ac-a^2\right)\)
\(\left(b-c\right)\left(c-a\right)\left(b^2-a^2+c\left(b-a\right)\right)=\left(b-c\right)\left(c-a\right)\left(b-a\right)\left(a+b+c\right)\)
Lời giải:
e)
\(27x^3-27x^2+18x-4\)
\(=(27x^3-9x^2)-(18x^2-6x)+(12x-4)\)
\(=9x^2(3x-1)-6x(3x-1)+4(3x-1)\)
\(=(9x^2-6x+4)(3x-1)\)
f)
\(a(b^3-c^3)+b(c^3-a^3)+c(a^3-b^3)\)
\(=a(b^3-c^3)-b[(b^3-c^3)+(a^3-b^3)]+c(a^3-b^3)\)
\(=a(b^3-c^3)-b(b^3-c^3)-b(a^3-b^3)+c(a^3-b^3)\)
\(=(a-b)(b^3-c^3)-(b-c)(a^3-b^3)\)
\(=(a-b)(b-c)(b^2+bc+c^2)-(b-c)(a-b)(a^2+ab+b^2)\)
\(=(a-b)(b-c)(b^2+bc+c^2-a^2-ab-b^2)\)
\(=(a-b)(b-c)(bc-ab+c^2-a^2)=(a-b)(b-c)[b(c-a)+(c-a)(c+a)]\)
\(=(a-b)(b-c)(c-a)(b+c+a)\)