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1 , \(x^5+x^4+1=\left(x^5+x^4+x^3\right)-\left(x^3+x^2+x\right)+\left(x^2+x+1\right)\)
= \(x^3\left(x^2+x+1\right)-x\left(x^2+x+1\right)+\left(x^2+x+1\right)\)=\(\left(x^2+x+1\right)\left(x^3-x+1\right)\)
2 , \(x\left(x+4\right)\left(x+6\right)\left(x+10\right)+128=\left(x^2+10x\right)\left(x^2+10x+24\right)+128\)(*)
Đặt x2 + 10 = a , a>0 (1)
=> (*) <=> a(a+24)+128=a2 + 24a+128=(a+8)(a+16) (**)
Thay (1) vào (**) ta được :
(*) <=> \(\left(x^2+10+8\right)\left(x^2+10+16\right)\)
a) \(5x-10x^2\) = \(5x\left(1-2x\right)\)
b) Mạn phép sửa đề:
\(\dfrac{1}{2}x\left(x^2-4\right)+4\left(x+2\right)\) = \(\left(x+2\right)\left[\dfrac{1}{2}x\left(x-2\right)+4\right]\)
= \(\left(x+2\right)\left(\dfrac{1}{2}x^2-x+4\right)\)
c) \(x^4-y^6=\left(x^2-y^3\right)\left(x^2+y^3\right)\)
e) \(x^3-4x^2+4x-1=x^3-x^2-3x^2+3x+x-1\)
= \(x^2\left(x-1\right)-3x\left(x-1\right)+\left(x-1\right)\)
= \(\left(x-1\right)\left(x^2-3x+1\right)\)
g) \(x^4+6x^3-12x^2-8x\)
= \(x\left(x^3-2x^2+8x^2-16x+4x-8\right)\)
= \(x\left[x^2\left(x-2\right)+8x\left(x-2\right)+4\left(x-2\right)\right]\)
= \(x\left(x-2\right)\left(x^2+8x+4\right)\)
h) \(\left(x^2+4x+8\right)^2+3x\left(x^2+4x+8\right)+2x^2\) (*)
Đặt \(x^2+4x+8=a\) => (*) trở thành:
\(a^2+3ax+2x^2\) = \(a^2+ãx+2ax+x^2\)
= \(a\left(a+x\right)+2x\left(a+x\right)\)
= \(\left(a+x\right)\left(a+2x\right)\) (1)
Thay \(a=x^2+4x+8\) vào (1) ta được:
\(\left(x^2+5x+8\right)\left(x^2+6x+8\right)\)
=\(\left(x^2+5x+8\right)\left(x^2+2x+4x+8\right)\)
= \(\left(x^2+5x+8\right)\left[x\left(x+2\right)+4\left(x+2\right)\right]\)
= \(\left(x+2\right)\left(x+4\right)\left(x^2+5x+8\right)\)
P/s: Còn câu f đang suy nghĩ!
a)\(\left(4x^3-xy^2+y^3\right)\left(x^2y+2xy^2-2y^3\right)\)
\(=x^2y\left(4x^3-xy^2+y^3\right)+2xy^2\left(4x^3-xy^2+y^3\right)\)
\(-2y^3\left(4x^3-xy^2+y^3\right)\)
\(=4x^5y-x^3y^3+x^2y^4+8x^4y^2-2x^2y^4+2xy^5\)
\(-8x^3y^3+2xy^5-2y^6\)
\(=-2y^6+4x^5y+\left(2xy^5+2xy^5\right)+8x^4y^2+\left(x^2y^4-2x^2y^4\right)\)
\(-\left(x^3y^3+8x^3y^3\right)\)
\(=-2y^6+4x^5y+4xy^5+8x^4y^2-x^2y^4-9x^3y^3\)
b)
(!) \(2\left(x+y\right)^2-7\left(x+y\right)+5\)
\(=2\left(x+y\right)^2-2\left(x+y\right)-5\left(x+y\right)+5\)
\(=2\left(x+y\right)\left(x+y-1\right)-5\left(x+y-1\right)\)
\(=\left(2x+2y-5\right)\left(x+y-1\right)\)
(!!) \(\left(x+y+z\right)^2-x^2-y^2-z^2\)
\(=\left(x^2+y^2+z^2+2xy+2yz+2zx\right)-x^2-y^2-z^2\)
\(=2\left(xy+yz+zx\right)\)
\(1,x^3-7x+6\)
\(=x^3+3x^2-3x^2-9x+2x+6\)
\(=x^2\left(x+3\right)-3x\left(x+3\right)+2\left(x+3\right)\)
\(=\left(x+3\right)\left(x^2-3x+2\right)\)
\(=\left(x+3\right)\left(x^2-2x-x+2\right)\)
\(=\left(x+3\right)\left(x-2\right)\left(x-1\right)\)
\(2,x^3-9x^2+6x+16\)
\(=x^3+x^2-10x^2-10x+16x+16\)
\(=x^2\left(x+1\right)-10x\left(x+1\right)+16\left(x+1\right)\)
\(=\left(x+1\right)\left(x^2-10x+16\right)\)
\(=\left(x+1\right)\left(x^2-2x-8x+16\right)\)
\(=\left(x+1\right)\left(x-8\right)\left(x-2\right)\)
mk ms lm hai câu thôi mà đã mệt r , bh mk lm bt mai đi học ,lúc khác lm đ cko bn
\(x^8+x^4+1\)
\(=\left(x^8+2x^4+1\right)-x^4\)
\(=\left(x^4+1\right)^2-x^4\)
\(=\left(x^4+1-x^2\right)\left(x^4+1+x^2\right)\)
\(=\left(x^4-x^2+1\right)\left(x^4+2x^2-x^2+1\right)\)
\(=\left(x^4-x^2+1\right)[\left(x^2+1\right)^2-x^2]\)
\(=\left(x^4-x^2+1\right)\left(x^2+1-x\right)\left(x^2+1+x\right)\)