Phân tích đa thức thành nhân tử:
a) $4x^{4}$+$4x^{3}$-$x^{2}$-x
b) $x^{6}$-$x^{4}$-$9x^{3}$+$9x^{2}$
c) $(xy+4)^{2}$-$4(x+y)^{4}$
d) $4x^{4}$+1
e) $x^{4}$+324
f) $x^{3}$-$x^{2}$-x-2
g) $x^{3}$-$9x^{2}$+6x+16
h) $6x^{2}$-11x+3
i) $2x^{2}$-5xy-$3y^{2}$
k) (x+2)(x+3)(x+4)(x+5)-24
l) $x^{2}$+2xy+$y^{2}$-x-y-12
a)4x4+4x3−x2−x=(4x4+4x3)−(x2+x)=4x3(x+1)−x(x+1)=x(x+1)(4x2−1)=x(x+1)(2x−1)(2x+1)
b)x6−x4−9x3+9x2=(x6−x4)−(9x3−9x2)=x4(x2−1)−9x2(x−1)=x4(x−1)(x+1)−9x2(x−1)=x2(x−1)[x2(x+1)−9]
c)(xy+4)2−4(x+y)4=(xy+4)2−[2(x+y)2]2=(xy+4−2(x+y)2)(xy+4+2(x+y)2)
d)4x4+1=4x4+4x2+1−4x2=(2x2+1)−(2x)2=(2x2+1−2x)(2x2+1+2x
)e)x4+342=x4+36x2+342−36x2=(x2+18)2−(6x)2=(x2+18−6x)(x2+18+6x)
f)x3−x2−x−2=x3−8−x2−x+6=(x3−8)−(x2+x−6)=(x−2)(x2+2x+4)−(x2−2x+3x−6)=(x−2)(x2+2x+4)−[x(x−2)+3(x−2)]=(x−2)(x2+2x+4)−(x−2)(x+3)=(x−2)(x2+2x+4−x−3)=(x−2)(x2+x+1)
dạo này mik trả lời toàn bị báo cáo thôi chẳng bt ai báo cáo nữa
\(\eqalign{
& a)\,\,4{x^4} + 4{x^3} – {x^2} – x \cr
& = \left( {4{x^4} + 4{x^3}} \right) – \left( {{x^2} + x} \right) \cr
& = 4{x^3}\left( {x + 1} \right) – x\left( {x + 1} \right) \cr
& = x\left( {x + 1} \right)\left( {4{x^2} – 1} \right) \cr
& = x\left( {x + 1} \right)\left( {2x – 1} \right)\left( {2x + 1} \right) \cr
& b)\,\,{x^6} – {x^4} – 9{x^3} + 9{x^2} \cr
& = \left( {{x^6} – {x^4}} \right) – \left( {9{x^3} – 9{x^2}} \right) \cr
& = {x^4}\left( {{x^2} – 1} \right) – 9{x^2}\left( {x – 1} \right) \cr
& = {x^4}\left( {x – 1} \right)\left( {x + 1} \right) – 9{x^2}\left( {x – 1} \right) \cr
& = {x^2}\left( {x – 1} \right)\left[ {{x^2}\left( {x + 1} \right) – 9} \right] \cr
& c)\,\,{\left( {xy + 4} \right)^2} – 4{\left( {x + y} \right)^4} \cr
& = {\left( {xy + 4} \right)^2} – {\left[ {2{{\left( {x + y} \right)}^2}} \right]^2} \cr
& = \left( {xy + 4 – 2{{\left( {x + y} \right)}^2}} \right)\left( {xy + 4 + 2{{\left( {x + y} \right)}^2}} \right) \cr
& d)\,\,4{x^4} + 1 \cr
& = 4{x^4} + 4{x^2} + 1 – 4{x^2} \cr
& = \left( {2{x^2} + 1} \right) – {\left( {2x} \right)^2} \cr
& = \left( {2{x^2} + 1 – 2x} \right)\left( {2{x^2} + 1 + 2x} \right) \cr
& e)\,\,{x^4} + 342 \cr
& = {x^4} + 36{x^2} + 342 – 36{x^2} \cr
& = {\left( {{x^2} + 18} \right)^2} – {\left( {6x} \right)^2} \cr
& = \left( {{x^2} + 18 – 6x} \right)\left( {{x^2} + 18 + 6x} \right) \cr
& f)\,\,{x^3} – {x^2} – x – 2 \cr
& = {x^3} – 8 – {x^2} – x + 6 \cr
& = \left( {{x^3} – 8} \right) – \left( {{x^2} + x – 6} \right) \cr
& = \left( {x – 2} \right)\left( {{x^2} + 2x + 4} \right) – \left( {{x^2} – 2x + 3x – 6} \right) \cr
& = \left( {x – 2} \right)\left( {{x^2} + 2x + 4} \right) – \left[ {x\left( {x – 2} \right) + 3\left( {x – 2} \right)} \right] \cr
& = \left( {x – 2} \right)\left( {{x^2} + 2x + 4} \right) – \left( {x – 2} \right)\left( {x + 3} \right) \cr
& = \left( {x – 2} \right)\left( {{x^2} + 2x + 4 – x – 3} \right) \cr
& = \left( {x – 2} \right)\left( {{x^2} + x + 1} \right) \cr
& g)\,\,{x^3} – 9{x^2} + 6x + 16 \cr
& = {x^3} + {x^2} – 10{x^2} – 10x + 16x + 16 \cr
& = {x^2}\left( {x + 1} \right) – 10x\left( {x + 1} \right) + 16\left( {x + 1} \right) \cr
& = \left( {x + 1} \right)\left( {{x^2} – 10x + 16} \right) \cr
& = \left( {x + 1} \right)\left( {{x^2} – 2x – 8x + 16} \right) \cr
& = \left( {x + 1} \right)\left[ {x\left( {x – 2} \right) – 8\left( {x – 2} \right)} \right] \cr
& = \left( {x + 1} \right)\left( {x – 2} \right)\left( {x – 8} \right) \cr
& h)\,\,6{x^2} – 11x + 3 \cr
& = 6{x^2} – 9x – 2x + 3 \cr
& = 3x\left( {2x – 3} \right) – \left( {2x – 3} \right) \cr
& = \left( {2x – 3} \right)\left( {3x – 1} \right) \cr
& i)\,\,2{x^2} – 5xy – 3{y^2} \cr
& = 2{x^2} – 6xy + xy – 3{y^2} \cr
& = 2x\left( {x – 3y} \right) + y\left( {x – 3y} \right) \cr
& = \left( {x – 3y} \right)\left( {2x + y} \right) \cr} \)
\(\eqalign{
& k)\,\,\left( {x + 2} \right)\left( {x + 3} \right)\left( {x + 4} \right)\left( {x + 5} \right) – 24 \cr
& = \left( {x + 2} \right)\left( {x + 5} \right)\left( {x + 3} \right)\left( {x + 4} \right) – 24 \cr
& = \left( {{x^2} + 7x + 10} \right)\left( {{x^2} + 7x + 12} \right) – 24 \cr
& = {\left( {{x^2} + 7x} \right)^2} + 22\left( {{x^2} + 7x} \right) + 120 – 24 \cr
& = {\left( {{x^2} + 7x} \right)^2} + 22\left( {{x^2} + 7x} \right) + 96 \cr
& = {\left( {{x^2} + 7x} \right)^2} + 6\left( {{x^2} + 7x} \right) + 16\left( {{x^2} + 7x} \right) + 96 \cr
& = \left( {{x^2} + 7x} \right)\left( {{x^2} + 7x + 6} \right) + 16\left( {{x^2} + 7x + 6} \right) \cr
& = \left( {{x^2} + 7x + 6} \right)\left( {{x^2} + 7x + 16} \right) \cr
& l)\,\,{x^2} + 2xy + {y^2} – x – y – 12 \cr
& = {\left( {x + y} \right)^2} – \left( {x + y} \right) – 12 \cr
& = {\left( {x + y} \right)^2} – 4\left( {x + y} \right) + 3\left( {x + y} \right) – 12 \cr
& = \left( {x + y} \right)\left( {x + y – 4} \right) + 3\left( {x + y – 4} \right) \cr
& = \left( {x + y – 4} \right)\left( {x + y + 3} \right) \cr} \)