a) 52x2y+35xy2+xy352x2y+35xy2+xy3
=5.5y22x2y.5y2+3.2xy5xy2.2xy+x.10x2y3.10x2=5.5y22x2y.5y2+3.2xy5xy2.2xy+x.10x2y3.10x2
=25y210x2y3+6xy10x2y3+10x310x2y3=25y210x2y3+6xy10x2y3+10x310x2y3
=25y2+6xy+10x310x2y3=25y2+6xy+10x310x2y3
b) x+12x+6+2x+3x(x+3)=x+12(x+3)+2x+3x(x+3)x+12x+6+2x+3x(x+3)=x+12(x+3)+2x+3x(x+3)
=x(x+1)2x(x+3)+2(2x+3)2x(x+3)=x2+x+4x+62x(x+3)=x(x+1)2x(x+3)+2(2x+3)2x(x+3)=x2+x+4x+62x(x+3)
=x2+5x+62x(x+3)=x2+2x+3x+62x(x+3)=x2+5x+62x(x+3)=x2+2x+3x+62x(x+3)
=x(x+2)+3(x+2)2x(x+3)=x(x+2)+3(x+2)2x(x+3)=(x+2)(x+3)2x(x+3)=x+22x=(x+2)(x+3)2x(x+3)=x+22x
c) 3x+5x2−5x+25−x25−5x=3x+5x2−5x+x−255x−253x+5x2−5x+25−x25−5x=3x+5x2−5x+x−255x−25
=3x+5x(x−5)+x−255(x−5)=5(3x+5)5x(x−5)+x(x−25)5x(x−5)=3x+5x(x−5)+x−255(x−5)=5(3x+5)5x(x−5)+x(x−25)5x(x−5)
=15x+25+x2−25x5x(x−5)=x2−10x+255x(x−5)=15x+25+x2−25x5x(x−5)=x2−10x+255x(x−5)
=(x−5)25x(x−5)=x−55x=(x−5)25x(x−5)=x−55x
d) x2+x4+11−x2+1=1+x2+x4+11−x2x2+x4+11−x2+1=1+x2+x4+11−x2
=(1+x2)(1−x2)1−x2+x4+11−x2=(1+x2)(1−x2)1−x2+x4+11−x2
=1−x4+x4+11−x2=21−x2=1−x4+x4+11−x2=21−x2
e) 4x2−3x+17x3−1+2x−1x2+x+1+61−x4x2−3x+17x3−1+2x−1x2+x+1+61−x
4x2−3x+17(x−1)(x2+x+1)+2x−1x2+x+1+−6x−14x2−3x+17(x−1)(x2+x+1)+2x−1x2+x+1+−6x−1
=4x2−3x+17(x−1)(x2+x+1)+(2x−1)(x−1)(x−1)(x2+x+1)=4x2−3x+17(x−1)(x2+x+1)+(2x−1)(x−1)(x−1)(x2+x+1)+−6(x2+x+1)(x−1)(x2+x+1)+−6(x2+x+1)(x−1)(x2+x+1)
=4x2−3x+17+(2x−1)(x−1)−6(x2+x+1)(x−1)(x2+x+1)=4x2−3x+17+(2x−1)(x−1)−6(x2+x+1)(x−1)(x2+x+1)
=4x2−3x+17+2x2−3x+1−6x2−6x−6(x−1)(x2+x+1)=4x2−3x+17+2x2−3x+1−6x2−6x−6(x−1)(x2+x+1)
=−12x+12(x−1)(x2+x+1)=−12(x−1)(x−1)(x2+x+1)=−12x+12(x−1)(x2+x+1)=−12(x−1)(x−1)(x2+x+1)
=−12x2+x+1=−12x2+x+1
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