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a) (x-2)(x+2)(x^2-10)-72=(x^2-4)(x^2-82)
b) x^8+x^6+x^4+x^2+1=x^2 (x^4+x^3+x^2+1+1/x^2)
c)(x+y)^4+x^4+y^4=(x+y)^4+(x+y)^4=2 (x+y)^4
a) (x-2)(x+2)(x^2 - 10) -72
= (x^2 - 4)(x^2 - 10) - 72
= x^4 - 4x^2 -10x^2 + 40 - 72
= x^4 - 14x^2 - 32
= x^4 - 16x^2 + 2x^2 - 32
= x^2(x^2 - 16) + 2(x^2 - 16)
= (x^2 - 16)(x^2 + 2)
= (x-4)(x+4)(x^2 + 2)
c) (x+y)4 + x4 + y4
= 2x4 + 4xy3 + 6x2y2 + 4x3y + 2y3
= 2(y4 + 2xy3 + 3x2y2 + 2x3y + x4)
= 2(y2 + xy + y2)2
a)x^2-(a+b)x+ab
= x^2 - ax - bx + ab
= (x^2 - ax) - (bx - ab)
= x(x-a) - b(x-a)
= (x-b)(x-a)
b)7x^3-3xyz-21x^2+9z
=
c)4x+4y-x^2(x+y)
= 4(x + y) - x^2(x+y)
= (4-x^2) (x+y)
= (2-x)(2+x)(x+y)
d) y^2+y-x^2+x
= (y^2 - x^2) + (x+y)
= (y-x)(y+x)+ (x+y)
= (y-x+1) (x+y)
e)4x^2-2x-y^2-y
= [(2x)^2 - y^2] - (2x +y)
= (2x-y)(2x+y) - (2x+y)
= (2x -y -1)(2x+y)
f)9x^2-25y^2-6x+10y
=
\(1,\\ 1,=15\left(x+y\right)\\ 2,=4\left(2x-3y\right)\\ 3,=x\left(y-1\right)\\ 4,=2x\left(2x-3\right)\\ 2,\\ 1,=\left(x+y\right)\left(2-5a\right)\\ 2,=\left(x-5\right)\left(a^2-3\right)\\ 3,=\left(a-b\right)\left(4x+6xy\right)=2x\left(2+3y\right)\left(a-b\right)\\ 4,=\left(x-1\right)\left(3x+5\right)\\ 3,\\ A=13\left(87+12+1\right)=13\cdot100=1300\\ B=\left(x-3\right)\left(2x+y\right)=\left(13-3\right)\left(26+4\right)=10\cdot30=300\\ 4,\\ 1,\Rightarrow\left(x-5\right)\left(x-2\right)=0\Rightarrow\left[{}\begin{matrix}x=2\\x=5\end{matrix}\right.\\ 2,\Rightarrow\left(x-7\right)\left(x+2\right)=0\Rightarrow\left[{}\begin{matrix}x=7\\x=-2\end{matrix}\right.\\ 3,\Rightarrow\left(3x-1\right)\left(x-4\right)=0\Rightarrow\left[{}\begin{matrix}x=\dfrac{1}{3}\\x=4\end{matrix}\right.\\ 4,\Rightarrow\left(2x+3\right)\left(2x-1\right)=0\\ \Rightarrow\left[{}\begin{matrix}x=-\dfrac{3}{2}\\x=\dfrac{1}{2}\end{matrix}\right.\)
\(a,\left(x-2\right)\left(x+2\right)\left(x^2-10\right)-72\)
\(=\left(x^2-4\right)\left(x^2-10\right)-72\)
\(=x^4-14x^2+40-72\)
\(=x^4-14x^2-32\)
\(=x^4-16x^2+2x^2-32\)
\(=\left(x^2-16\right)\left(x^2+2\right)\)
\(=\left(x-4\right)\left(x+4\right)\left(x^2+2\right)\)
\(b,x^8+x^6+x^4+x^2+1\)
\(=x^8+x^7+x^6+x^5+x^4-x^7-x^6-x^5-x^4-x^3+x^6+x^5+x^4+x^3+x^2-x^5-x^4-x^3-x^2-x+x^4+x^3+x^2+x+1\)
\(=x^4\left(x^4+x^3+x^2+x+1\right)-x^3\left(x^4+x^3+x^2+x+1\right)+x^2\left(x^4+x^3+x^2+x+1\right)-x\left(x^4+x^3+x^2+x+1\right)+\left(x^4+x^3+x^2+x+1\right)\)
\(=\left(x^4+x^3+x^2+x+1\right)\left(x^4-x^3+x^2-x+1\right)\)
\(c,\left(x+y\right)^4+x^4+y^4\)
\(=x^4+4xy^3+6x^2y^2+4x^3y+y^4+x^4+y^4\)
\(=2x^4+2y^4+4xy^3+4x^3y+6x^2y^2\)
\(=2\left(x^4+y^4+2xy^3+2x^3y+3x^2y^2\right)\)
\(=2\left(x^2+y^2+xy\right)^2\)
\(d,\left(x+1\right)^4+\left(x^2+x+1\right)^2\)
\(=x^4+4x^3+4x+6x^2+1+x^4+x^2+1+2x^3+2x+2x^2\)
\(=2x^4+6x^3+9x^2+6x+2\)
\(=2x^4+2x^3+x^2+4x^3+4x^2+2x+4x^2+4x+2\)
\(=x^2\left(2x^2+2x+1\right)+2x\left(2x^2+2x+1\right)+2\left(2x^2+2x+1\right)\)
\(=\left(2x^2+2x+1\right)\left(x^2+2x+2\right)\)