BÀI TẬP
Bài 2.26 : Phân tích các đa thức sau thành nhân tử:
a) $x^2$– 6x + 9 – $y^2$;
b) 4$x^2$ – $y^2$ + 4y – 4;
c) xy +$z^2$ + xz + yz;
d) $x^2$ – 4xy + 4$y^2$+ xz – 2yz.
Trả lời:
a) $x^2$ – 6x + 9 – $y^2$ = $(x-3)^2$ – $y^2$ = (x + y – 3)(x – y – 3);
b) 4$x^2$ – $y^2$ + 4y – 4 = 4$x^2$– ($y^2$– 4y + 4)
= $x^2$–$(y-2)^2$ = (2x + y – 2)(2x – y + 2);
c) Cách 1: xy + $z^2$ + xz + yz = (xy + xz) + (yz + $z^2$)
= x(y + z) + z(y + z) = (x + z)(y + z);
Cách 2: xy + $z^2$+ xz + yz = (xy + yz) + ($z^2$ + xz)
= y(x + z) + z(z + x) = (x + z)(y + z).
d) $x^2$– 4xy + 4$y^2$ + xz – 2yz = ($x^2$ – 4xy + 4$y^2$) + (xz – 2yz)
= $(x-2y)^2$ + z(x – 2y) = (x – 2y)(x – 2y + z).
Bài 2.27: Phân tích các đa thức sau thành nhân tử:
a) $x^3$ + $y^3$ + x + y;
b) $x^3$ – $y^3$ + x – y;
c) $(x-y)^3$ + $(x+y)^3$ ;
d) $x^3$ – $3x^2y$ + $3xy^2$ – $y^3$ + $y^2$ –$x^2$ .
Trả lời:
a) $x^3$ + $y^3$ + x + y = ($x^3$ + $y^3$ ) + (x + y)
= (x + y)($x^2$ – xy + $y^2$ ) + (x + y)
= (x + y)($x^2$ – xy + $y^2$ + 1);
b) $x^3$ – $y^3$ + x – y = ($x^3$ – $y^3$ ) + (x – y)
= (x – y)($x^2$ + xy +$y^2$ ) + (x – y)
= (x – y)(x2 + xy + y2 + 1);
c) $(x-y)^3$ + $(x+y)^3$
= [(x – y) + (x + y)] [$(x-y)^2$ – (x – y)(x + y) +$(x+y)^2$]
= (x – y + x + y) [$(x-y)^2$– $(x^2-y^2)$ + $(x+y)^2$]
= 2x($x^2$– 2xy + $y^2$– $x^2$ + $y^2$ + $x^2$ + 2xy + $y^2$)
= 2x($x^2$ + 3$y^2$);
d) $x^3$ – 3$x^2$y + 3x$y^2$ – $y^3$ + $y^2$– $x^2$
= ($x^3$– 3$x^2$y + 3x$y^2$ – $y^3$) – ($x^2$ – $y^2$)
= $(x-y)^3$ – (x – y)(x + y)
= (x – y) [ $(x-y)^2$– (x + y)]
= (x – y) ($x^2$– 2xy + $y^2$– x – y).
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