BÀI TẬP
8.13. Khai triển các đa thức
a) $(x – 2)^{4}$;
b) $(x + 2)^{5}$;
c) $(2x + 3y)^{4}$;
d) $(2x – y)^{5}$.
Bài Làm:
a) $(x – 2)^{4} = [x + (– 2)^{4}]$
$=C_{4}^{0}\times x^{4}+C_{4}^{1}\times x^{3}\times (−2)+C_{4}^{2}\times x^{2}\times (−2)^{2}+C_{4}^{3}\times x\times (−2)^{3}+C_{4}^{4}\times (−2)^{4}$
$= 1\times x^{4} + 4\times x^{3}\times (–2) + 6\times x^{2}\times 4 + 4\times x\times (–8) + 1\times 16$
$= x^{4} – 8x^{3} + 24x^{2} – 32x + 16.$
b)$(x+2)^{5}=C_{5}^{0}\times x^{5}+C_{5}^{1}\times x^{4}\times 2+C_{5}^{2}\times x^{3}\times 2^{2}+C_{5}^{3}\times x^{2}\times 2^{3}+C_{5}^{4}\times x\times 2^{4}+C_{5}^{5}\times 2^{5}$
$= 1\times x^{5} + 5\times x^{4}\times 2 + 10\times x^{3}\times 4 + 10\times x^{2}\times 8 + 5\times x\times 16 + 1\times 32$
$= x^{5} + 10x^{4} + 40x^{3} + 80x^{2} + 80x + 32.$
c) $(2x + 3y)^{4}$
$=C_{4}^{0}\times (2x)^{4}+C_{4}^{1}\times (2x)^{3}\times 3y+C_{4}^{2}\times (2x)^{2}\times (3y)^{2}+C_{4}^{3}\times 2x\times (3y)^{3}+C_{4}^{4}\times (3y)$
$= 1\times 16x^{4} + 4\times 8x^{3}\times 3y + 6\times 4x^{2}\times 9y^{2} + 4\times 2x\times 27y^{3} + 1\times 81y^{4}$
$= 16x^{4} + 96x^{3}y + 216x^{2}y^{2} + 216xy^{3} + 81y^{4}.$
d) $(2x – y)^{5} = [2x + (– y)^{5}]$
$=C_{5}^{0}\times (2x)^{5}+C_{5}^{1}\times (2x)^{4}\times (−y)+C_{5}^{2}\times (2x)^{3}\times (−y)^{2}+C_{5}^{3}\times (2x)^{2}\times (−y)^{3}$
$= 1\times 32x^{5} + 5\times 16x^{4}\times (–y) + 10\times 8x^{3}\times y^{2} + 10\times 4x^{2}\times (–y)^{3} + 5\times 2x\times y^{4} + 1\times (–y)^{5}$
$= 32x^{5} – 80x^{4}y + 80x^{3}y^{2} – 40x^{2}y^{3} + 10xy^{4} – y^{5}.$