多項式乘法 [20點]

(a) (x+1) (x^2-x-4)

= x ( x^2 - x - 4 ) + 1 ( x^2 - x - 4 )

= x^3 - x^2 - 4x + x^2 - x - 4

= x^3 - 5x - 4

(b) (2+a) (a^2+3a-5)

= 2 ( a^2 + 3a - 5 ) + a ( a^2 + 3a - 5 )

= 2a^2 + 6a - 10 + a^3 + 3a^2 - 5a

= a^3 + 5a^2 + a - 10

(c) (y^2+3y-3) (y-2)

= y ( y^2 + 3y - 3 ) - 2 ( y^2 + 3y - 3 )

= y^3 + 3y^2 - 3y - 2y^2 - 6y + 6

= y^3 + y^2 - 9y + 6

(d) (a^2+2a+1) (3-a)

= 3 ( a^2 + 2a + 1 ) - a ( a^2 + 2a + 1 )

= 3a^2 + 6a + 3 - a^3 - 2a^2 - a

= -a^3 + a^2+ 5a + 3

(a) (x+1)(x^2-x-4)
=x^3-x^2-4x+x^2-x-4
=x^3-5x-4
(b) (2+a)(a^2+3a-5)
=a^3+3a^2-5a+2a^2+6a-10
=a^3+5a^2+a-10
(c) (y^2+3y-3)(y-2)
=y^3-2y^2+3y^2-6y-3y+6
=y^3+y^2-9y+6
(d) (a^2+2a+1)(3-a)
=3a^2-a^3+6a-2a^2+3-a
=-a^3+a^2+5a+3

a)
(x+1)(x^2-x-4)
=x(x^2-x-4)+1(x^2-x-4)
=x^3-x^2-4x+x^2-x-4
=x^3-5x-4

b)
(2+a)(a^2+3a-5)
=2(a^2+3a-5)+a(a^2+3a-5)
=2a^2+6a-10+a^3+3a^2-5a
=a^3+5a^2+a-10

c)
(y^2+3y-3)(y-2)
=y^2(y-2)+3y(y-2)-3(y-2)
=y^3-2y^2+3y^2-6y-3y+6
=y^3+y^2-9y+6

d)
(a^2+2a+1)(3-a)
=a^2(3-a)+2a(3-a)+1(3-a)
=3a^2-a^3+6a-2a^2+3-a
=-a^3+a^2+5a+3