Question

Phân tích đa thức sau thành nhân tử:

(x + 2)(x + 3)(x + 4)(x + 5) - 24

Answer 1

\(A=\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)-24\)

\(A=\left(x^2+7x+10\right)\left(x^2+7x+12\right)-24\)

\(A=\left(x^2+7x+11\right)^2-1-24\)

\(A=\left(x^2+7x+6\right)\left(x^2+7x+17\right)\)

\(A=\left(x+1\right)\left(x+6\right)\left(x^2+7x+17\right)\)

Answer 2

(x+2)(x+3)(x+4)(x+5)-24

=(x²+7x+10)(x²+7x+12)-24

Đặt x²+7x+11=a

(a-1)(a+1)-24

=a²-1-24

=a²-25

=(a-5)(a+5)

=(x²+7x+11-5)(x²+7x+11+5)

=(x²+7x+6)(x²+7x+16)

Answer 3

( x + 2 ) . ( x + 3 ) . ( x + 4 ) . ( x + 5 ) - 24

= [ ( x + 2 ) . ( x + 5 ) ] . [ ( x + 3 ) . ( x + 4 ) ] - 24

=( x² + 7x + 10 ) . ( x² + 7x + 12 ) - 24

Đặt x² + 7x + 11 = y ta có

( y - 1 ) . ( y + 1 ) - 24 = y² - 1 - 24

= y² - 25

= ( y + 5 ) . ( y - 5 )

Thay y = x² + 7x + 11 vào biểu thức ta có :

( x² + 7x + 11 + 5 ) . ( x² + 7x + 11 - 5 )

= ( x² + 7x + 16 ) . ( x² + 7x + 6 )

= ( x² + 7x + 16 ) . ( x² + x + 6x + 6 )

= ( x² + 7x + 16 ) . [ ( x² + x ) + ( 6x + 6 )]

= ( x² + 7x + 16 ) . [ x . ( x + 1 ) + 6 . ( x + 1 ) ]

= ( x² + 7x + 16 ) . ( x + 1 ) . ( x + 6 )