Question

Phân tích các đa thức sau thành nhân tử:

a) x³ - x² - 5x + 125

b) x³ + 2x² - 6x - 27

c) 12x³ + 4x² - 27x - 9

d) x⁴ - 25x² + 20x - 4

e) x². (x² - 6) - x² + 9

g) x⁴ - 4x³ + 8x² - 16x + 16

h) (a + b + c)² + (a - b + c)² - 4b²

i) (x² + 8x - 34)² - (3x² - 8x - 2)²

Answer 1

a) x³ - x² - 5x + 125 = ( x\(^3\) + 125 ) - ( x\(^2\) + 5x ) = ( x + 5 ) ( x\(^2\) - 5x + 25 ) - x ( x + 5 )

= ( x + 5 ) ( x\(^2\) - 5x + 25 - x ) = ( x + 5 ) ( x\(^2\) - 6x + 25 )

b) x³ + 2x² - 6x - 27 = ( x\(^3\) - 27 ) + ( 2x\(^2\) - 6x ) = ( x - 3 ) ( x\(^2\) + 3x + 9 ) + 2x ( x - 3 )

= ( x- 3 ) ( x\(^2\) + 3x + 9 + 2x ) = ( x - 3 ) ( x\(^2\) + 5x + 9 )

c) 12x³ + 4x² - 27x - 9 = (12x3 + 4x2 ) - ( 27x + 9 ) = 4x\(^2\)( 3x + 1 ) - 9 ( 3x + 1 )

= ( 4x\(^2\) - 9 ) ( 3x -1 ) = ( 2x - 3 ) ( 2x + 3 ) ( 3x - 1 )

i) (x² + 8x - 34)² - (3x² - 8x - 2)²

= (x2 + 8x - 34- 3x\(^2\) + 8x + 2) ( x\(^2\) + 8x - 34+3x2 - 8x - 2)

= ( -2x\(^2\) + 16x - 32 ) ( 4x\(^2\) - 36 )

= ( -2x\(^2\) + 16x - 32 ) ( 2x - 6 ) ( 2x + 6 )

Answer 2

a, x³- x²- 5x+ 125

=- (x²+ 5x)+ (x³+125)

=-x(x+5)+ (x+5)(x²- 5x+ 25)

=(x+5)(-x+x²- 5x+ 25)

=(x+ 5)(x²- 6x+ 25)

Answer 3

g, x⁴- 4x³+ 8x²- 16x+ 16

=(x⁴+ 8x²+16)- (4x³+16x)

=(x²+4)- 4x(x²+4)

=(x²+4)(1- 4x)

Answer 4

h, (a+ b+ c)²+ (a- b+ c)²- 4b²

=(a+ b+ c)²+ (a- b+c- 2b)(a- b+ c+ 2b)

=(a+b+c)²+(a-3b+c)(a+b+c)

=(a+b+c)(a+b+c+a-3b+c)

=(a+b+c)(2a-2b+2c)

=2(a+b+c)(a-b+c)