Question

1.Zapisz w postaci sumu algebraicznej
a) (1-2x)^3

b)(5x+1)^3

c)(x-\sqrt{2})^3

d)(x+2)^2(x^2-2x+4)

e)(3x-1)^2(9x^2+3x+1)

f)(x^2+5x+25)(x-5)^2

g)(x+2)(x^2+4)(x-2)

h)(x-1)(x^2+x+1)(x^3+1)

i)(x+2)(x^4+4x^2+16)(x-2)

Answer 1

a) (1-2x)^3 = 1  - 6x + 12x²  - 8x³

 b)(5x+1)^3 = 125x³ + 75x² + 15x + 1

 c)(x-√2)^3 = x³  - 3√2x² + 6x - 2√2

 d)(x+2)^2(x^2-2x+4) = (x² + 4x + 4)(x² - 2x + 4) = x⁴ - 2x³ + 4x² + 4x³ - 8x² + 16x + 4x² - 8x + 16 = x⁴ + 2x³ + 8x + 16

 e)(3x-1)^2(9x^2+3x+1) = (9x² - 6x + 1)(9x² + 3x + 1) = 81x⁴ + 27x³ + 9x² - 54x³ - 18x² - 6x + 9x² + 3x + 1 = 81x⁴ -27x³ - 3x + 1

 f)(x^2+5x+25)(x-5)^2 = (x²+5x+25)(x² - 10x + 25) = x⁴ - 10x³ + 25x² + 5x³ - 50x² + 125x + 25x² - 250x + 625 = x⁴ -5x³ - 125x + 325

 g)(x+2)(x^2+4)(x-2) = (x²-4)(x²+4) = x⁴ - 16

h)(x-1)(x^2+x+1)(x^3+1) = (x³ + x² + x - x² - x - 1)(x³+1) = x⁵ - 1

i)(x+2)(x^4+4x^2+16)(x-2) = (x²-4)(x⁴+4x²+16) = x⁶ + 4x⁴ + 16x² - 4x⁴ - 16x² - 64 = x⁶ - 64

Answer 2

a) (1-2x)³ = 1³  - 3*2x + 3*4x²  - 8x³ = 1 -6x + 12x² - 8x³

 

 b)(5x+1)³ =(5x)³ + 3*25x² + 3*5x + 1³=125x³ + 75x² + 15x + 1

 

c)(x-√2)³ =x³-3*x²*√2+ 3*x*(√2)²-(√2)³=x³-3√2x²+6x - 2√2

 

 d)(x+2)²(x²-2x+4) = (x²+4x+4)(x²- 2x+4)=

=x⁴-2x³+4x²+4x³-8x²+16x+4x²-8x+16 = x⁴+2x³+8x+16

 

 e)(3x-1)²(9x²+3x+1) = (9x²-6x+1)(9x²+3x+1) =

=81x⁴+27x³+9x²-54x³-18x²-6x+9x²+3x+1=

=81x⁴+27x³-54x³+9x²-18x²+9x²-6x+3x+1=81x⁴-27x³-3x+1

 

 f)(x²+5x+25)(x-5)² = (x²+5x+25)(x²-10x+25) =

=x⁴-10x³+25x²+5x³-50x²+125x+25x²-250x+625=

=x⁴-10x³+5x³+25x²-50x²+25x²+125x-250x+625=

=x⁴-5x³ -125x+625

 

 g)(x+2)(x²+4)(x-2) =(x+2)(x-2)(x²+4)= (x²-4)(x²+4) =(x²)²-4²=x⁴-16

 

h)(x-1)(x²+x+1)(x³+1) = (x³+x²+x-x²-x-1)(x³+1) =

=(x³-1)(x³+1)=(x³)²-1²=x⁶ -1                      (x³ * x³ = x³⁺³ = x⁶)

 

 

i)(x+2)(x⁴+4x²+16)(x-2) = (x²-4)(x⁴+4x²+16) = x⁶+4x⁴+16x²-4x⁴-16x²-64=

=x⁶+4x⁴-4x⁴+16x²-16x²-64=x⁶-64

 

 

(a ±b)² = a²± 2ab + b²
a² - b² = (a - b)(a + b) 
(a ± b)³= a³ ± 3a²b + 3ab² ± b³