Question

1.Wykonaj potęgowanie

a) (x³)⁷ , (2y⁵)³, (- x⁷)³, (-2x³)⁴
b) (2x – 3)², (3 – 5x)², (0,2x – 1)², (1/3x – 3)²
c) (x – 4)², (2x + √2)², (x - √2 + 1)² , (x + 3 - √3)²
d) (3√2x – 2√2)², (√3x + 2√3)², (Ułamek licznik √5 mianownik 5, x – 2√5)² , (2√7x + ułamek licznik 7x, mianownik √7)²

2.Wykonaj potęgowanie

a) (2x + 1)³ , (2x – 3)³, (2x – 1)³
b) (4x – 3)³ , (3x – 4)³, (4x – 1)³
c) (x2 – 1)³ , (x2 + 2)³ , (x2 + 1)³
d)(2x2 – 5x3)³ , (3x3 – 1)³ , (2z3 – 5)³

3. Oblicz iloczyn wielomianów S i R
a) S (x) = 2x² – 3 R (x) = x - 1
b) S (x) = -3x² + 5x – 1 R (x) = 2x + 1
c) S (x) = 2x² + 3x – 5 R (x) = -x² + x + 2
d) S (x) = 3x³ + 2x² + 1 R (x) = - 2x³ + x – 3

4. Dane są wielomiany W (x) = - x³ + 2x² – x + 5, P (x) = 2x² – x + 1
Q (x) = 5x + 2. Oblicz:
a) 2W (x) + 3P (x) – Q (x)
b) 2 [W (x) – P (x)] + Q (x)
c) W (x) – [P(x) + 3Q (x)]
d) 2W (x) + 3 [ Q (x) – 2P (x)]
e) 5Q (x) – [W (x) + 2P (x)]
f) 3 [W (x) – 2P (x)] – [P (x) + 3W (x)]

Answer 1

A) (x³)⁷=x²¹ , (2y⁵)³=8y¹⁵, (- x⁷)³=-x²¹, (-2x³)⁴=16x²
b) (2x – 3)²=4x²-12x+9
(3 – 5x)²=9-30x+25x²
(0,2x – 1)²=0,04x²-0,4x+1
(1/3x – 3)²=1/9x²-2x+9
c) (x – 4)²=x²-8x+16
(2x + √2)²=4x²+4√2x+2
(x - √2 + 1)²=x²+2+1-2√2x+2x-2√2=x²+2x-2√2x-2√2+3
(x + 3 - √3)²=x²+9+3-2√3x+6x-6√3=x²+6x-2√3x-6√3+12
d) (3√2x – 2√2)²=18x²-24x+8
(√3x + 2√3)²=3x²+12x+12
(Ułamek licznik √5 mianownik 5, x – 2√5)²=⅕x²-4x+20
(2√7x + ułamek licznik 7x, mianownik √7)²=28x²+28x+7x²=35x²+28

2.Wykonaj potęgowanie

a) (2x + 1)³=8x³+12x²+6x+1
(2x – 3)³=8x³-36x²+56x-27
(2x – 1)³=8x³-12x²+6x-1
b) (4x – 3)³ =64x³-144x²+108x-27
(3x – 4)³=27x³-68x²+144x-64
(4x – 1)³=64x³-48x²+12x-1
c) (x2 – 1)³=x⁶-3x⁴+2x²=1
(x2 + 2)³=x⁶+6x⁴+6x+8
(x2 + 1)³=x⁶+3x⁴+3x²+1
d)(2x2 – 5x3)³=8x⁶+60x⁷+150x¹¹-125x⁹
(3x3 – 1)³=27x⁹ -9x⁹+9x³-1=18x⁹+9x³-1
(2z3 – 5)³=24√3-180+150√3-125=164√3-305

3. Oblicz iloczyn wielomianów S i R
a) ( 2x² – 3 ) ( x - 1)=2x³-2x²-3x+3
b) (-3x² + 5x – 1)(2x + 1)=-6x³+10x²-2x-3x²+5x-1= 6x³+7x²+3x-1
c) (2x² + 3x – 5 )(-x² + x + 2)= -2x⁴+2x³+4x²-3x³+3x²+6x+5x²-5x-10= -2x⁴-x³+12x²+x-10
d) ( 3x³ + 2x² + 1)( - 2x³ + x – 3)= -9x⁶+3x⁴-9x³-4x⁵+ 2x³-6x²- 2x³+x-3= -9x⁶-4x⁵+3x⁴-9x³-6x²+x-3

4. Dane są wielomiany W (x) = - x³ + 2x² – x + 5, P (x) = 2x² – x + 1
Q (x) = 5x + 2. Oblicz:
a) 2( - x³ + 2x² – x + 5) + 3(2x² – x + 1) – (5x + 2)=
=-2x³+4x²-2x+10+6x²-3x+3-5x-2=
=-2x³+10x²-10x+11

b) 2 [ - x³ + 2x² – x + 5–( 2x² – x + 1)] +5x + 2=
=2( - x³ + 2x² – x + 5– 2x² + x - 1)+5x+2=
=2(-x³+4)+5x+2=-2x³+8+5x+2=-2x³+5x+10

c) W (x) – [P(x) + 3Q (x)]
- x³ + 2x² – x + 5-[2x² – x + 1+3( 5x + 2)=
= -x³ + 2x² – x + 5-(2x² – x + 1+15x+6)=
=-x³ + 2x² – x + 5-(2x²+14x+7)=
=-x³ + 2x² – x + 5-2x²-14x-7= -x³-15x-2

d) 2(- x³ + 2x² – x + 5)+ 3 [ 5x+2 – 2(2x²-x+1)]=
=-2x³+4x²-2x+10+3(5x+2-4x²+2x-2)=
=-2x³+4x²-2x+10+3(-4x²+7x)=
=-2x³+4x²-2x+10-12x²+21x=
=2x³-8x²+19x+10

e) 5(5x+2) – [-x³+x²-x+5 + 2(2x²-x+1)=
=25x+10-(-x³+2x²-x+5+4x²-2x+2)=
=25x+10-(-x³+4x²-3x+7)=
25x+10+x³-4x²+3x-7=x³-4x²++28x+3

f) 3 [-x³+x²-x+5– 2(2x²-x+1)] – [2x²-x+1 + 3(-x³+2x²-x+5)]=
=-3x³+6x²-3x+15-4x²+2x-2-(2x²-x+1-3x³+6x²-3x+15)=
=-3x³+2x²-x+13-(-3x³+8x²-4x+16)=
=-3x³+2x²-x+13+3x³-8x²+4x-16=
=-6x²+3x-3