Question

1. Zamień poniższe sumy algebraiczne na iloczyny.

a) (2x-3)² - (3y-2)²

b) -25x² + (b-x)²

c) 9b² - (2a - 5b)²

d) (5a - 2b)² - (a - 4b)²

e) (4x - y)² - (2x + y)²

f) (x + 3y)² - 16x²

g) 4p²q² - (2pq + 1)

h) (3x - y)² - (2x + y)²

i) (3b + 5a)² - (b - 2a)²

j) (1 - 3b)² - (a + 2b)²

2. Zamień poniższe sumy algebraiczne na iloczyn trzech czynników, wyłączając w pierwszym kroku wspólny czynnik przed nawias.

a) 2a² - 12a + 18

b) 32a²x - 18b²x

c) -3c + 48a²c

d) 2x² - 4xy + 2y²

e) 12a² - 12ab + 3b²

f) 9b²m + m - 6bm

g) 3a² - 6ab + 3b²

Answer 1

zad.1.

 

a) (2x-3)do kwadratu - (3y-2)do kwadratu=

   = 4x do kwadratu-12x+9 - (9y do kwadratu-12y do kwadratu+4

   =  4x do kwadratu-12x+9 - 9y do kwadratu+12y do kwadratu-4

      4x do kwadratu-12x- 9y do kwadratu+12y do kwadratu+5

b) -25x² + (b-x)²

    =-25x do kwadratu+b do kwadratu-2bx+x do kwadratu

    =-24xdo kwadratu+b do kwadratu-2bx

c) 9b² - (2a - 5b)²  =9b do kwadratu -(4a do kwadratu-20ab+25b do kwadratu)

                            =9b do kwadratu -4a do kwadratu+20ab-25b do kwadratu

                             =-16b do kwadratu-4a do kwadratu+20ab

d) (5a - 2b)² - (a - 4b)²

 =25a do kwadratu-20ab+4b do kwadratu-(a do kwadratu-8ab+16b do kwadratu)

=25a do kwadratu-20ab+4b do kwadratu-a do kwadratu+8ab-16b do kwadratu

=24a do kwadratu-12ab-12b do kwadratu

e) (4x - y)² - (2x + y)²

=16x do kwadratu-8xy+y do kwadratu-(4x do kwadratu+4xy+y do kwadratu)

=16x do kwadratu-8xy+y do kwadratu-4x do kwadratu-4xy-y do kwadratu

=12 x do kwadratu-4xy

f) (x + 3y)² - 16x²

=x do kwadratu+6xy+9y do kwadratu-16x do kwadratu

= -15x do kwadratu+6xy+9y

g) 4p²q² - (2pq + 1)

=4p do kwadratu q do kwadratu - 2pq-1

h) (3x - y)² - (2x + y)²

=9x do kwadratu-6xy-(4x do kwadratu+4xy+y do kwadratu)

=9x do kwadratu-6xy-4x do kwadratu-4xy-y do kwadratu

=5x do kwadratu-2xy

i) (3b + 5a)² - (b - 2a)²

=9b do kwadratu+30ba+25a do kwadratu-(b do kwadratu-4ba+4a do kwadratu)

=9b do kwadratu+30ba+25a do kwadratu-b do kwadratu+4ba-4a do kwadratu

=8b do kwadratu+34ba+21a do kwadratu

j) (1 - 3b)² - (a + 2b)²

=1-6b+9b do kwadratu-(a do kwadratu+4ab+4b do kwadratu)

=1-6b+9b do kwadratu-a do kwadratu-4ab-4b do kwadratu

=1-6b+8b do kwadratu-a do kadratu-4ab do kwadratu

zad.2.

a) 2a² - 12a + 18

=2(a do kwadratu-6a+9)

=2(a-3)do kwadratu

b) 32a²x - 18b²x

=2(16a do kwadatu*x-9b do kwadratu*x)

=2x(4a)do kwadratu-2x(3b)do kwadratu

c) -3c + 48a²c- nie umiem

 

d) 2x² - 4xy + 2y²

2(x-y)do kwadratu

e) 12a² - 12ab + 3b²

3(2a-b)do kwadratu

f) 9b²m + m - 6bm

=m(3b-1)do kwadratu

g) 3a² - 6ab + 3b²

=3(a-b)do kwadratu

Answer 2

Zadanie 1

a) (2x-3)² - (3y-2)²=[(2x-3)-(3y-2)]·[(2x-3)+(3y-2)]=(2x-3y-1)(2x+3y-5)

b) -25x² + (b-x)²=[(b-x)-5x]·[(b-x)+5x]=(b-6x)(b+4x)

c) 9b² - (2a - 5b)²=[3b-(2a-5b)]·[3b+(2a-5b)]=(-2a-2b)(2a-2b)=-4(a+b)(a-b)

d) (5a - 2b)² - (a - 4b)²=[(5a-2b)-(a-4b)]·[(5a-2b)+(a-4b)]=(4a+2b)(6a-6b)=12(2a+b)(a-b)

e) (4x - y)² - (2x + y)²=[(4x-y)-(2x+y)]·[(4x-y)+(2x+y)]=(2x-2y)(6x)=12(x-y)x

f) (x + 3y)² - 16x²=[(x+3y)-4x]·[(x+3y)+4x]=[-3x+3y][5x+3y]=3(y-x)(5x+3y)

g) 4p²q² - (2pq + 1)²=//zakładam że zapomniano dopisać kwadrtu//

wtedy mamy

=(2pq-2pq-1)(2pq+2pq+1)=-(4pq+1)

 

 

h) (3x - y)² - (2x + y)²=[(3x-y)-(2x+y)]·[(3x-y)+(2x+y)]=(x-2y)5x

i) (3b + 5a)² - (b - 2a)²=[(3b+5a)-(b-2a)]·[(3b+5a)+(b-2a)]=(2b7a)(4b+3a)

j) (1 - 3b)² - (a + 2b)²=[(1-3b)-(a+2b)]·[(1-3b)+(a+2b)]=(1-5b-a)(1-b+a)

 

Zadanie 2

a) 2a² - 12a + 18=2(a²-6a+9)=2(a-3)²2(a-3)(a-3)

b) 32a²x - 18b²x=2x(16a²-9b²)2x(4a-3b)(4a+3b)

c) -3c + 48a²c=-3c(1-16a²)=-3c(1-4a)(1+4a)

d) 2x² - 4xy + 2y²=2(x²-2xy+y²)=2(x-y)²=2(x-y)(x+y)

e) 12a² - 12ab + 3b²=3(4a²-4ab+b²)=3(2a-b)²=3(2a-b)(2a+b)

f) 9b²m + m - 6bm=m(9b²+1-6b)=m(3b-1)²=m(3b-1)(3b+1)

g) 3a² - 6ab + 3b²=3(a²-2ab+b²)=3(a-b)²=3(a-b)(a-b)