Question

Fatore completamente os polinômios:
a) 6x2 + 11xy – 10y2
b) 5x4y2 – 40xy5
c) x3 – 2x2 – 9x + 18
d) 2x2 – 3x – 9

Answer 1

a) não é possível fatorar

$6{x}^{2}+ 11xy-10{y}^{2}$

b)

$5{x}^{4}{y}^{2}-40x{y}^{5}=5x{y}^{2}({x}^{3}-8{y}^{3})\\=5x{y}^{2}(x-2y)({x}^{2}+2xy+4{y}^{2})$

c)

${x}^{3}-2{x}^{2}-9x + 18\\={x}^{2}(x-2)-9(x-2)$

$(x-2)({x}^{2}-9)=(x-2)(x-3)(x+3)$

d)

$2{x}^{2}-3x- 9=2{x}^{2}-6x+3x-9\\=2x(x-3)+3(x-3)=(x-3)(2x+3)$

Answer 2

Resposta:

a)

6x² + 11xy- 10y²

6x² + 11xy+4xy-4xy- 10y²

6x² +15xy-4xy- 10y²

3x(2x+5y) -2y(2x+5y)

=(3x-2y)*(2x+5y)

6x² +15xy-4xy- 10y²  = (3x-2y)*(2x+5y)

b) 5x⁴y² – 40xy⁵

5xy²*(x³ -8y³) =5xy²*[x³ - (2y)³]

***(x-2y)³ =x³-6x²y+12xy²- (2y)³

***(x-2y)³ =x³-6xy*(x-2y)- (2y)³

***[x³- (2y)³] =(x-2y) *[(x-2y)²+6xy]

***[x³- (2y)³] =(x-2y) *x²-4xy+4y+6xy]

***[x³- (2y)³] =(x-2y) *(x²+4y+2xy)

5x⁴y² – 40xy⁵ = 5xy²*(x²+4y+2xy)

c) x³ – 2x² – 9x + 18

x²(x-2) -9(x-2)  =(x-2)*(x²-9)

x³ – 2x² – 9x + 18  =(x-2)*(x+3)*(x-3)

d)

2x² – 3x – 9

2x²-3x-9= 0 ==> x'=-3/2  e x''=3

Observe ==:> P(x)=ax²+bx+c=a*(x-x')*(x-x'')

2x² – 3x – 9  = 2*(x+3/2)*(x-3)