Question

Q)Factorize:

1) 6 - x - x2
2) x3 - 23x2 + 142x - 120

Q) If x + y = 10, and xy = 25, find value of x3 + y3

Q) If x + y = 8, xy= 15, then find value of x4 + x2y2 + y4

Q) If 9x2 + 25y2 = 181, xy = -6. Find value of 3x + 5y

Answer 1

hello,
(1)-6-x-x²
=6-3x+2x-x²
3(2-x)+x(2-x)
(2-x)(3+x)
(2)x³-23x²+142x-120
let p(x)=x³-23x²+142x-120
let x=1
p(1)=1³-23×1²+142×1-120
=143-143
=0
∴x-1 is a factor of p(x)
now divide p(x) by x-1
after division, you will remainder 0 and quotient as x²-22x+120
p(x)=(x-1)(x²-22x+120)
x-1(x²-12x-10x-120)
x-1(x(x-12)-10(x-12)
(x-10)(x-12)(x-10)

(3)x+y=10   xy=25
(x+y)³=x³+y³+3xy(x+y)
10³=x³+y³+3×25(10)
1000=x³+y³+750
x³+y³=250

(4)x+y=8    xy=15
(x+y)²=x²+y²+2xy
8²=x²+y²+2×15
x²+y²=64-30
x²+y²=34
(x²+y²)²=x⁴+y⁴+2(xy)²
(34)²-(xy)²=x⁴+y⁴+(xy)²
(34+15)(34-15)=x⁴+y⁴+x²y²
=931

(5)9x2+25y2=181   xy=-6
(3x)2+(5y)2=181
(3x+5y)²=(3x)2+(5y)2+2.3x.5y
=9x2+25y2+30xy
=181+30.-6
181-180
=1
=(3x+5y)2
3x+5y=1