5. Factorise :
1. x³- 2x²- x+2
2.2y³+y²-2y-1
Answers
Answer:
Answer:
(i)
Let the given polynomial be,
p(x)=x3−2x2−x+2
By trial method,
p(1)=(1)3−2(1)2−1+2=1−2−1+2=0
So, (x−1) is the factor of the given polynomial.
Divide the given polynomial by (x−1) by long division.
x−1x2−x−2x3−2x2−x+2x3−x2−−−−−− −x2−x −x2+x−−−−−−−−− −2x+2 −2x+2−−−−−−−−−−− 0−−−−−−−−
It is known that,
Dividend=Divisor×Quotient+Remainderx3−2x2−x+2=(x−1)(x2−x−2)+0=(x−1)(x2−2x+x−2)=(x−1)[x(x−2)+1(x−2)]
Further simplify,
x3−2x2−x+2=(x−1)(x+1)(x−2)
(ii)
Let the given polynomial be,
p(y)=2y3+y2−2y−1
By trial method,
p(1)=2(1)3+(1)2−2(1)−1=2+1−2−1=0
So, (y−1) is the factor of the given polynomial.
Divide the given polynomial by (y−1) by long division.
y−12y2+3y+12y3+y2−2y−12y3−2y2−−−−−−−− 3y2−2y 3y2−3y−−−−−−−−−− y−1 y−1−−−−−−−−−− 0−−−−−−−−
It is known that,
Dividend=Divisor×Quotient+Remainder2y3+y2−2y−1=(y−1)(2y2+3y+1)+0=(y−1)(2y2+2y+y+1)=(y−1)[2y(y+1)+1(y+1)]
Further simplify,
2y3+y2−2y−1=(y−1)(y+1)(2y+1)