Question

Using factor theorem, factorize each of the following polynomial:
x³-2x²-x+2

Answer 1

The factorised given polynomial is $x^3-2x^2-x+2=(x-1)(x+1)(x-2)$

Step-by-step explanation:

Given that the polynomial $x^3-2x^2-x+2$

To factorise the given polynomial  :

Let f(x) be the given polynomial

$f(x)=x^3-2x^2-x+2$

By using the Factor theorem here

Put x=1 in the polynomial f(x) we get

• $f(1)=1^3-2(1)^2-1+2$
• $=1-2-1+2$
• $=0$
• $f(1)=0$
• Therefore x-1 is a factor and it satisfies the given polynomial.( by factor theorem )

                   $x^2-x-2$

              ____________________________

       x-1) $x^3-2x^2-x+2$

                 $x^3-x^2$

                  ___(-)__(+)__________________________

                         $-x^2-x$

                         $-x^2+x$

           _________(+)__(-)_______________

                               $-2x+2$

                               $-2x+2$

                     ____ (+)__(-)___________

           

                                          0  

                                ____________

Hence we have the quadratic equation $x^2-x-2=0$  

$(x+1)(x-2)=0$

x+1=0 or x-2=0

Put x=-1 in f(x)

• $f(-1)=(-1)^3-2(-1)^2-(-1)+2$
• $=-1-2+1+2$
• $=0$
• $f(-1)=0$
• Therefore x+1 is a factor and it satisfies the given polynomial.( by factor theorem )

Put x=2 in f(x)

• $f(2)=2^3-2(2)^2-2+2$
• $=8-8-2+2$
• $=0$
• $f(2)=0$
• Therefore x-2 is a factor and it satisfies the given polynomial.( by factor theorem )
• Therefore the factors are x-1,x+1,x-2

The factorised given polynomial is $x^3-2x^2-x+2=(x-1)(x+1)(x-2)$

                                                                                           

Answer 2

Answer:

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