Question

Show that 2x-3 is a factor of
x+2x³-9x²+12​

Answer 1

Answer:

2x3+9x2+7x−6

=2x3+4x2+5x2+7x−6

=2x2(x+2)+5x2+10x−3x−6

=2x2(x+2)+5x(x+2)−3(x+2)

=(x+2)(2x2+5x−3)

=(x+2)(2x2+6x−x−3)

=(x+2)[2x(x+3)−1(x+3)]

=(x+2)(x+3)(x−1)

Answer 2

Answer:

The factor theorem states that a polynomial f(x) has a factor (x−a) if and only if f(a)=0

Let p(x)=x+2x

3

−9x

2

+12 and g(x)=2x−3

g(x)=2x−3=0 gives x=

2

3

g(x) will be factor of p(x) if p(

2

3

)=0 (Factor theorem)

Now, p(

2

3

)=

2

3

+2(

2

3

)

3

−9(

2

3

)

2

+12=

2

3

+2(

8

27

)−9(

4

9

)+12

=

2

3

+

4

27

−

4

81

+12=

4

6+27−81+48

=

4

0

=0

Since, p(

2

3

)=0, so g(x) is a factor of p(x).