Question

prove that x+2 is a factor of 6x^3+19x^2+16x+4 by using the factor theorem and also factorise 6x^3+19x^2+16x+4.

Answer 1

0 h iska ans isme x = -2 put krne se. Ye solve hoga

Answer 2

GIVEN :

The expression x+2 is a factor of the cubic polynomial  $6x^3+19x^2+16x+4$

TO PROVE :

x+2 is a factor of $6x^3+19x^2+16x+4$ by using the factor theorem and also factorise

SOLUTION :

By using the Factor theorem we can prove that the polynomial is divisible by x+2 then we get the remainder is zero.

             $x^2+7x+2$

       ______________________

x+2 ) $6x^3+19x^2+16x+4$

        $6x^3+12x^2$

      (-)___(-)_______________

                 $7x^2+16x$

                 $7x^2+14x$

    _____(-)___(-)___________

                            $2x+4$

                            $2x+4$

                          _(-)__(-)____

                                    0

                           _________

Hence the given cubic equation is completely divided by the factor x+2 and it leaves remainder 0.

Hence proved

Now factorise the given cubic polynomial

Equating the given cubic polynomial to zero we get

$6x^3+19x^2+16x+4=0$

By factor theorem we can write it as

$(x+2)(6x^2+7x+2)=0$

$(x+2)(6x^2+4x+3x+2)=0$

$(x+2)(2x(3x+2)+1(3x+2))=0$

$(x+2)(2x+1)(3x+2)=0$

∴ the factors are (x+2) , (2x+1) and (3x+2)

∴ the given cubic polynomial is factorised as $6x^3+19x^2+16x+4=(x+2)(2x+1)(3x+2)$