Question

Using factor theorem, factorize each of the following polynomial:
x⁴-7x³+9x²+7x-10

Answer 1

The factorised polynomial to the given polynomial is $x^4-7x^3+9x^2+7x-10=(x-1)(x+1)(x-2)(x-5)$

Step-by-step explanation:

Given polynomial is $x^4-7x^3+9x^2+7x-10$

To factorise the given polynomial by Factor method :

Let f(x) be the given polynomial

Therefore $f(x)=x^4-7x^3+9x^2+7x-10$  

Since the degree of the given polynomial is 4  

Therefore it has 4 factors

• Put x=1 in the above polynomial we get

$f(1)=1^4-7(1)^3+9(1)^2+7(1)-10$

$=1-7+9+7-10$

Therefore f(1)=0

Therefore x-1 is  a factor

By factor theorem " If f(x) is a polynomial and f(1)=0 then x-1 is a factor of f(x) "

        $x^3-6x^2+3x+10$

      ____________________

 x-1) $x^4-7x^3+9x^2+7x-10$

        $x^4-x^3$

      (-)__(+)______________

            $-6x^3+9x^2$

            $-6x^3+6x^2$

_____ (+)__(-)___________

                      $3x^2+7x$

                      $3x^2-3x$

__________(-)__(+)___________

                               $10x-10$

                               $10x-10$

                           _(-)__(+)_________

                                           0

                          ____________________

Therefore we have  $x^3-6x^2+3x+10$

• Put x=-1 in the above polynomial we get

$f(-1)=(-1)^4-7(-1)^3+9(-1)^2+7(-1)-10$

$=1+7+9-7-10$

Therefore f(-1)=0

Therefore x+1 is also a factor

By Factor theorem we have that  

         $x^2-7x+10$

       ____________________

 x+1) $x^3-6x^2+3x+10$

        $x^3+x^2$

      _(-)_(-)______________

             $-7x^2+3x$

              $-7x^2-7x$

   _____(+)__(+)___________

                       $10x+10$

                       $10x+10$

                    _(-)__(-)_________

                                   0

                          _________________

Therefore we have the quadratic equation $x^2-7x+10=0$

$x^2-7x+10=0$

$(x-2)(x-5)=0$

Therefore x-2 and x-5 are also factors of f(x)

By using the factor theorem we have factorised the given polynomial as (x-1)(x+1)(x-2)(x-5)

The factorised polynomial to the given polynomial is $x^4-7x^3+9x^2+7x-10=(x-1)(x+1)(x-2)(x-5)$