Question

In each of the following, use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x) or, not:
f(x) = 3x⁴+17x³+9x²-7x-10, g(x) = x+5

Answer 1

The expression g(x)=x+5 is a factor for the polynomial $f(x)=3x^4+17x^3+9x^2-7x-10$ is verified.

Step-by-step explanation:

Given that the polynomial $f(x)=3x^4+17x^3+9x^2-7x-10$

and g(x)=x+5

To verify that polynomial g(x) is a factor of polynomial f(x) or, not :

By using the Factor theorem here

• Since x+5 is a factor (given)
• so that x=-5

Put x=-5 in the polynomial f(x) we get

• $f(-5)=3(-5)^4+17(-5)^3+9(-5)^2-7(-5)-10$
• $=3(625)+17(-125)+9(25)+35-10$
• $=1875-2125+225+25$
• $=2125-2125$
• $=0$
• f(-5)=0
• Therefore x+5 is a factor and it satisfies the given polynomial.( by factor theorem )
• Hence g(x)=x+5 is a factor for the polynomial f(x)

                   $3x^3+2x^2-x-2$

              ____________________________

       x+5) $3x4+17x^3+9x^2-7x-10$

                  $3x^4+15x^3$

                  ___(-)___(-)__________________________

                            $2x^3+9x^2$

                            $2x^3+10x^2$

           ______(-)____(-)_______________

                                      $-x^2-7x$

                                      $-x^2-5x$

        _____________(+)__(+)_________________

                                               -2x-10

                                               -2x-10

                                            _(+)_(+)________

                                                       0  

                                             _____________

Hence the expression g(x)=x+5 is a factor for the polynomial f(x)

Hence verified