Determine the values of m and n so that the polynomials 2x^3 +mx^2 -3 and x^3 - 3mx^2 + 2nx +4 are divisble by x-2

Question

please help me I'm very confused

Answer 1

Hi Beverly,

 

Let's begin by knowing what it means for these polynomials to be divisible by (x-2).  It means that

(x-2) is a factor of these polynomials, which in turn means that (x=2) is a root or "zero" of these polynomials.  This is true because the roots or zeroes of any polynomial are found by setting each of it's factors to zero and solving each one for x.

 

Therefore, we can find the values of m and n in each of these polynomials by plugging in 2 for x and setting each polynomial to zero and solving for m and n.

 

Let's go ahead and do this for our 1st polynomial:

    2x³ + mx² - 3 = 0

    2(2³) + m(2²) - 3 = 0

    16 + 4m - 3 = 0

                4m = 3 - 16

                4m = -13

                  m = -13/4

                  m = -3.25

 

We can verify our solution for m by plugging it back into our polynomial equation along with our root of 2 for x and see if it evaluates to zero:

    2x³ + mx² - 3 = 0

    2(2³) + (-3.25)(2²) - 3 = 0

    16 - 13 - 3 = 0

                 0 = 0 (evaluates to zero)

 

Since our polynomial equation evaluates to zero, we are confident that our value for m is correct.  

 

Now let's follow the same procedure for our 2nd polynomial using m = -3.25 from our 1st solution and our root of 2 for x:

    x³ - 3mx² + 2nx + 4 = 0

    2³ - 3(-3.25)(2²) + 2n(2) + 4 = 0

    8 + 39 + 4n + 4 = 0

               4n + 51 = 0

                      4n  = -51

                        n  = -51/4

                        n  = -12.75

 

We can verify our solutions for n by plugging it back into our polynomial equation along with out values for m and x and see if it evaluates to zero: 

    x³ - 3mx² + 2nx + 4 = 0

    2³ - 3(-3.25)(2²) + 2(-12.75)(2) + 4 = 0

    8 + 39 - 51 + 4 = 0

             39 - 39 = 0

                       0 = 0 (evaluates to zero)

 

Since our polynomial equation evaluates to zero, we are confident that our rule for n in terms of m is correct.

 

The key point to remember here is that if (x-2) is a polynomial factor then it follows that (x=2) is a polynomial root.

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan.

Answer 2

If a polynomial function f(x) is divisible by x - 2, then x-2 is a factor of f(x) and, by the Factor Theorem,  f(2) must equal zero.  

 

So, 2(2)³ + m(2)² - 3 = 0

 

       16 + 4m - 3 = 0

 

               4m = -13     m = -13/4

 

Plugging 2 in for x and -13/4 for m into the second function and setting the result equal to zero, we get:       

   2³ - 3(-13/4)(2)² + 4n + 4 = 0

 

   8 +39 + 4n + 4 = 0

 

                     4n = -51

 

                      n = -51/4