Finding the value of a constant

Question

Determine the value of the constant k so that x-2 is a factor of the function g(x)=2x^4+kx^3+3x+4

Answer 1

Since x-2 is a factor, g(2) = 0

g(2) = 2(2^4) + k(2^3) + 3(2) + 4 = 0

2(16) + k(8) + 6 + 4 = 0

42 + 8k = 0

k = -42/8

k = -5.25

Answer 2

Just a note to all students and tutors:   PLZ, always check your answers; I've often found trivial calculation errors.  Thankfully, Tutors can edit their Answers and several have made corrections.  THX.

 

For example, last night I wrote comments on two answers (which differed, but now are the same -- THX) and voted to indicate the correct answer.

 

My comments:

 

      And, checking:     Does  g(2) = 2*(2^4)+(-5.25)*(2^3)+3*(2)+4   =  0    ?    Yes.

 

      And, checking:       Does   g(2) =2*2^4+(-0.25)*2^3+3*(2)+4 = 0  ?  No, it equals 40, so k ≠ -1/4.

 

p.s., Tutors cannot change Comments.

Answer 3

We can use synthetic division to determine the value of k.  This method divides coefficient of the polynomial by the root of the factor.  Also, the remainder must be zero.  In the synthetic division, the last digit is the remainder.

 

The coefficients of the polynomial is:    2  k   0   3    4.

The root of the factor (x - 2) is 2.  This is because if we set x-2 = 0, x is 2.

 

Set ups the division.

 

2 |  2      k        0       3        4

 

 

__________________________

 

 

 

Bring down the first coefficient of the polynomial below the line.  Place a zero below the line under the polynomials last digit.

 

 

2 |  2      k        0       3        4
 
 
__________________________

      2                                  0

 

 

 

Multiply the root by the first digit under the line.  This product will be the addend in the succeeding column.

 

2 |  2      k        0       3        4
 
             4                           -4
__________________________
      2                                  0
 
 

Next, we need to go backwards.  That is the reason why I inserted a -4 as the addend in the last column because 4 and -4 gets us zero. So if we multiply going forward, we must divide going backward.  Divide the addend by the root and place under the line in the preceding column.  Repeat this step.

 

 

2 |  2      k            0           3        4
 
             4         - 5/2        -5       -4
______________________________
      2    -5/4       -5/2        -2         0

 

 

Now we focus on the column that has k.

 

Write an equation since we know we have to add to get the sum of -5/4.

 

 

k + 4 = -5 / 4

 

 

Subtract 4 on both sides of equation.

 

k = (-5 / 4) - 4

 

k = (-5 / 4) - (16 / 4)

 

k = -21 /4