Use synthetic division to determine whether the number k is an upper or lower bound (as specified for the real zeros of the function f). k = 4; f(x) = 4x4 - 3x3 - 2x2 - 5x - 4; upper bound?

is an upper bound.

Step-by-step explanation:

We have been given a polynomial . We are asked to use synthetic division to find whether is an upper bound our lower bound for our given polynomial.

To use synthetic division, we will divide coefficients of each term of our given polynomial by 4 as:

     |   4                -3           -2             -5                  -4

4    |   ↓               16           52          200               780        

          4               13          50            195              776

First of all, we brought down the leading coefficient 4 to bottom row and then we multiplied 4 by 4. We will write the product (16) beneath the second coefficient of our polynomial that is -3 and we will write the sum of -3 and 16 in the bottom row.

Now, we will multiply 13 by 4 and write the product (52) beneath the third coefficient of our polynomial that is -2 and we will write the sum of -2 and 52 in the bottom row.

Again, we we will multiply 50 by 4 and write the product (200) beneath the fourth coefficient of our polynomial that is -5 and we will write the sum of -5 and 200 in the bottom row.

Finally, we will multiply 195 by 4 and write the product (780) beneath the constant number of our polynomial that is -4 and we will write the sum of -4 and 780 in the bottom row. Our remainder is 776.

We know that k will be upper bound, when and all the coefficients of quotient and remainder are positive.

Since the value of k is 4 and all the coefficients of quotient and remainder are positive for our given polynomial, therefore, is an upper bound.

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