When the polynomial p(x) = 5x3 − 51x2 + kx − 9 is divided by x − 9, the remainder is 0. which of the following is also a factor of p(x)?
a- x − 5
b- x + 1
c- x − 1
d- 5x + 1

answer C:  (x - 1) is also a factor of P(x).

Step-by-step explanation:

Synthetic division is the best approach here.  Given that one factor is x - 9, we know that 9 is the appropriate divisor in synthetic division:

9    )    5    -51      k       -9

                 45    -54      (9k - 486)

           5     -6   (k - 54)  (-9 + 9k - 486)

and this remainder must = 0.  Find k:  -9 + 9k - 486 = 0, or

                                                                      9k = 486 + 9 = 495

                                                                      Then k = 495/9 = 55

Look at the last line of synthetic division, above:

5   -6   (k - 54)    0

Substituting 55 for k, we get:

5   -6     1          

These are the coefficients of the quotient obtained by

dividing P(x) by (x - 9).    They correspond to 5x^2 - 6x + 1.

We must factor this result.

Let's start with 5x + 1, and check whether this is a factor of 5x^2 - 6x + 1 or not.  If 5x + 1 is a factor, then the related root is -1/5.  Let's use -1/5 as the divisor in synthetic div.:

-1/5    )       5       -6        1

                           -1        7/5

                 5        -7       12/5       Here the remainder is not zero, so -1/5 is

                                                    not a root and 5x + 1 is not a factor.

Now try x - 5.  Is this a factor of 5x^2 - 6x + 1?  Use 5 as divisor in synth. div.:

5    )     5      -6      1

                    25    95

         5         19     96  Same conclusion:  x - 5 is not a factor.

Try x = -1:

-1     )    5     -6     1

                    -5     11

            5       -11     12.  The remainder is not zero, so (x + 1) is not a factor.

Finally, try x = 1:

1     )    5    -6     1

                  5    -1

          5     -1      0

Finally, we get a zero remainder, and thus we know that x - 1 is a factor of P(x)

Answer C is correct:  x - 1 is a factor of P(x)

answer 1 : option "c"

answer 2 : option "d"

answer 3 : option "b"

answer 4 : option "d"

answer 5 : option "b"

answer 6 : option "a"

answer 7 : option "c"

answer 8 : option "d"

answer 9 : option "c"

answer 10 : option "d"

answer 11 : option "c"

answer 12 option "b"

answer 13 option "c"

answer 14 option "c"

answer 15 option "d"

answer 16 option "a"

answer 17 option "b"

answer 18 option "c"

answer 19 option "a"

answer 20 option "a"

answer 21 option "c"

d