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What is the remainder in the synthetic division problem below?

[tex]\[ 1 \longdiv {4x^6 - 1} \][/tex]

A. 5
B. 3
C. 7
D. 9

Answer :

We are given the polynomial

[tex]$$
4x^2 + 6x - 1
$$[/tex]

and asked to perform synthetic division using [tex]$x=1$[/tex]. The synthetic division steps go as follows:

1. Write down the coefficients of the polynomial: [tex]$4$[/tex], [tex]$6$[/tex], and [tex]$-1$[/tex].

2. Bring the first coefficient down:
[tex]$$
b_0 = 4.
$$[/tex]

3. Multiply [tex]$b_0$[/tex] by [tex]$1$[/tex] (since [tex]$x=1$[/tex]):
[tex]$$
4 \times 1 = 4.
$$[/tex]

4. Add this result to the second coefficient:
[tex]$$
b_1 = 6 + 4 = 10.
$$[/tex]

5. Multiply [tex]$b_1$[/tex] by [tex]$1$[/tex]:
[tex]$$
10 \times 1 = 10.
$$[/tex]

6. Add this to the third coefficient to obtain the remainder:
[tex]$$
\text{Remainder} = -1 + 10 = 9.
$$[/tex]

Thus, the remainder is

[tex]$$
\boxed{9}.
$$[/tex]

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