Verified answer

Step-by-step explanation:

To find the remainder using long division method, we divide the dividend by the divisor step by step.

First, let's write the dividend and divisor in descending order of powers of x:

Dividend: 3x^3 + 3x^2 + 3x + 1

Divisor: 2x + 5

Now, let's perform long division:

_________________________

2x + 5 | 3x^3 + 3x^2 + 3x + 1

We start by dividing the highest power term of the dividend (3x^3) by the highest power term of the divisor (2x). The result is (3/2)x^2. We then multiply this result by the entire divisor and subtract it from the dividend:

(3/2)x^2

_________________________

2x + 5 | 3x^3 + 3x^2 + 3x + 1

- (3/2)x^2 + (15/2)x

Next, we bring down the next term from the dividend, which is just "0" in this case:

(3/2)x^2

_________________________

2x + 5 | 3x^3 + 0*x^2 + 15/2*x + 1

- (3/2)x^2 + (15/2)x

Now, we repeat the process. We divide (-15/4)x by (2x), which gives us (-15/8). We then multiply this result by the entire divisor and subtract it from what we have so far:

(3/4)x - (15/8)

_________________________

2x+5 | x³+0*x²+(15/2)*x+1

- (3/2)x² + (15/2)x

_________________________

- (3/4)x + 1

We bring down the next term from the dividend, which is just "1" in this case:

(3/4)x - (15/8)

_________________________

2x+5 | x³+0*x²+(15/2)*x+1

- (3/2)x² + (15/2)x

_________________________

- (3/4)x + 1

Now, we divide (-4) by (2x), which gives us (-2/x). We then multiply this result by the entire divisor and subtract it from what we have so far:

(3/4)x - (15/8) - (-2/x)

_________________________

2x+5 | x³+0*x²+(15/2)*x+1

- (3/2)x² + (15/2)x

_________________________

- (3/4)x + 1 + (-2/x)

_________________________

0

Since the remainder is zero, we can conclude that:

(3x^3 + 3x^2 + 3x + 1) ÷ (2x + 5) = (3/4)x - (15/8) - (-2/x)

Therefore, the remainder is zero.

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