Question

Use the long division method to find the result when 3, x, cubed, plus, 8, x, squared, plus, 8, x, plus, 53x
3
+8x
2
+8x+5 is divided by 3, x, plus, 53x+5.

Answer 1

To perform polynomial long division for the expression \( 3x^3 + 8x^2 + 8x + 53 \) divided by \( 3x + 5 \), we follow the systematic approach of polynomial division.

**Step 1: Set up the long division.**
We want to divide \( 3x^3 + 8x^2 + 8x + 53 \) by \( 3x + 5 \).

**Step 2: Divide the leading terms.**
Take the leading term of the dividend \( 3x^3 \) and divide it by the leading term of the divisor \( 3x \):

\[
\frac{3x^3}{3x} = x^2
\]

**Step 3: Multiply the entire divisor by this result.**
Now, multiply \( 3x + 5 \) by \( x^2 \):

\[
x^2(3x + 5) = 3x^3 + 5x^2
\]

**Step 4: Subtract this result from the original polynomial.**
Now, subtract \( (3x^3 + 5x^2) \) from \( 3x^3 + 8x^2 + 8x + 53 \):

\[
(3x^3 + 8x^2 + 8x + 53) - (3x^3 + 5x^2) = (8x^2 - 5x^2) + 8x + 53 = 3x^2 + 8x + 53
\]

**Step 5: Repeat the process.**
Now, we take the new polynomial \( 3x^2 + 8x + 53 \) and divide it by \( 3x + 5 \).

Divide the leading term:

\[
\frac{3x^2}{3x} = x
\]

Multiply \( 3x + 5 \) by \( x \):

\[
x(3x + 5) = 3x^2 + 5x
\]

Now, subtract:

\[
(3x^2 + 8x + 53) - (3x^2 + 5x) = (8x - 5x) + 53 = 3x + 53
\]

**Step 6: Repeat the process again.**
Now take \( 3x + 53 \) and divide by \( 3x + 5 \):

Divide the leading term:

\[
\frac{3x}{3x} = 1
\]

Multiply \( 3x + 5 \) by \( 1 \):

\[
1(3x + 5) = 3x + 5
\]

Now subtract:

\[
(3x + 53) - (3x + 5) = 53 - 5 = 48
\]

**Step 7: Compile the results.**
At this point, we cannot divide further because the degree of the remainder \( 48 \) is less than the degree of the divisor \( 3x + 5 \).

Therefore, the result of the long division is:

\[
\text{Quotient: } x^2 + x + 1
\]
\[
\text{Remainder: } 48
\]

Thus, when divided:

\[
\frac{3x^3 + 8x^2 + 8x + 53}{3x + 5} = x^2 + x + 1 + \frac{48}{3x + 5}
\]