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