Bot GPT 4-o mini To perform the division of the polynomial \( 24x^3 - 14x^2 + 49x - 36 \) by \( 6x - 5 \), we can use polynomial long division.
Step 1: Set up the division.
We divide \( 24x^3 - 14x^2 + 49x - 36 \) by \( 6x - 5 \).
Step 2: Divide the leading terms.
Divide the leading term of the dividend \( 24x^3 \) by the leading term of the divisor \( 6x \):
\[ \frac{24x^3}{6x} = 4x^2 \]
Step 3: Multiply and subtract.
Multiply \( 6x - 5 \) by \( 4x^2 \):
\[ 4x^2(6x - 5) = 24x^3 - 20x^2 \]
Now, subtract this from the original polynomial:
\[ (24x^3 - 14x^2) - (24x^3 - 20x^2) = 6x^2 \]
Bring down the next term \( +49x \):
\[ 6x^2 + 49x \]
Step 4: Repeat the process.
Now divide the leading term \( 6x^2 \) by \( 6x \):
\[ \frac{6x^2}{6x} = x \]
Multiply \( 6x - 5 \) by \( x \):
\[ x(6x - 5) = 6x^2 - 5x \]
Subtract this from the current polynomial:
\[ (6x^2 + 49x) - (6x^2 - 5x) = 54x \]
Bring down the next term \( -36 \):
\[ 54x - 36 \]
Step 5: Continue the process.
Now divide \( 54x \) by \( 6x \):
\[ \frac{54x}{6x} = 9 \]
Multiply \( 6x - 5 \) by \( 9 \):
\[ 9(6x - 5) = 54x - 45 \]
Subtract this from the current polynomial:
\[ (54x - 36) - (54x - 45) = 9 \]
Step 6: Write the final answer.
Now we have no more polynomial left to bring down, and we have reached a remainder of \( 9 \).
So, the result of the division is:
\[ 4x^2 + x + 9 + \frac{9}{6x - 5} \]
Final Answer:
The quotient in standard form, including any remainder, is:
\[ \boxed{4x^2 + x + 9 + \frac{9}{6x - 5}} \]
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