Question

Perform the division:



(
24
x
3
−
14
x
2
+
49
x
−
36
)
÷
(
6
x
−
5
)
(24x
3
−14x
2
+49x−36)÷(6x−5)



Write the quotient in standard form, representing any remainders as a fraction.

Answer 1

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:

\[
\boxed{4x^2 + x + 9 + \frac{9}{6x - 5}}
\]