Question
long division the polynomial 24x^4 -24^3 -18x^3 by 4x^3 +2x^2 standard form
Answers
GPT-4o mini
To perform long division on the polynomials \( 24x^4 - 24x^3 - 18x^3 \) by \( 4x^3 + 2x^2 \), we first need to rearrange the dividend in standard form. The polynomial in the dividend can be simplified and written as:
\[
24x^4 - 24x^3 - 18x^3 = 24x^4 - 42x^3
\]
Now we will divide \( 24x^4 - 42x^3 \) by \( 4x^3 + 2x^2 \).
**Step 1: Divide the leading term of the dividend by the leading term of the divisor.**
\[
\frac{24x^4}{4x^3} = 6x
\]
**Step 2: Multiply the entire divisor by this result (6x).**
\[
6x(4x^3 + 2x^2) = 24x^4 + 12x^3
\]
**Step 3: Subtract this from the original polynomial.**
\[
(24x^4 - 42x^3) - (24x^4 + 12x^3) = -42x^3 - 12x^3 = -54x^3
\]
**Step 4: Now bring down the next term. In this case, there are no more terms to bring down, so we continue with \(-54x^3\).**
**Step 5: Divide the leading term of the current dividend by the leading term of the divisor.**
\[
\frac{-54x^3}{4x^3} = -\frac{54}{4} = -\frac{27}{2}
\]
**Step 6: Multiply the entire divisor by this result.**
\[
-\frac{27}{2}(4x^3 + 2x^2) = -54x^3 - 27x^2
\]
**Step 7: Subtract this from the current polynomial.**
\[
(-54x^3) - (-54x^3 - 27x^2) = 0 + 27x^2 = 27x^2
\]
Since there are no more terms to bring down, we can stop here and conclude the division.
The final result of the long division is:
\[
6x - \frac{27}{2} \quad \text{with a remainder of } 27x^2
\]
Thus, we can write the result as:
\[
\frac{24x^4 - 42x^3}{4x^3 + 2x^2} = 6x - \frac{27}{2} + \frac{27x^2}{4x^3 + 2x^2}
\]
\[
24x^4 - 24x^3 - 18x^3 = 24x^4 - 42x^3
\]
Now we will divide \( 24x^4 - 42x^3 \) by \( 4x^3 + 2x^2 \).
**Step 1: Divide the leading term of the dividend by the leading term of the divisor.**
\[
\frac{24x^4}{4x^3} = 6x
\]
**Step 2: Multiply the entire divisor by this result (6x).**
\[
6x(4x^3 + 2x^2) = 24x^4 + 12x^3
\]
**Step 3: Subtract this from the original polynomial.**
\[
(24x^4 - 42x^3) - (24x^4 + 12x^3) = -42x^3 - 12x^3 = -54x^3
\]
**Step 4: Now bring down the next term. In this case, there are no more terms to bring down, so we continue with \(-54x^3\).**
**Step 5: Divide the leading term of the current dividend by the leading term of the divisor.**
\[
\frac{-54x^3}{4x^3} = -\frac{54}{4} = -\frac{27}{2}
\]
**Step 6: Multiply the entire divisor by this result.**
\[
-\frac{27}{2}(4x^3 + 2x^2) = -54x^3 - 27x^2
\]
**Step 7: Subtract this from the current polynomial.**
\[
(-54x^3) - (-54x^3 - 27x^2) = 0 + 27x^2 = 27x^2
\]
Since there are no more terms to bring down, we can stop here and conclude the division.
The final result of the long division is:
\[
6x - \frac{27}{2} \quad \text{with a remainder of } 27x^2
\]
Thus, we can write the result as:
\[
\frac{24x^4 - 42x^3}{4x^3 + 2x^2} = 6x - \frac{27}{2} + \frac{27x^2}{4x^3 + 2x^2}
\]