Question

Use long division to divide the polynomials: 4x3−21x2+27xx−3 .
4x2−33x+126
4x2−33x+126+−378x−3
4x2−33x−252
252 4x2−9x

Answer 1

To perform the long division of the polynomials \( 4x^3 - 21x^2 + 27x - 3 \) by \( 4x^2 - 33x + 126 \), we will follow the steps of polynomial long division.

1. **Set up the division**: Write \( 4x^3 - 21x^2 + 27x - 3 \) under the long division symbol and \( 4x^2 - 33x + 126 \) outside.

2. **Divide the first term**: Divide the leading term of the dividend \( 4x^3 \) by the leading term of the divisor \( 4x^2 \):
\[
\frac{4x^3}{4x^2} = x
\]
Write \( x \) above the division line.

3. **Multiply**: Multiply \( x \) by the entire divisor:
\[
x(4x^2 - 33x + 126) = 4x^3 - 33x^2 + 126x
\]

4. **Subtract**: Subtract this result from the original polynomial:
\[
(4x^3 - 21x^2 + 27x - 3) - (4x^3 - 33x^2 + 126x) = (12x^2 - 99x - 3)
\]

5. **Repeat**: Now, divide the leading term \( 12x^2 \) by the leading term of the divisor \( 4x^2 \):
\[
\frac{12x^2}{4x^2} = 3
\]
Write \( +3 \) above the division line next to \( x \).

6. **Multiply and subtract again**:
Multiply \( 3 \) by the entire divisor:
\[
3(4x^2 - 33x + 126) = 12x^2 - 99x + 378
\]
Now subtract this from the previous remainder:
\[
(12x^2 - 99x - 3) - (12x^2 - 99x + 378) = -381
\]

After this step, we can summarize the division process so far:

Quotient: \( x + 3 \)

Remainder: \( -381 \)

Finally, we can express the result of the polynomial division as:
\[
\frac{4x^3 - 21x^2 + 27x - 3}{4x^2 - 33x + 126} = x + 3 - \frac{381}{4x^2 - 33x + 126}
\]

And that is the complete solution to the polynomial long division.