Question

Use long division to divide the polynomials: 4x3−21x2+27x/x−3 .(1 point)

Answer 1

To divide the polynomial \( 4x^3 - 21x^2 + 27x \) by \( x - 3 \) using long division, follow these steps:

1. **Set Up the Division**: Write the dividend \( 4x^3 - 21x^2 + 27x \) and the divisor \( x - 3 \) in long division format.

2. **Divide the Leading Terms**: Divide the leading term of the dividend \( 4x^3 \) by the leading term of the divisor \( x \):
\[
\frac{4x^3}{x} = 4x^2
\]

3. **Multiply and Subtract**:
- Multiply \( 4x^2 \) by the entire divisor \( x - 3 \):
\[
4x^2(x - 3) = 4x^3 - 12x^2
\]
- Subtract this from the original polynomial:
\[
(4x^3 - 21x^2 + 27x) - (4x^3 - 12x^2) = -21x^2 + 12x^2 + 27x = -9x^2 + 27x
\]

4. **Repeat the Process**: Now, divide the leading term of the new polynomial \( -9x^2 \) by \( x \):
\[
\frac{-9x^2}{x} = -9x
\]
- Multiply \( -9x \) by the divisor \( x - 3 \):
\[
-9x(x - 3) = -9x^2 + 27x
\]
- Subtract this from the current polynomial:
\[
(-9x^2 + 27x) - (-9x^2 + 27x) = 0
\]

5. **Final Steps**: Since there are no remaining terms, the quotient of the division is:
\[
\text{Quotient} = 4x^2 - 9x
\]
and there is no remainder.

Thus, the result of dividing \( 4x^3 - 21x^2 + 27x \) by \( x - 3 \) is:
\[
\boxed{4x^2 - 9x}
\]