Question

Divide using long division. Check your answer.
(5x^2 -13x+1)/(x-3)
The quotient is ____ with remainder ____.

Answer 1

To perform the long division of \( 5x^2 - 13x + 1 \) by \( x - 3 \), follow these steps:

1. **Divide the first term:** Divide the leading term of the numerator (\( 5x^2 \)) by the leading term of the denominator (\( x \)):
\[
5x^2 \div x = 5x.
\]
This gives the first term of the quotient: \( 5x \).

2. **Multiply and subtract:** Now multiply \( 5x \) by the entire divisor \( x - 3 \):
\[
5x(x - 3) = 5x^2 - 15x.
\]
Next, subtract this from the original polynomial:
\[
(5x^2 - 13x + 1) - (5x^2 - 15x) = -13x + 15x + 1 = 2x + 1.
\]

3. **Repeat the process:** Now divide \( 2x \) by \( x \):
\[
2x \div x = 2.
\]
This gives the second term of the quotient: \( 2 \).

4. **Multiply and subtract again:** Now multiply \( 2 \) by the divisor \( x - 3 \):
\[
2(x - 3) = 2x - 6.
\]
Subtract this from \( 2x + 1 \):
\[
(2x + 1) - (2x - 6) = 1 + 6 = 7.
\]

At this stage, there are no more \( x \) terms left to bring down, so we have completed the division.

Thus, the quotient is \( 5x + 2 \) and the remainder is \( 7 \).

### Final Answer:
The quotient is \( 5x + 2 \) with remainder \( 7 \).

### Check the answer:
To verify, we can reconstruct the original polynomial using the quotient and remainder:
\[
(x - 3)(5x + 2) + 7.
\]
Calculating,
\[
(x - 3)(5x + 2) = 5x^2 + 2x - 15x - 6 = 5x^2 - 13x - 6.
\]
Now adding the remainder:
\[
5x^2 - 13x - 6 + 7 = 5x^2 - 13x + 1.
\]
This confirms that our long division is correct.

Thus, the answer \( 5x + 2 \) with remainder \( 7 \) is indeed verified.