Question

Divide as indicated. Check the answer by showing that the product of the divisor and the​ quotient, plus the​ remainder, is the dividend.
StartFraction 7 x squared minus 23 x minus 20 Over x minus 4 EndFraction

Answer 1

To divide \( \frac{7x^2 - 23x - 20}{x - 4} \), we can use polynomial long division.

1. **Long Division Steps:**

- Divide the leading term of the dividend \( 7x^2 \) by the leading term of the divisor \( x \):
\[
7x^2 \div x = 7x
\]
- Multiply the entire divisor \( x - 4 \) by \( 7x \):
\[
7x(x - 4) = 7x^2 - 28x
\]
- Subtract \( 7x^2 - 28x \) from the dividend \( 7x^2 - 23x - 20 \):
\[
(7x^2 - 23x - 20) - (7x^2 - 28x) = -23x + 28x - 20 = 5x - 20
\]

2. **Next Step:**
- Now take \( 5x - 20 \) and divide \( 5x \) by \( x \):
\[
5x \div x = 5
\]
- Multiply \( x - 4 \) by \( 5 \):
\[
5(x - 4) = 5x - 20
\]
- Subtract:
\[
(5x - 20) - (5x - 20) = 0
\]

3. **Result:**
- The quotient is \( 7x + 5 \) and the remainder is \( 0 \).
- So we have:
\[
\frac{7x^2 - 23x - 20}{x - 4} = 7x + 5 \quad \text{with a remainder of } 0.
\]

4. **Check the Answer:**
- To verify, we use the formula \( \text{(divisor)} \times \text{(quotient)} + \text{(remainder)} = \text{(dividend)} \):
\[
(x - 4)(7x + 5) + 0
\]
- Multiply it out:
\[
= 7x^2 + 5x - 28x - 20 = 7x^2 - 23x - 20
\]
- This equals the original dividend \( 7x^2 - 23x - 20 \).

Thus, the final answer is:
\[
\frac{7x^2 - 23x - 20}{x - 4} = 7x + 5
\] with a remainder of \( 0 \).