Question

The image shows the first few steps of the polynomial division (x^2−4)÷(x−1) . Complete the polynomial division process, then give the quotient and the remainder expressed as a fraction.

Answer 1

To complete the polynomial division of \( (x^2 - 4) \div (x - 1) \), we can follow the polynomial long division method.

### Step 1: Divide
We start with \( x^2 \) divided by \( x \):
\[
\frac{x^2}{x} = x
\]
This is our first term of the quotient.

### Step 2: Multiply
Next, multiply \( x \) by the divisor \( (x - 1) \):
\[
x(x - 1) = x^2 - x
\]

### Step 3: Subtract
Now, subtract this result from the original polynomial:
\[
(x^2 - 4) - (x^2 - x) = -4 + x = x - 4
\]

### Step 4: Repeat
Now we take the new polynomial \( x - 4 \) and divide it by \( x - 1 \):
\[
\frac{x}{x} = 1
\]
This is the next term of the quotient.

### Step 5: Multiply
Multiply \( 1 \) by \( (x - 1) \):
\[
1(x - 1) = x - 1
\]

### Step 6: Subtract
Subtract this from \( x - 4 \):
\[
(x - 4) - (x - 1) = -4 + 1 = -3
\]

### Final Step: Conclusion
Now we can't divide anymore since the degree of the remainder \(-3\) is less than the degree of the divisor \((x - 1)\).

Thus, the quotient is:
\[
x + 1
\]
and the remainder is:
\[
-3
\]

### Result as Fraction
Now, we express the remainder as a fraction:
\[
\text{Remainder} = \frac{-3}{x - 1}
\]

### Final Answer
So, the complete result of the polynomial division \( (x^2 - 4) \div (x - 1) \) is:
- **Quotient:** \( x + 1 \)
- **Remainder:** \( \frac{-3}{x - 1} \)