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: \( \frac{-3}{x - 1} \)