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.
To perform polynomial division for (x^2 - 4) ÷ (x - 1), we set up the long division as follows:
```
x + 1
_______________
x - 1 | x^2 - 4 + 0x
- (x^2 - x)
___________
-5x - 4
```
To find the next term of the quotient, we divide x^2 by (x - 1), which gives us x. We then multiply x by (x - 1) to get x^2 - x, which we subtract from the dividend (x^2 - 4). This leaves us with -5x - 4.
To find the next term of the quotient, we divide -5x by (x - 1), which gives us -5. We then multiply -5 by (x - 1) to get -5x + 5, which we subtract from the remaining polynomial (-5x - 4). This leaves us with -9.
We cannot divide -9 by (x - 1) anymore, so the polynomial division is complete.
The quotient is x + 1 and the remainder is -9.
Expressing the quotient and remainder as a fraction:
Quotient: x + 1 can be written as (x + 1)/1.
Remainder: -9 can be written as -9/1.
Therefore, the quotient and remainder expressed as fractions are:
Quotient: (x + 1)/1
Remainder: -9/1
To complete the polynomial division process for (x^2-4) ÷ (x-1), let's follow these steps:
Step 1: Divide the first term of the numerator by the first term of the denominator. In this case, it is x^2 ÷ x, which equals x.
```
x
_______
x - 1 | x^2 - 4
```
Step 2: Multiply the quotient obtained in the previous step (x) by the denominator (x-1), and write the result below the numerator. Then subtract this result from the numerator.
```
x
___________
x - 1 | x^2 - 4
- (x^2 - x)
```
Step 3: Bring down the next term from the numerator, which is -4, and write it beside the result obtained in the previous step.
```
x - 1
___________
x - 1 | x^2 - 4
- (x^2 - x)
-4
```
Step 4: Repeat steps 1 to 3 until all terms have been processed. In this case, since there are no more terms, we can stop.
The quotient is x, and the remainder is -4.
Expressing the quotient and remainder as a fraction, it becomes:
Quotient = x
Remainder = -4
Therefore, the division result is x - 4/(x - 1).