x squared minus 4 is divided by x minus 1 using the long division method. The first expression of divisibility, left parenthesis x squared minus x right parenthesis, is subtracted from the dividend. The remainder is x minus 4. The quotient is x.
The image shows the first few steps of the polynomial division (x2−4)÷(x−1)
. Complete the polynomial division process,
To complete the polynomial division process, we need to continue dividing the remaining term x - 4 by x - 1.
The next step is to bring down the next term, which is -4:
```
_____________________
x - 1 | x^2 - 4
- (x^2 - x)
______________
- x + 4
```
Now we divide - x + 4 by x - 1:
```
_____________________
x - 1 | x^2 - 4
- (x^2 - x)
______________
- x + 4
- (-x + 1)
______________
3
```
The remainder is 3.
Therefore, the completed polynomial division is:
```
Quotient: x
Remainder: 3
```
So the final result is x + 3.
To complete the polynomial division process of (x^2 - 4) ÷ (x - 1), we need to continue dividing until there are no more terms left to divide.
The first step is already given, where we subtracted (x^2 - x) from the dividend (x^2 - 4). This gives us a remainder of (x - 4).
The next step is to bring down the next term, which is -4. The new dividend becomes (x - 4, -4).
Now, we need to determine how many times (x - 1) divides into (x - 4).
By looking at the leading terms, we can see that (x - 1) divides into (x^2) x times.
Next, we multiply (x - 1) by x, which gives us (x^2 - x). We then subtract this from the current dividend (x - 4, -4).
(x - 1) * x = x^2 - x
(x - 4, -4) - (x^2 - x) = (-x - 4, -4)
Next, we bring down the next term, which is -4. The new dividend becomes (-x - 4, -4).
Now, we need to determine how many times (x - 1) divides into (-x - 4).
By looking at the leading terms, we can see that (x - 1) divides into (-x) -1 times.
Next, we multiply (x - 1) by -1, which gives us (-x + 1). We then subtract this from the current dividend (-x - 4, -4).
(x - 1) * -1 = -x + 1
(-x - 4, -4) - (-x + 1) = (-3, -3)
At this point, we have no more terms to bring down, so (-3, -3) is the final remainder.
The quotient is found by combining the coefficients of the divided terms. Since the leading term of the original dividend is x^2, and we divided by (x - 1) which has a degree of 1, the quotient will have a degree of 1.
Therefore, the quotient is x.
In summary,
(x^2 - 4) ÷ (x - 1) = x + (-3, -3)
or
(x^2 - 4) ÷ (x - 1) = x - 3 remainder - 3
To complete the polynomial division process for (x^2 - 4) ÷ (x - 1), we can follow the steps of long division:
Step 1: Write the dividend (x^2 - 4) and the divisor (x - 1), and then divide the first term of the dividend by the first term of the divisor. In this case, it is (x^2 ÷ x) which equals x. Write x above the division bar as the leading term of the quotient.
x | x^2 - 4
....
Step 2: Multiply the divisor (x - 1) by the quotient term (x), and write the result underneath the dividend. In this case, it is (x * (x - 1)) which equals (x^2 - x).
x | x^2 - 4
- (x^2 - x)
.....
Step 3: Subtract (x^2 - x) from the dividend (x^2 - 4). This step is similar to subtracting column by column in regular arithmetic.
x | x^2 - 4
- (x^2 - x)
__________
+ x
Step 4: Bring down the next term from the dividend (-4). Now we have (x - x^2 + x - 4) as the new dividend.
x | x^2 - 4
- (x^2 - x)
__________
x - 4
Step 5: Repeat the process by dividing the new dividend (x - 4) by the divisor (x - 1). In this case, the division is straightforward since there are no additional terms. The result is 1.
x | x^2 - 4
- (x^2 - x)
__________
x - 4
- (x - 4)
__________
0
Since we've reached a remainder of 0, the division process is complete. The quotient is x, and there is no remainder.
Therefore, the answer to (x^2 - 4) ÷ (x - 1) is x.