Divide using long division and then check your answer. (4x^2 - 2x + 6) divided by (x - 1).
The quotient is ? with a remainder of ?
To divide (4x^2 - 2x + 6) by (x - 1) using long division, we start by dividing the highest degree term of the numerator by the highest degree term of the denominator.
_______
(x - 1) | 4x^2 - 2x + 6
We divide 4x^2 by x, which gives us 4x. Next, we multiply (x - 1) by 4x to get 4x^2 - 4x.
4x
_______
(x - 1) | 4x^2 - 2x + 6
- (4x^2 - 4x)
We subtract (4x^2 - 4x) from (4x^2 - 2x) to get 2x. We bring down the next term, which is +6.
4x + ?
_______
(x - 1) | 4x^2 - 2x + 6
- (4x^2 - 4x)
__________
2x + 6
Now, we divide 2x by x, which gives us 2. Next, we multiply (x - 1) by 2 to get 2x - 2.
4x + 2
_______
(x - 1) | 4x^2 - 2x + 6
- (4x^2 - 4x)
__________
2x + 6
- (2x - 2)
We subtract (2x - 2) from (2x + 6) to get 8. Since we have no more terms to bring down, we have completed the long division.
4x + 2
_______
(x - 1) | 4x^2 - 2x + 6
- (4x^2 - 4x)
__________
2x + 6
- (2x - 2)
__________
8
Therefore, the quotient is 4x + 2 with a remainder of 8.
To check our answer, we can multiply the divisor (x - 1) by the quotient (4x + 2) and add the remainder (8).
(x - 1)(4x + 2) + 8
Expanding the expression, we get:
4x^2 + 2x - 4x - 2 + 8
Combining like terms, we have:
4x^2 + 6
This matches the original numerator, 4x^2 - 2x + 6, confirming that our long division was done correctly.
To divide (4x^2 - 2x + 6) by (x - 1) using long division, follow these steps:
Step 1: Start by dividing the first term of the numerator (4x^2) by the first term of the denominator (x). This will give you the first term of the quotient.
4x^2 / x = 4x
Step 2: Multiply the denominator (x - 1) by the quotient term obtained in step 1 (4x), and write the result below the numerator.
4x * (x - 1) = 4x^2 - 4x
Step 3: Subtract the result obtained in step 2 from the numerator (4x^2 - 2x + 6). This will give you a new polynomial.
(4x^2 - 2x + 6) - (4x^2 - 4x) = 2x + 6
Step 4: Repeat steps 1 to 3 with the new polynomial (2x + 6) as the numerator.
Step 5: Divide the first term of the new numerator (2x) by the first term of the denominator (x). This will give you the next term of the quotient.
2x / x = 2
Step 6: Multiply the denominator (x - 1) by the quotient term obtained in step 5 (2), and write the result below the new numerator.
2 * (x - 1) = 2x - 2
Step 7: Subtract the result obtained in step 6 from the new numerator (2x + 6). This will give you a new polynomial.
(2x + 6) - (2x - 2) = 8
Step 8: The new numerator (8) is a constant term, and the denominator (x - 1) is a linear term. Since there are no like terms, we cannot simplify the equation further.
Therefore, the quotient is 4x + 2, and the remainder is 8.
To divide (4x^2 - 2x + 6) by (x - 1) using long division, follow these steps:
Step 1: Divide the first term of the numerator (4x^2) by the first term of the denominator (x).
Answer: 4x
Step 2: Multiply the entire denominator (x - 1) by the answer obtained in Step 1 (4x), and write the result below the numerator.
(4x) * (x - 1) = 4x^2 - 4x
Step 3: Subtract the result obtained in Step 2 from the numerator (4x^2 - 2x + 6).
(4x^2 - 2x + 6) - (4x^2 - 4x) = 2x + 6
Step 4: Repeat Steps 1-3 until there are no more terms to bring down.
Since there are no more terms, the quotient is 4x and the remainder is 2x + 6.
To check the answer, multiply the quotient (4x) by the divisor (x - 1) and add the remainder (2x + 6).
(4x) * (x - 1) + (2x + 6)
= 4x^2 - 4x + 2x + 6
= 4x^2 - 2x + 6
As this result matches the original numerator, the answer is correct.