Divide using long division and then check your answer. (x^3 + 3x^2 + x + 3) divided by (x - 1).
The quotient is ? with a remainder of ?
To divide (x^3 + 3x^2 + x + 3) by (x - 1), we will use long division:
x^2 + 4x + 5
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x - 1 | x^3 + 3x^2 + x + 3
- (x^3 - x^2)
_______________
4x^2 + x
- (4x^2 - 4x)
_______________
5x + 3
- (5x - 5)
_______________
8
Therefore, the quotient is x^2 + 4x + 5 with a remainder of 8.
To divide (x^3 + 3x^2 + x + 3) by (x - 1) using long division, follow these steps:
Step 1: Divide the first term of the dividend (x^3) by the first term of the divisor (x), which gives us x^2 as the first term of the quotient.
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(x - 1) | x^3 + 3x^2 + x + 3
- (x^3 - x^2)
_______________
4x^2 + x + 3
Step 2: Multiply the divisor (x - 1) by the first term of the quotient (x^2) and subtract the result from the dividend.
(x - 1)(x^2) = x^3 - x^2
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(x - 1) | x^3 + 3x^2 + x + 3
- (x^3 - x^2)
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(4x^2 + x + 3) - (x^3 - x^2)
Simplifying further, we get:
4x^2 + x + 3 - x^3 + x^2
= -x^3 + 5x^2 + x + 3
Step 3: Repeat the process with the simplified dividend.
Divide the first term of the simplified dividend (-x^3) by the first term of the divisor (x), which gives us -x^2 as the next term of the quotient.
__________
(x - 1) | x^3 + 3x^2 + x + 3
- (x^3 - x^2)
________________
(4x^2 + x + 3) - (x^3 - x^2)
- (-x^3 + 5x^2 + x + 3)
_______________________
(4x^2 + x + 3) - (-x^3 + x^2)
__________
= (-x^3 + 5x^2 + x + 3) - (-x^3 + x^2)
= (-x^3 + 5x^2 + x + 3) + (x^3 - x^2)
Simplifying further, we get:
= (-x^3 + x^3) + (5x^2 - x^2) + (x + 3)
= 0 + 4x^2 + x + 3
= 4x^2 + x + 3
Step 4: Repeat the process with the simplified dividend once again.
Divide the first term of the simplified dividend (4x^2) by the first term of the divisor (x), which gives us 4x as the next term of the quotient.
_________
(x - 1) | x^3 + 3x^2 + x + 3
- (x^3 - x^2)
______________
(4x^2 + x + 3) - (x^3 - x^2)
- (-x^3 + 5x^2 + x + 3)
_________________________
(4x^2 + x + 3) - (-x^3 + x^2)
- (-x^3 + x^3) + (5x^2 - x^2) + (x + 3)
______________________________________
(-x^3 + x^3) + (4x^2 - x^2) + (x + 3)
Simplifying further, we get:
= (-x^3 + x^3) + (4x^2 - x^2) + (x + 3)
= 0 + 3x^2 + x + 3
= 3x^2 + x + 3
Now, we have obtained the quotient as 4x + 3 and the remainder as 3x^2 + x + 3.
To check our answer, we can multiply the divisor by the quotient and add the remainder to see if it equals the original dividend:
(x - 1)(4x + 3) + (3x^2 + x + 3) = x^2 - x + 12x + 9 + 3x^2 + x + 3
= 4x^2 + 12x + 9 + 3x^2 + x + 3
= 7x^2 + 13x + 12
Since the result matches the original dividend (x^3 + 3x^2 + x + 3), we can conclude that our quotient (4x + 3) with a remainder of (3x^2 + x + 3) is correct.
To divide (x^3 + 3x^2 + x + 3) by (x - 1) using long division, follow these steps:
Step 1: Write the dividend and divisor in descending order of exponents.
x^3 + 3x^2 + x + 3 ÷ x - 1
Step 2: Divide the first term of the dividend by the first term of the divisor (x^3 ÷ x).
- The result is x^2, which will be the first term of the quotient.
- Multiply the whole divisor by the x^2 term and write it below the dividend.
___________________
x - 1 | x^3 + 3x^2 + x + 3
- (x^3 - x^2)
_______________
4x^2 + x + 3
Step 3: Now, divide the first term of the new dividend (4x^2) by the first term of the divisor (x).
- The result is 4x, which will be the second term of the quotient.
- Multiply the entire divisor by the 4x and write it below the new dividend.
___________________
x - 1 | x^3 + 3x^2 + x + 3
- (x^3 - x^2)
4x + 4 | 4x^2 + x + 3
- (4x^2 - 4x)
_______________
5x + 3
Step 4: Finally, divide the first term of the new dividend (5x) by the first term of the divisor (x).
- The result is 5, which will be the third term of the quotient.
- Multiply the entire divisor by 5 and write it below the new dividend.
___________________
x - 1 | x^3 + 3x^2 + x + 3
- (x^3 - x^2)
4x + 4 | 4x^2 + x + 3
- (4x^2 - 4x)
5x + 3 | 5x + 3
- (5x - 5)
__________
8
The quotient is x^2 + 4x + 5 with a remainder of 8.
To check the answer, multiply the quotient (x^2 + 4x + 5) by the divisor (x - 1) and add the remainder (8) to the result. The resulting expression should be equal to the original dividend (x^3 + 3x^2 + x + 3).