When a certain polynomial is divided by x2 x 1, its quotient is x2 x 1 and its remainder is – 1. What is the polynomial?
To find the polynomial, we will multiply the divisor (x^2 + x + 1) by the quotient (x^2 - x + 1) and add the remainder (-1).
Let's begin:
Divisor: x^2 + x + 1
Quotient: x^2 - x + 1
Remainder: -1
Multiplying the divisor and quotient:
(x^2 + x + 1) * (x^2 - x + 1) = x^4 + x^3 + x^2 - x^3 - x^2 + x + x^2 - x + 1
Simplifying:
= x^4 + (x^3 - x^3) + (x^2 + x^2 - x^2) + (x - x) + 1
= x^4 + 1
Adding the remainder:
x^4 + 1 - 1
Therefore, the polynomial is x^4.
To find the polynomial, we need to use the polynomial division method. Let's start by dividing the given polynomial by the divisor, which is x^2 + x + 1.
Step 1: Write the polynomial in standard form with missing terms filled as 0.
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x^2 + x + 1 | dividend
Step 2: Divide the first term in the dividend (x^2) by the first term in the divisor (x^2) to get x (quotient). Multiply the entire divisor by x and write the result below.
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x^2 + x + 1 | x^2
- (x^2 + x +1)
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- x - 1
Step 3: Bring down the next term from the dividend (0x) and divide it by the first term in the divisor (x^2) to get 0 (quotient). Multiply the entire divisor by 0 and write the result below.
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x^2 + x + 1 | x^2 - x
- (x^2 + x +1)
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-2x - 1
Step 4: Continue this process until you have brought down all the terms from the dividend. In this case, the next term is 0, and we divide it by the first term in the divisor (x^2) to get 0 (quotient). Multiply the entire divisor by 0 and write the result below.
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x^2 + x + 1 | x^2 - x + 1
- (x^2 + x +1)
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2
Step 5: At this point, we have used all the terms from the dividend. The result we are looking for is the quotient and the remainder.
The quotient we obtained is x^2 - x + 1, and the remainder is 2.
Now, to find the original polynomial, we need to multiply the divisor by the quotient and add the remainder.
Divisor (x^2 + x + 1) x Quotient (x^2 - x + 1) = x^4 - x^3 + x^2 - x + x^2 - x + 1 = x^4 - x^3 + 2x^2 - 2x + 1
Finally, add the remainder (2) to the above result to get the complete polynomial:
x^4 - x^3 + 2x^2 - 2x + 1 + 2 = x^4 - x^3 + 2x^2 - 2x + 3
Therefore, the polynomial is x^4 - x^3 + 2x^2 - 2x + 3.