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Xalos
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Suppose a polynomial is divided by (x-1) and remainder=2 and when the same polynomial is divided by (x-2), remainder is 3. What is the remainder when the polynomial is divided by (x-1)(x-2)? Why?
Xalos said:Suppose a polynomial is divided by (x-1) and remainder=2 and when the same polynomial is divided by (x-2), remainder is 3. What is the remainder when the polynomial is divided by (x-1)(x-2)? Why?
HallsofIvy said:Actually, I wrote out a long response arguing that this was a bad problem because there were too many possibilities and the couldn't all give the same remainder.
Until I did the critical division at the end and found out they did!
The purpose of dividing a polynomial by (x-1)(x-2) is to find the remainder when the polynomial is divided by the expression (x-1)(x-2). This can be helpful in simplifying the polynomial and understanding its behavior.
Yes, a polynomial can be divided by (x-1)(x-2) without a remainder if (x-1)(x-2) is a factor of the polynomial. This means that the polynomial is divisible by (x-1)(x-2) and the remainder will be equal to zero.
To find the remainder when dividing a polynomial by (x-1)(x-2), you can use the polynomial long division method. This involves dividing the polynomial by (x-1)(x-2) and then finding the remainder by subtracting the product of the divisor and quotient from the original polynomial.
The remainder in polynomial division represents the leftover terms that cannot be divided evenly by the divisor. It is the part of the polynomial that does not fit into the divisor and is left as a remainder after the division process.
Understanding polynomial division and finding the remainder can help in solving real-world problems that involve polynomial equations. For example, it can be used to determine the maximum or minimum value of a polynomial function, or to find the roots or zeros of a polynomial. It can also be helpful in evaluating limits or finding the inverse of a polynomial function.