Final answer:
The expression (x^3 - 4x^2 + 9x - 10) / (x - 2) simplifies to x^2 + 2x + 13 with a remainder of 16 using long division. This is accomplished by dividing each term sequentially, multiplying, and subtracting until a remainder of lesser degree than the divisor is obtained.
Step-by-step explanation:
Long Division of a Polynomial
To simplify the expression (x^3 - 4x^2 + 9x - 10) / (x - 2) using long division, we will divide the polynomial x^3 - 4x^2 + 9x - 10 by the binomial x - 2. We begin by dividing the leading term of the numerator, x^3, by the leading term of the denominator, x, to get x^2. We then multiply this quotient by the entire divisor and subtract the result from the polynomial. We continue this process, dealing with the resulting polynomial each time, until we have no remainder or a remainder of a degree less than the degree of the divisor. The full step-by-step solution looks like this:
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- Divide: x^3 by x to get x^2.
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- Multiply: x^2 by (x - 2) and subtract from x^3 - 4x^2 to get 2x^2.
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- Bring down the next term: creating 2x^2 + 9x.
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- Divide: 2x^2 by x to get 2x.
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- Multiply: 2x by (x - 2) and subtract from 2x^2 + 9x to get 13x.
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- Bring down the next term: creating 13x - 10.
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- Divide: 13x by x to get 13.
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- Multiply: 13 by (x - 2) and subtract from 13x - 10 to arrive at the remainder 16.
The final simplified form of the expression is x^2 + 2x + 13 with a remainder of 16, or x^2 + 2x + 13 + 16/(x - 2).