Question

Write the expression that is equivalent to 2x³-15x²+17x+3/x-5.

Answer 1

For the expression 2x³-15x²+17x+3/x-5, polynomial long division or synthetic division must be performed to simplify the expression if x-5 is a factor. If there's no remainder after the division, the expression can be simplified; otherwise, the result will include the remainder as a fraction.

Step-by-step explanation:

The expression 2x³-15x²+17x+3/x-5 cannot be simplified in an obvious way, so we must perform polynomial division or use synthetic division if x-5 is a factor of the polynomial. Since the question indicates simplifying the expression, we need to check if (x-5) is a factor. If (x-5) is a factor, there will be no remainder after the division, and we obtain a simpler expression that's equivalent. However, if there is a remainder, the result will be of the form Q(x) + R/(x-5), where Q(x) is the quotient and R is the remainder.

To solve this, we have to follow these steps:

1. Set up the division, with 2x³-15x²+17x+3 being the dividend and x-5 the divisor.
2. Divide the first term of the dividend by the first term of the divisor, which is 2x³/x. This gives us 2x².
3. Multiply the divisor by this result (2x²) and subtract it from the dividend.
4. Bring down the next term from the dividend and repeat the process until all terms are accounted for.
5. If there's a remainder, express it as a fraction over (x-5).

This process is commonly used in algebra and is essential for polynomial long division or synthetic division techniques. Remember that this explanation is provided without performing the actual division, which would give the specific result for this particular expression.