Questions: What polynomial must be added to -3c^2 + 3 - 6c in order to get a sum of 2c^2 - c + 5?

Solution Steps

To find the polynomial that must be added to \(-3c^2 + 3 - 6c\) to get \(2c^2 - c + 5\), we need to subtract the first polynomial from the second polynomial. This will give us the polynomial that, when added to the first, results in the second.

We are given two polynomials:

1. The polynomial to which we need to add another polynomial: \(-3c^2 - 6c + 3\).
2. The desired sum of the two polynomials: \(2c^2 - c + 5\).

To find the polynomial that must be added, we set up the equation: \[ \text{First Polynomial} + \text{Polynomial to Add} = \text{Desired Sum} \] Substituting the given polynomials, we have: \[ (-3c^2 - 6c + 3) + \text{Polynomial to Add} = 2c^2 - c + 5 \]

To find the polynomial to add, we subtract the first polynomial from the desired sum: \[ \text{Polynomial to Add} = (2c^2 - c + 5) - (-3c^2 - 6c + 3) \]

Simplifying the expression, we get: \[ \text{Polynomial to Add} = 2c^2 - c + 5 + 3c^2 + 6c - 3 \] Combine like terms: \[ \text{Polynomial to Add} = (2c^2 + 3c^2) + (-c + 6c) + (5 - 3) \] \[ \text{Polynomial to Add} = 5c^2 + 5c + 2 \]

Final Answer