Simplify the expression 2x^3 + x^3 + x^2 + 2x - 6.

Step-by-step solution

1. 1

   Sort the terms by degree. The terms present are $2x^3$ and $x^3$ (degree 3), $x^2$ (degree 2), $2x$ (degree 1) and $-6$ (degree 0). Only terms of the same degree in the same variable can be combined.

2. 2

   Add the coefficients of the like terms. The two cubic terms share the variable part $x^3$, so their coefficients add:

   $2x^3 + x^3 = (2+1)x^3 = 3x^3$

   The implicit coefficient of $x^3$ is $1$, which is easy to overlook and would give $2x^3$ instead.

3. 3

   Leave the remaining terms alone. There is only one $x^2$ term, one $x$ term and one constant, so none of them can be merged with anything else.

4. 4

   Write the simplified polynomial in descending order.

   $3x^3 + x^2 + 2x - 6$

   It is a cubic with leading coefficient $3$ and constant term $-6$.

5. 5

   Check by evaluating both forms. At $x = 2$: the original gives $16 + 8 + 4 + 4 - 6 = 26$, and the simplified form gives $24 + 4 + 4 - 6 = 26$. At $x = -1$: original $-2 - 1 + 1 - 2 - 6 = -10$, simplified $-3 + 1 - 2 - 6 = -10$.