Question

What is the sum of the polynomial expressions 2x^3 + 5x^2 - 3x - 1 and x^3 - 2x^2 - 4x + 2?

Answer 1

To find the sum of the polynomial expressions \(2x^3 + 5x^2 - 3x - 1\) and \(x^3 - 2x^2 - 4x + 2\), we will combine like terms.

First, we write the two expressions:

1. \(2x^3 + 5x^2 - 3x - 1\)
2. \(x^3 - 2x^2 - 4x + 2\)

Next, we add the coefficients of each corresponding term:

1. For the \(x^3\) terms:
\[
2x^3 + x^3 = (2 + 1)x^3 = 3x^3
\]

2. For the \(x^2\) terms:
\[
5x^2 - 2x^2 = (5 - 2)x^2 = 3x^2
\]

3. For the \(x\) terms:
\[
-3x - 4x = (-3 - 4)x = -7x
\]

4. For the constant terms:
\[
-1 + 2 = -1 + 2 = 1
\]

Now, we can write the sum of the two polynomial expressions:
\[
3x^3 + 3x^2 - 7x + 1
\]

Thus, the sum of the polynomial expressions is:
\[
\boxed{3x^3 + 3x^2 - 7x + 1}
\]