Sara subtracted two polynomials. She asked a friend to check her solution and work. What should her friend tell her? Is Sara correct? If she is not correct, explain which step the error was made AND what the error was.

Question

(2x3−7x2+4x)−(9x−5x3+9x2)

Step 1: 2x3−7x2+4x−9x−5x3+9x2

Step 2: 2x3−5x3−7x2+9x2+4x−9x

Step 3: −3x3+2x2−5x
(2 points)

Answer 1

Let's go through Sara's steps and verify her work on the subtraction of the polynomials.

We start with the expression:

\[ (2x^3 - 7x^2 + 4x) - (9x - 5x^3 + 9x^2) \]

Step 1:

Sara rewrites the expression:

\[ 2x^3 - 7x^2 + 4x - 9x + 5x^3 - 9x^2 \]

The second polynomial is distributed correctly (the negative sign is applied), so this step is correct.

Step 2:

In this step, she combines like terms:

\[ 2x^3 - 5x^3 - 7x^2 + 9x^2 + 4x - 9x \]

This notation is also correct; she is gathering the terms correctly according to their degrees.

Step 3:

Now, she combines the coefficients:

1. Combine \(2x^3\) and \(-5x^3\):

\[ 2x^3 - 5x^3 = -3x^3 \]

2. Combine \(-7x^2\) and \(9x^2\):

\[ -7x^2 + 9x^2 = 2x^2 \]

3. Combine \(4x\) and \(-9x\):

\[ 4x - 9x = -5x \]

Thus, the final combined polynomial is:

\[ -3x^3 + 2x^2 - 5x \]

Conclusion:

Sara's final result in Step 3: \[ -3x^3 + 2x^2 - 5x \]

is indeed correct based on her work in the previous steps.

Her friend should tell her she is correct with her results, as there are no mistakes in the steps shown.