A student subtracted two polynomials. They asked a friend to check their solution and work. What should their friend tell them? Are they correct? If they are not correct, in which step did the student make an error? (3x^4−8x^2+4x)−(9x−5x^4+2x^2)

Question

A student subtracted two polynomials. They asked a friend to check their solution and work. What should their friend tell them? Are they correct? If they are not correct, in which step did the student make an error? (3x^4−8x^2+4x)−(9x−5x^4+2x^2) 
Step 1: 3x^4−8x^2+4x−9x+5x^4+2x^2 
Step 2: 3x^4+5x^4−8x^2+2x^2+4x−9
Step 3: 8x^4 - 6x^2 - 5x
Funal Awnser: 8x^4 - 6x^ - 5x

GPT-4o mini · Answer

Let's go through the student's work step by step and see where the error occurred in their polynomial subtraction.

Original Expression: \[ (3x^4 - 8x^2 + 4x) - (9x - 5x^4 + 2x^2) \]

Step 1: The student rewrote the expression correctly: \[ 3x^4 - 8x^2 + 4x - 9x + 5x^4 + 2x^2 \] This step is correct.

Step 2: Here, they combined like terms:

For \(x^4\): \(3x^4 + 5x^4 = 8x^4\)
For \(x^2\): \(-8x^2 + 2x^2 = -6x^2\)
For \(x\): \(4x - 9x = -5x\)

So, it should be: \[ 8x^4 - 6x^2 - 5x \] This step is also correct.

Step 3: The final answer should be: \[ 8x^4 - 6x^2 - 5x \] However, the student wrote "8x^4 - 6x^ - 5x" which contains a typographical error ("x^" is incomplete).

Thus, while their calculations for the polynomial subtraction are correct, the final presentation has a typographical error in the answer.

Final Conclusion: Their friend should tell them that the calculations in the steps are correct, but they made a mistake in writing the final answer due to a typo. The correct final answer is: \[ 8x^4 - 6x^2 - 5x \]