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

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^2 - 5x

Answer 1

Let's verify the steps to see if the student performed the subtraction correctly.

Given polynomials: \[ (3x^4−8x^2+4x)−(9x−5x^4+2x^2) \]

Step 1: Rewrite the expression correctly. \[ 3x^4 - 8x^2 + 4x - (9x - 5x^4 + 2x^2) \] Distributing the negative sign (subtracting the whole second polynomial): \[ 3x^4 - 8x^2 + 4x - 9x + 5x^4 - 2x^2 \]

The expression should be: \[ 3x^4 + 5x^4 - 8x^2 - 2x^2 + 4x - 9x \]

Step 2: Combine like terms.

1. Combine \(3x^4\) and \(5x^4\): \[ 3x^4 + 5x^4 = 8x^4 \]
2. Combine \(-8x^2\) and \(-2x^2\): \[ -8x^2 - 2x^2 = -10x^2 \]
3. Combine \(4x\) and \(-9x\): \[ 4x - 9x = -5x \]

Putting it all together, we have: \[ 8x^4 - 10x^2 - 5x \]

Verification of the student's final answer: The student's final answer was \(8x^4 - 6x^2 - 5x\). The correct answer after correctly processing the subtraction is \(8x^4 - 10x^2 - 5x\).

Conclusion: The student made an error in Step 2 when adding \(-8x^2\) and \(2x^2\). The correct result should have been \(-10x^2\), not \(-6x^2\). Therefore, their friend should tell them that they made an error in combining like terms, specifically with the \(x^2\) terms, and the final answer is not correct. The correct answer is \(8x^4 - 10x^2 - 5x\).