What is the error made when subtracting these two polynomials

Question

(2x^3-7x^2+4x)-(9x-5x^3+9x^2)
Step 1: 2x^3-7x^2+4x-9x-5x^3+9x^2
Step 2: 2x^3-5x^3-7x^2+9x^2+4x-9x
Step 3: -3x^3+2x^2-5x
Final answer: -3x^3+2x^2-5x

Answer 1

Let's go through the polynomial subtraction step by step to identify any errors.

1. Start with the original expression: \[ (2x^3 - 7x^2 + 4x) - (9x - 5x^3 + 9x^2) \] This can be rewritten as: \[ 2x^3 - 7x^2 + 4x - 9x + 5x^3 - 9x^2 \]

2. Combine like terms (in the order you did, your work is generally correct): \[ (2x^3 + 5x^3) + (-7x^2 - 9x^2) + (4x - 9x) \]

3. Calculate each part:

   • For the \(x^3\) terms: \(2x^3 + 5x^3 = 7x^3\)
   • For the \(x^2\) terms: \(-7x^2 - 9x^2 = -16x^2\)
   • For the \(x\) terms: \(4x - 9x = -5x\)

4. Therefore, combining all the results yields: \[ 7x^3 - 16x^2 - 5x \]

Thus, the corrected final answer should be: \[ 7x^3 - 16x^2 - 5x \]

The error in your solution was in the combination of the \(x^3\) and \(x^2\) terms. You subtracted instead of adding the coefficients correctly.

Answer 2

In which step was there an error

Answer 3

Let's break down each step of your polynomial subtraction to pinpoint where the error occurred:

1. Original Step: \[ (2x^3 - 7x^2 + 4x) - (9x - 5x^3 + 9x^2) \] You correctly rewrite this as: \[ 2x^3 - 7x^2 + 4x - 9x + 5x^3 - 9x^2 \]

2. Step 1: You arranged the terms: \[ 2x^3 - 7x^2 + 4x - 9x - 5x^3 + 9x^2 \] This step appears correct.

3. Step 2: You combined like terms: \[ 2x^3 - 5x^3 - 7x^2 + 9x^2 + 4x - 9x \] The arrangement of terms looks fine here as well.

4. Step 3: Then combined them: \[ -3x^3 + 2x^2 - 5x \] Here's where the error occurred:

   • For the \(x^3\) terms:
     • You should have \(2x^3 + 5x^3 = 7x^3\), but instead you wrote \(-3x^3\).
   • For the \(x^2\) terms:
     • You should have \(-7x^2 - 9x^2 = -16x^2\), but instead you wrote \(2x^2\).

5. Final answer: Your answer of \(-3x^3 + 2x^2 - 5x\) is incorrect due to both mishaps in the combination of the \(x^3\) and \(x^2\) terms.

The correct final answer, after checking all the steps, should indeed be: \[ 7x^3 - 16x^2 - 5x \]

In summary, the errors were in Step 3, specifically in the calculations of the coefficients of \(x^3\) and \(x^2\).

Answer 4

why am I supposed to add 2x^3 and 5x^3 instead of subtract?

Answer 5

In the expression you provided, you are subtracting one polynomial from another. Let's clarify how to manage the signs when performing subtraction.

When you subtract a polynomial, you distribute the negative sign (or the minus sign) across all the terms of the polynomial being subtracted. This alters the signs of those terms. Here's a breakdown of this process:

Original Expression

You started with:

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

Step 1: Distributing the Negative Sign

When you subtract the second polynomial, you distribute the negative sign through its terms:

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

As you can see, the term \(-5x^3\) comes from the original polynomial \( -( -5x^3)\) which becomes \(+5x^3\).

Step 2: Combine Like Terms

Now, you combine like terms. Here’s how you combine them:

1. Combine the \(x^3\) terms:

   • You have \(2x^3\) from the first polynomial and \(+5x^3\) from the second polynomial after distribution. So: \[ 2x^3 + 5x^3 = 7x^3 \]

2. Combine the \(x^2\) terms:

   • You have \(-7x^2\) from the first polynomial and \(-9x^2\) from the second polynomial after distributing. Thus: \[ -7x^2 - 9x^2 = -16x^2 \]

3. Combine the \(x\) terms:

   • You have \(4x\) from the first polynomial and \(-9x\) from the second polynomial after distributing. Therefore: \[ 4x - 9x = -5x \]

Result

Combining these, you arrive at: \[ 7x^3 - 16x^2 - 5x \]

Conclusion

You were supposed to add \(2x^3\) and \(5x^3\) because when you distribute the negative sign, the term \(-5x^3\) becomes \(+5x^3\). When combining the coefficients of like terms, that leads you to addition, not subtraction for the \(x^3\) terms.

This is why understanding the distribution of the negative sign in polynomial subtraction is crucial for correctly combining the terms.