Question

Write two examples of a polynomial expression each with a minimum of three terms and a degree no larger than four give each polynomial expression a different degree then find the sun and difference of your two polynomials what is a common mistake when subtracting polynomials

Answer 1

Sure! Here are two examples of polynomial expressions with different degrees:

1. **Polynomial A (Degree 3):**
\[ P(x) = 2x^3 + 5x^2 - 3x + 4 \]

2. **Polynomial B (Degree 4):**
\[ Q(x) = 4x^4 - x^3 + 2x - 7 \]

### Sum of the Polynomials

To find the sum, we add the like terms of both polynomials.

\[
P(x) + Q(x) = (2x^3 + 5x^2 - 3x + 4) + (4x^4 - x^3 + 2x - 7)
\]

Now, arranging and combining like terms:

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

### Difference of the Polynomials

To find the difference, we subtract polynomial A from polynomial B.

\[
Q(x) - P(x) = (4x^4 - x^3 + 2x - 7) - (2x^3 + 5x^2 - 3x + 4)
\]

Now, rearranging and combining like terms:

\[
= 4x^4 + (-x^3 - 2x^3) + (-5x^2) + (2x + 3x) + (-7 - 4)
\]
\[
= 4x^4 - 3x^3 - 5x^2 + 5x - 11
\]

### Summary of Results

- **Sum:** \( S(x) = 4x^4 + x^3 + 5x^2 - x - 3 \)
- **Difference:** \( D(x) = 4x^4 - 3x^3 - 5x^2 + 5x - 11 \)

### Common Mistake When Subtracting Polynomials

A common mistake when subtracting polynomials is forgetting to change the signs of all the terms in the polynomial being subtracted. For example, when subtracting polynomial A from polynomial B, one must distribute the negative sign to each term of polynomial A before combining like terms. Failing to do so will lead to incorrect results in the resultant polynomial.

For instance, if we mistakenly wrote:

\[
Q(x) - P(x) = (4x^4 - x^3 + 2x - 7) - (2x^3 + 5x^2 - 3x + 4)
\]

and did not change the signs of the terms in \( P(x) \):

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

That would lead to combining as:

\[
= 4x^4 + (-1x^3 - 2x^3) + (-5x^2) + (2x + 3x) + (-7 + 4)
\]

This results in an incorrect polynomial.