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.