Question

cory writes the polynomial $x^7 + 3x^5 + 3x + 1$. melissa writes the polynomial $x^7 + 5x + 10$. is there a difference between the degree of the sum and the degree of the difference of the polynomials?

• adding their polynomials together or subtracting one polynomial from the other both result in a polynomial with degree 7.
• adding their polynomials together or subtracting one polynomial from the other both result in a polynomial with degree 5.
• adding their polynomials together results in a polynomial with degree 14, but subtracting one polynomial from the other results in a polynomial with degree 5.
• adding their polynomials together results in a polynomial with degree 7, but subtracting one polynomial from the other results in a polynomial with degree 5.

Explanation:

Step1: Calculate sum of polynomials

$(x^7 + 3x^5 + 3x + 1) + (x^7 + 5x + 10)$
$=x^7+x^7 + 3x^5 + 3x+5x + 1+10$
$=2x^7 + 3x^5 + 8x + 11$

Step2: Find degree of sum

The highest power of $x$ is 7, so degree = 7.

Step3: Calculate difference of polynomials

$(x^7 + 3x^5 + 3x + 1) - (x^7 + 5x + 10)$
$=x^7 - x^7 + 3x^5 + 3x-5x + 1-10$
$=3x^5 - 2x - 9$

Step4: Find degree of difference

The highest power of $x$ is 5, so degree = 5.

Step5: Compare degrees

Sum degree = 7, difference degree = 5.

Answer:

Adding their polynomials together results in a polynomial with degree 7, but subtracting one polynomial from the other results in a polynomial with degree 5.