Cory writes the polynomial [tex]$x^7+3x^5+3x+1$[/tex]. Melissa writes the polynomial [tex]$x^7+5x+10$[/tex].

Is there a difference between the degree of the sum and the degree of the difference of the polynomials?

A. Adding their polynomials together or subtracting one polynomial from the other both result in a polynomial with degree 7.
B. Adding their polynomials together or subtracting one polynomial from the other both result in a polynomial with degree 5.
C. 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.
D. 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.

Question

09-08-2024
Mathematics

Answered

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Cory writes the polynomial [tex]$x^7+3x^5+3x+1$[/tex]. Melissa writes the polynomial [tex]$x^7+5x+10$[/tex].

Is there a difference between the degree of the sum and the degree of the difference of the polynomials?

A. Adding their polynomials together or subtracting one polynomial from the other both result in a polynomial with degree 7.
B. Adding their polynomials together or subtracting one polynomial from the other both result in a polynomial with degree 5.
C. 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.
D. 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.

Sagot :

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GinnyAnsweridn GinnyAnsweridn · Answer 1 · 2024-08-09T11:33:14-04:00

Let's analyze the problem step by step:

1. Identify the given polynomials:
- Cory's polynomial: [tex]$ x^7 + 3x^5 + 3x + 1 $[/tex]
- Melissa's polynomial: [tex]$ x^7 + 5x + 10 $[/tex]

2. Sum of the polynomials:

Adding the two polynomials:

[tex]\[ (x^7 + 3x^5 + 3x + 1) + (x^7 + 5x + 10) \][/tex]

Combine like terms:

[tex]\[ x^7 + x^7 + 3x^5 + 3x + 5x + 1 + 10 = 2x^7 + 3x^5 + 8x + 11 \][/tex]

The degree of the resulting polynomial is determined by the highest power of [tex]$ x $[/tex]. In this case, the highest power is [tex]$ x^7 $[/tex].

Therefore, the degree of the sum is 7.

3. Difference of the polynomials:

Subtracting Melissa's polynomial from Cory's polynomial:

[tex]\[ (x^7 + 3x^5 + 3x + 1) - (x^7 + 5x + 10) \][/tex]

Combine like terms:

[tex]\[ x^7 - x^7 + 3x^5 + 3x - 5x + 1 - 10 = 3x^5 - 2x - 9 \][/tex]

The degree of the resulting polynomial is determined by the highest power of [tex]$ x $[/tex]. In this case, the highest power is [tex]$ x^5 $[/tex].

Therefore, the degree of the difference is 5.

4. Comparison of the degrees:

- Degree of the sum: 7
- Degree of the difference: 5

From the analysis, we see that adding their polynomials results in a polynomial with degree 7, while subtracting one polynomial from the other results in a polynomial with degree 5.

Therefore, the correct statement is:

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.

Sagot :

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