Answer :

To determine the degree of a polynomial, you need to find the term with the highest power of the variable (in this case, [tex]\(x\)[/tex]). Here's how you can do it step-by-step for the polynomial [tex]\(7x^6 - 6x^5 + 2x^3 + x - 8\)[/tex]:

1. Identify Each Term: The polynomial is made up of several terms, each with a coefficient and a power of [tex]\(x\)[/tex]. The terms here are:
- [tex]\(7x^6\)[/tex]
- [tex]\(-6x^5\)[/tex]
- [tex]\(2x^3\)[/tex]
- [tex]\(x\)[/tex] (which is [tex]\(1x^1\)[/tex])
- [tex]\(-8\)[/tex] (constant term, which is [tex]\(8x^0\)[/tex])

2. Find the Power of [tex]\(x\)[/tex] in Each Term: Look at the exponent of [tex]\(x\)[/tex] in each term:
- The exponent in [tex]\(7x^6\)[/tex] is 6.
- The exponent in [tex]\(-6x^5\)[/tex] is 5.
- The exponent in [tex]\(2x^3\)[/tex] is 3.
- The exponent in [tex]\(x\)[/tex] (or [tex]\(1x^1\)[/tex]) is 1.
- The constant term [tex]\(-8\)[/tex] has an [tex]\(x\)[/tex] exponent of 0.

3. Identify the Highest Power: Compare all the exponents. The highest exponent here is 6, which comes from the term [tex]\(7x^6\)[/tex].

4. Conclusion: The degree of the polynomial is the highest exponent found, which is 6.

Therefore, the degree of the polynomial [tex]\(7x^6 - 6x^5 + 2x^3 + x - 8\)[/tex] is [tex]\(\boxed{6}\)[/tex].

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