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Answer :
To determine the degree of the polynomial [tex]\(9x^6 + 5x^2 - x + 7\)[/tex], we need to identify the term with the highest power of the variable [tex]\(x\)[/tex].
Here's a detailed, step-by-step solution:
1. Understand the polynomial structure: The given polynomial is:
[tex]\[
9x^6 + 5x^2 - x + 7
\][/tex]
Each term in the polynomial is composed of a coefficient (a numerical constant) and a variable [tex]\(x\)[/tex] raised to a power (the exponent).
2. Identify the exponent of each term:
- The first term is [tex]\(9x^6\)[/tex]. Here, [tex]\(x\)[/tex] is raised to the power of 6.
- The second term is [tex]\(5x^2\)[/tex]. Here, [tex]\(x\)[/tex] is raised to the power of 2.
- The third term is [tex]\(-x\)[/tex]. This is the same as [tex]\(-1x^1\)[/tex], so [tex]\(x\)[/tex] is raised to the power of 1.
- The fourth term is [tex]\(7\)[/tex]. This is a constant term, which can be written as [tex]\(7x^0\)[/tex], where the exponent is 0.
3. Determine the highest exponent:
- Among the exponents (6, 2, 1, and 0), the highest exponent is 6.
4. Conclusion:
- The degree of a polynomial is defined as the highest power of the variable in the polynomial.
- Therefore, the degree of the polynomial [tex]\(9x^6 + 5x^2 - x + 7\)[/tex] is 6.
So, the degree of the polynomial is [tex]\(6\)[/tex].
Here's a detailed, step-by-step solution:
1. Understand the polynomial structure: The given polynomial is:
[tex]\[
9x^6 + 5x^2 - x + 7
\][/tex]
Each term in the polynomial is composed of a coefficient (a numerical constant) and a variable [tex]\(x\)[/tex] raised to a power (the exponent).
2. Identify the exponent of each term:
- The first term is [tex]\(9x^6\)[/tex]. Here, [tex]\(x\)[/tex] is raised to the power of 6.
- The second term is [tex]\(5x^2\)[/tex]. Here, [tex]\(x\)[/tex] is raised to the power of 2.
- The third term is [tex]\(-x\)[/tex]. This is the same as [tex]\(-1x^1\)[/tex], so [tex]\(x\)[/tex] is raised to the power of 1.
- The fourth term is [tex]\(7\)[/tex]. This is a constant term, which can be written as [tex]\(7x^0\)[/tex], where the exponent is 0.
3. Determine the highest exponent:
- Among the exponents (6, 2, 1, and 0), the highest exponent is 6.
4. Conclusion:
- The degree of a polynomial is defined as the highest power of the variable in the polynomial.
- Therefore, the degree of the polynomial [tex]\(9x^6 + 5x^2 - x + 7\)[/tex] is 6.
So, the degree of the polynomial is [tex]\(6\)[/tex].
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