To determine the degree of the polynomial [tex]\(5x^4 - 3x^2 + 2x + 1\)[/tex], we need to find the highest power of the variable [tex]\(x\)[/tex] that appears in the polynomial.
1. Observe the terms in the polynomial:
- The first term is [tex]\(5x^4\)[/tex].
- The second term is [tex]\(-3x^2\)[/tex].
- The third term is [tex]\(2x\)[/tex].
- The fourth term is the constant [tex]\(1\)[/tex].
2. Identify the powers of [tex]\(x\)[/tex] in each term:
- For the term [tex]\(5x^4\)[/tex], the power of [tex]\(x\)[/tex] is [tex]\(4\)[/tex].
- For the term [tex]\(-3x^2\)[/tex], the power of [tex]\(x\)[/tex] is [tex]\(2\)[/tex].
- For the term [tex]\(2x\)[/tex], the power of [tex]\(x\)[/tex] is [tex]\(1\)[/tex].
- The term [tex]\(1\)[/tex] is a constant, which can be written as [tex]\(1x^0\)[/tex], so the power of [tex]\(x\)[/tex] is [tex]\(0\)[/tex].
3. Determine the highest power of [tex]\(x\)[/tex] among all terms:
- The powers of [tex]\(x\)[/tex] we identified are [tex]\(4\)[/tex], [tex]\(2\)[/tex], [tex]\(1\)[/tex], and [tex]\(0\)[/tex].
The highest power of [tex]\(x\)[/tex] is [tex]\(4\)[/tex].
Therefore, the degree of the polynomial [tex]\(5x^4 - 3x^2 + 2x + 1\)[/tex] is [tex]\( \boxed{4} \)[/tex].
