To determine the maximum number of x-intercepts that a polynomial can have, you should look at the highest exponent of the variable [tex]\(x\)[/tex] in the polynomial. This highest exponent is known as the polynomial's degree.
Consider your polynomial:
[tex]\[ 3x^4 + 9x^2 - 1 \][/tex]
1. Identify the degree of the polynomial:
- The polynomial has terms [tex]\(3x^4\)[/tex], [tex]\(9x^2\)[/tex], and [tex]\(-1\)[/tex].
- The exponents of [tex]\(x\)[/tex] in these terms are 4, 2, and 0, respectively.
- The highest exponent is 4.
2. Understand the significance of the degree:
- The degree of a polynomial tells us the maximum number of x-intercepts (real roots) it can have.
- Therefore, a fourth-degree polynomial (degree 4) can have at most 4 x-intercepts.
Thus, the polynomial [tex]\(3x^4 + 9x^2 - 1\)[/tex] will have at most 4 x-intercepts.
