To determine how many roots the polynomial [tex]\(7 + 5x^4 - 3x^2\)[/tex] has, we need to examine its degree. A polynomial's degree is the highest power of the variable [tex]\(x\)[/tex] in that polynomial.
Let's break down the given polynomial:
[tex]\[7 + 5x^4 - 3x^2\][/tex]
Here are the terms of the polynomial:
- The constant term is [tex]\(7\)[/tex],
- The term with [tex]\(x^2\)[/tex] has a coefficient of [tex]\(-3\)[/tex],
- The term with [tex]\(x^4\)[/tex] has a coefficient of [tex]\(5\)[/tex].
The term with the highest power of [tex]\(x\)[/tex] is [tex]\(5x^4\)[/tex], and the highest power here is [tex]\(4\)[/tex]. This means the polynomial is of degree 4.
According to the Fundamental Theorem of Algebra, a polynomial of degree [tex]\(n\)[/tex] has exactly [tex]\(n\)[/tex] roots in the complex number system, counting multiplicities. This implies that all roots, real and complex, are included in this count.
So for our polynomial with a degree of 4, we can conclude that the polynomial [tex]\(7 + 5x^4 - 3x^2\)[/tex] has exactly [tex]\(4\)[/tex] roots in total.
Thus, the polynomial [tex]\(7 + 5x^4 - 3x^2\)[/tex] has 4 roots, which can be real or complex.
