Yes, that's correct! The degree of a polynomial is indeed the highest power of the variable in the polynomial expression, and it plays a crucial role in determining the number of roots (or zeros) as well as the number of factors the polynomial can have.
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Degree and Roots: According to the Fundamental Theorem of Algebra, a polynomial of degree \( n \) can have up to \( n \) complex roots (counting multiplicities). This means that if a polynomial is of degree 3, it can have up to 3 roots. If some of the roots are repeated (for example, a root of multiplicity 2 means it counts as two), that still counts toward the total number of roots.
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Factors: A polynomial of degree \( n \) can be factored into \( n \) linear factors in the case of complex coefficients. For example, the polynomial \( P(x) \) of degree 3 can be represented in factored form as: \[ P(x) = a(x - r_1)(x - r_2)(x - r_3) \] where \( r_1, r_2, r_3 \) are the roots of the polynomial and \( a \) is a non-zero constant.
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Real and Complex Roots: If we consider only real coefficients, then complex roots must occur in conjugate pairs. Therefore, if a polynomial has real coefficients, the number of real roots must be odd or even such that the total (real and complex) counts up to its degree.
In summary, the degree of a polynomial effectively determines its maximum number of roots and factors, making it a fundamental characteristic in understanding polynomial behavior in algebra.
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