Why Does an Equation Have the Same Number of Solutions as Their Highest Degree Number?

The statement that a polynomial equation has the same number of solutions (counting multiplicities) as its highest degree is a profound and fundamental concept in algebra. This principle is rooted in the Fundamental Theorem of Algebra, which provides a clear and definitive answer to this question.

The Fundamental Theorem of Algebra

Theorem: Every non-constant polynomial equation of degree n has exactly n roots in the complex number system, counting multiplicities. This theorem is a cornerstone of algebra and is pivotal in understanding the nature of polynomial equations.

Key Concepts

1. Degree of a Polynomial: The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial P(x) x3 - 2x 1, the degree is 3.

2. Roots/Solutions: The solutions or roots of a polynomial are the values of x for which P(x) 0. These solutions can be real or complex numbers.

3. Multiplicity: If a root occurs more than once, it is counted according to its multiplicity. For instance, in the polynomial P(x) (x - 1)2(x 2), the root x 1 has a multiplicity of 2 since it appears twice, and the root x -2 has a multiplicity of 1. This polynomial therefore has a total of 3 roots: 2 from x 1 and 1 from x -2.

Why This is True

Complex Roots: Polynomials can have complex roots, and these roots occur in conjugate pairs if the coefficients of the polynomial are real. This ensures that the total number of solutions (roots) matches the degree of the polynomial. For example, the polynomial P(x) x2 1 has roots x ±i, where i is the imaginary unit.

Graphical Interpretation: The graph of a polynomial function of degree n can intersect the x-axis at most n times, corresponding to the n roots of the equation. This graphical representation helps visualize why the degree of a polynomial is equal to the number of its solutions.

Example

Consider the polynomial P(x) x3 - 6x2 11x - 6: The degree is 3. Factoring gives P(x) (x - 1)(x - 2)(x - 3), which has three distinct roots: x 1, 2, 3. Thus, it has exactly 3 solutions in line with its degree.

In summary, the equality between the number of solutions and the degree of a polynomial ensures that all possible roots, both real and complex, are accounted for, as stated by the Fundamental Theorem of Algebra.

Further Considerations

To delve deeper, a polynomial of degree n can be written as the product of binomials of the form ax - c, where x is the variable of the polynomial, a is some real constant, and c is some constant that could be purely real, purely imaginary, or have both components. The number of such factors must be equal to the degree of the polynomial. When you multiply several such factors together, the highest power of x will be equal to the number of binomials. This leads to the very essence of why the degree of a polynomial equals the number of its solutions when expressed in its factored form.

The zero product rule states that if you multiply several quantities together and get zero, at least one of those quantities must be zero. This is crucial in understanding why each factor in the factored form of a polynomial equation must equal zero to satisfy the equation as a whole. This ensures that every solution is correctly identified and accounted for in the polynomial equation.