When you solve a polynomial equation, you are finding the value(s) of the variable (typically 'x') that make the equation true, specifically making the polynomial equal to zero. These values are also known as the roots, zeros, or solutions of the polynomial equation.
Here's a breakdown:
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The Equation: A polynomial equation is an equation in the form p(x) = 0, where p(x) is a polynomial. For example, x2 - 4x + 3 = 0 is a polynomial equation.
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Finding the Solution(s): Solving the equation means finding the values of 'x' that, when substituted into the polynomial, result in zero.
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Roots, Zeros, Solutions: These terms are interchangeable. If 'a' is a root (or zero or solution) of the polynomial equation p(x) = 0, then p(a) = 0.
Example:
Consider the polynomial equation x2 - 4x + 3 = 0.
We can factor this equation as (x - 3)(x - 1) = 0.
This equation is true if either (x - 3) = 0 or (x - 1) = 0.
Solving for x, we get x = 3 and x = 1.
Therefore, the roots (or zeros or solutions) of the polynomial equation x2 - 4x + 3 = 0 are x = 1 and x = 3. These are the x-values where the polynomial crosses the x-axis if graphed.
In summary, solving a polynomial equation p(x) = 0 is the process of determining the value(s) of 'x' for which the polynomial expression p(x) evaluates to zero.
