Write Polynomial Equations - Algebra-1

1. Fundamental Concepts

Polynomial Equation: An equation containing one or more unknowns, where the highest degree of the unknowns is a non-negative integer, and each term consists of the product of a constant and a power of the unknown. Its general form is (where , n is a non-negative integer, and are constants).
Core Components: Including the constant term (the term without the unknown, e.g., ), the first-degree term (the term with the unknown to the power of 1, e.g., ), the second-degree term (the term with the unknown to the power of 2, e.g., ), etc. Notably, the denominator cannot contain the unknown, and the radicand (the expression under the square root) cannot contain the unknown.

Degree of a Polynomial: The highest degree of the unknown in the equation, which determines the type of the equation (e.g., a polynomial equation with degree 1 is a first-degree polynomial equation, and one with degree 2 is a second-degree polynomial equation).
Number of Terms of a Polynomial: The number of monomials in the equation (excluding duplicate terms after combining like terms). For example, has 3 terms.
Relationship Between Roots and the Equation: If substituting into the equation makes the equality hold, then k is a root of the polynomial equation, and the polynomial can be factored into the product of and other factors.

Given that the roots of a polynomial equation are and (a = 1), write a second-degree polynomial equation.

Steps: From the roots, the factors are and . Multiply these factors to get the equation: , which expands to .

Given a third-degree polynomial equation with roots , , and , and the coefficient of the second-degree term is 5, write the equation.

Steps:
1. From the roots, the factors are x, , and . The initial equation is (where a is a constant).
2. Expand the initial equation: .
3. It is known that the coefficient of the second-degree term is 5, so . Substitute into the expanded equation to get the final equation: .

Writing Equations from Roots: If the roots of a polynomial equation are known, the equation can be expressed as (where , and a is determined by the given coefficient conditions). Then, expand and arrange the equation to get the final form.
Finding Coefficients by Substituting Points: If the degree of the equation and the points through which its graph passes are known, substitute the coordinates of the points into the equation to form a system of equations about the coefficients. Solve the system to find the coefficients, and then determine the equation.
Combining Like Terms and Arranging: After writing the equation, combine like terms (e.g., ) and arrange the terms in descending order of the degree of the unknown to ensure the equation is in a standard form.
Verification Method: After writing the equation, substitute the known roots or the coordinates of the points into the equation to check if the equality holds, so as to verify the correctness of the equation.