A polynomial in x has degree 3 . The coefficient of x^2 is 3...

A polynomial in $x$ has degree $3 .$ The coefficient of $x^{2}$ is 3 less than the coefficient of $x^{3} .$ The coefficient of $x$ is three times the coefficient of $x^{2} .$ The remaining constant is 2 more than the coefficient of $x^{3} .$ The sum of the coefficients is $-4 .$ Find the polynomial.

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Key Concepts

Sum of Coefficients

The sum of the coefficients of a polynomial is obtained by substituting 1 into the polynomial for the variable. This calculation simplifies the expression to a numerical value and is a useful method for linking all coefficient values, especially when a condition is provided on the sum.

Setting Up and Solving Equations

Translating relationships and conditions among polynomial coefficients into algebraic equations is a vital problem-solving strategy. By setting up these equations based on given conditions, one can solve for unknown coefficients systematically, ensuring that all the specified conditions are satisfied.

Degree of a Polynomial

The degree of a polynomial is defined as the highest exponent of the variable with a non-zero coefficient. Understanding the degree of a polynomial helps in identifying its fundamental characteristics, such as the maximum number of roots and the end behavior of the polynomial function.

Coefficients of a Polynomial

Coefficients are the numerical factors that multiply each power of the variable in a polynomial. They play a critical role in determining the shape and position of the polynomial's graph, and relationships or conditions among these coefficients can be used to characterize and solve for unknown elements of the polynomial.

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