Precalculus Midterm

Front

Define a polynomial of least degree with integral coefficients.

Back

A polynomial of least degree with integral coefficients is the simplest polynomial that has the given roots and is expressed in standard form, where the coefficients are whole numbers.

Front

What are the zeros of a polynomial?

Back

The zeros of a polynomial are the values of x for which the polynomial evaluates to zero. They are also known as the roots of the polynomial.

Front

State Descartes's Rule of Signs.

Back

Descartes's Rule of Signs provides a way to determine the possible number of positive and negative real roots of a polynomial based on the number of sign changes in the polynomial's coefficients.

Front

How do you find the inverse of a function?

Back

To find the inverse of a function, swap the x and y variables in the equation and solve for y.

Front

What is an extraneous solution?

Back

An extraneous solution is a solution that emerges from the process of solving an equation but does not satisfy the original equation.

Front

What is the standard form of a polynomial?

Back

Front

How do you check for extraneous solutions?

Back

To check for extraneous solutions, substitute the solution back into the original equation to see if it holds true.

Front

What is the significance of rational coefficients in polynomials?

Back

Rational coefficients in polynomials ensure that the polynomial can be expressed in a form that is easier to manipulate and solve, particularly when finding roots.

Front

What is the process for solving rational equations?

Back

To solve rational equations, first eliminate the denominators by multiplying through by the least common denominator, then solve the resulting polynomial equation.

Front

What is the relationship between the degree of a polynomial and the number of roots?

Back

The degree of a polynomial indicates the maximum number of roots (real or complex) it can have, according to the Fundamental Theorem of Algebra.