TAKE HOME!! *AA C&C FLASHCARD: Graphs of Polynomials (Parts 1-3)

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form is $$p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$$ , where $$a_n, a_{n-1}, ..., a_0$$ are constants and $$n$$ is a non-negative integer.

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.

To find the zeros of a polynomial, set the polynomial equal to zero and solve for $$x$$ . This can involve factoring, using the quadratic formula, or other algebraic methods.

The end behavior of a polynomial function describes how the function behaves as $$x$$ approaches positive or negative infinity. It is determined by the leading term of the polynomial.

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. It affects the end behavior of the polynomial function.

The degree of a polynomial is the highest power of the variable in the polynomial expression. It determines the shape and number of zeros of the graph.

The multiplicity of a zero indicates how many times that zero is repeated in the polynomial. A zero with an even multiplicity will touch the x-axis, while a zero with an odd multiplicity will cross the x-axis.

The maximum number of turning points of a polynomial function is one less than its degree. For example, a polynomial of degree 4 can have up to 3 turning points.

A polynomial of degree $$n$$ can have up to $$n$$ real zeros, counting multiplicities. It may have fewer real zeros if some are complex.

The factored form of a polynomial expresses it as a product of its linear factors. For example, $$p(x) = (x - r_1)(x - r_2)...(x - r_n)$$ , where $$r_1, r_2, ..., r_n$$ are the zeros.