Develop, generalize, explain, and apply long division for dividing polynomials by binomials of the form $x-a,a∈I$.

Compare long division of polynomial expressions by binomial expressions to synthetic division, and explain why synthetic division works.

Divide a polynomial expression by a binomial expression of the form $x-a,a∈I$ using synthetic division.

Explain the relationship between the linear factors of a polynomial expression and the zeros of the corresponding polynomial function.

Generalize, through inductive reasoning, the relationship between the remainder when a polynomial expression is divided by $x-a,a∈I$ and the value of the polynomial expression at $x=a$ (The Remainder Theorem).

Explain and apply the factor theorem to express a polynomial expression as a product of factors.

Categorize, with justification, a set of functions into polynomial functions and non-polynomial functions.

Analyze graphs of polynomial functions to determine the impact of changing the values of the constant term and leading coefficient in the equation of a polynomial function with respect to the graph of the function.

Generalize and apply strategies for graphing polynomial functions of an odd or even degree.

Explain the relationship between:

• the zeros of a polynomial function,
• the roots of the corresponding polynomial equation,
• the x-intercepts of the graph of the polynomial function.

Explain and apply strategies for determining the behaviour of the graph of a polynomial function at zeros with different multiplicities.

Sketch, with or without the use of technology, the graph of a polynomial function.

Solve situational questions by modelling the situations with polynomial functions and analyzing the graphs of the functions.