Polynomial Functions - Video Notes

Polynomial Functions - Video Notes

• What is a polynomial function?

  • A polynomial is an expression built from variables, coefficients, constants, and nonnegative integer exponents.
  • Example given: $y = 2x^2 + 3$ is a polynomial; when we view it as a function, we often write it as $f(x) = 2x^2 + 3$.
  • Distinction between polynomial (the algebraic expression) and polynomial function (the mapping from input to output): a function has a one-to-one relation between the independent variable (usually x) and the dependent variable (usually y).

• Key parts of a polynomial (and what they do):

  • Variables: the parts that can change (e.g., x).
  • Coefficients: the numbers in front of the variable terms (e.g., the 2 in $2x^2$ or the 5 in $5x$).
  • Constant term: a term without a variable (e.g., the $-2$ in $4x^3 + 5x - 2$).
  • Exponents: the powers attached to the variables (e.g., the 2 in $x^2$, the 3 in $x^3$). These determine the degree.
  • Degree: the highest exponent of a variable in the polynomial (for one-variable polynomials, the maximum exponent among its terms with nonzero coefficient).
  • Example: In $f(x) = 4x^3 + 5x - 2$, the degree is $3$.

• Why we write $f(x)$ for a function

  • It indicates a function with input (independent variable) $x$ and output (dependent variable) $f(x) = y$.
  • A function has a one-to-one relationship between the input and output in the sense that each input value maps to exactly one output value.
  • Graphical test: the vertical line test.
  • If a vertical line intersects the graph at more than one point, the relation is not a function.
  • If it intersects at exactly one point for every vertical line, it is a function.

• What makes something a polynomial function? (Rules for exponents)

  • Exponents must be whole numbers (natural numbers): $1, 2, 3, \, \dots$
  • Exponents cannot be zero for the variable term if you want a nonconstant polynomial, though a zero exponent yields a constant term (e.g., $x^0 = 1$).
  • If an exponent is not a nonnegative integer (e.g., $x^{1/2}$ or negative or fractional exponents), the expression may be a function but not a polynomial.
  • Examples of polynomials by this rule:
  • $2x$ (degree 1)
  • $2x^2 + 3x$ (degree 2)
  • $4x^3 + x^2 + 7x - 9$ (degree 3)
  • $5x^4 + 6x^3 - x^2 - 12x + 14$ (degree 4)
  • Note on the zero exponent: any term with $x^0$ is a constant term.

• Standard form of a polynomial

  • Mathematicians write a general polynomial in standard form as a sum of coefficients times powers of $x$ in descending order of degree:
  • $P(x) = a<em>0 x^n + a</em>1 x^{n-1} + a<em>2 x^{n-2} + \cdots + a</em>{n-1} x + a_n,$
  • where $n$ is a nonnegative integer (the degree) and $a_0 \neq 0$ (the leading coefficient).
  • This form can be thought of as starting from $x^n$ and multiplying by coefficients that step down the exponent by 1 each term until the constant term (the $x^0$ term).
  • If you start with a specific degree (e.g., $n=4$), you have terms from $x^4$ down to $x^0$: $a<em>0 x^4 + a</em>1 x^3 + a<em>2 x^2 + a</em>3 x + a_4.$
  • The video’s language about a generic “a0, a1, …” aligns with this standard form idea.

• Degrees and names of polynomials (in this course)

  • Degree 0: constant function (e.g., $f(x) = 7$).
  • Degree 1: linear (e.g., $f(x) = 7x$) – also called linear.
  • Degree 2: quadratic – also called parabolic (e.g., $f(x) = 9x^2 + 3x - 1$).
  • Degree 3: cubic (e.g., $f(x) = x^3 + x^2 + x$).
  • Degrees 4, 5: quartic, quintic, etc. (e.g., $f(x) = 5x^4 + 6x^3 - x^2 - 12x + 14$ is quartic).
  • The video notes that higher degrees exist, but the focus here is on degrees up to 3.

• Graphical overview: functions, polynomials, and non-polynomials

  • Function (vertical line test): a graph where every vertical line crosses the graph at most once.
  • Polynomial function: a graph that is a function and whose equation is a polynomial.
  • A graph that is a function but not a polynomial example: $y = \sqrt{x}$ (or $y = x^{1/2}$) is a function but not a polynomial because the exponent is not a natural number.
  • A graph that fails the vertical line test is not a function (even if it looks piecewise or has a break).

• Domain and range concepts (as discussed in the video)

  • Domain: the set of all possible input values (the x-values) for which the function is defined.
  • Range: the set of all possible output values (the y-values).
  • For many polynomials, the domain is all real numbers: $(-\infty, \infty)$.
  • If you have a scatter-like relation or discrete values, the domain could be all integers: {-\infty < x < \infty,\; x \in \mathbb{Z}}.
  • In some graphs, the function may be restricted by a condition (e.g.,