Polynomial Functions Study Guide

Definition and Components of Polynomial Functions

• Definition of a Polynomial Function:

  • A polynomial function is a function in the form:     $f(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \dots + a_2 x^2 + a_1 x + a_0$
  • The exponent $n$ is any positive integer.
  • The coefficients $a_n, a_{n-1}, a_{n-2}, \dots, a_2, a_1, a_0$ are real numbers.
  • In words, a polynomial function is a function with decreasing powers of $x$ and with real coefficients.

• Examples Evaluated for Polynomial Classification:

  • Example 31: $f(x) = -2x^3 + 7x^2 + 5x - 4$ is a polynomial function with decreasing powers of $x$ and real coefficients.
  • Example 32: $g(x) = 7x^5 + \frac{2}{3}x^3 - 14x^2 + 6x - 4$ is a polynomial function.
  • Example 33: $h(x) = 11x^2 + 8x - 3 + \frac{7}{x}$ is not a polynomial function because it contains the term $\frac{7}{x} = 7x^{-1}$, which has a negative exponent.

• Terms of a Polynomial Function:

  • The individual components of a polynomial function are called terms.
  • Terms are always added to or subtracted away from each other.

• Degrees of Terms:

  • The exponent that goes with each term is called the degree of the term.
  • Example 34: Analysis of terms in $8x^3 - 12x^2 + 3x - 7$:
  • Term $8x^3$ has degree $3$.
  • Term $-12x^2$ has degree $2$.
  • Term $3x$ has degree $1$.
  • Term $-7$ has degree $0$

• Leading Term, Leading Coefficient, and Polynomial Degree:

  • The term containing the highest degree, represented as $a_n x^n$, is called the leading term.
  • The coefficient $a_n$ of the leading term is called the leading coefficient.
  • The degree of the polynomial function is the exponent $n$ of the leading term.
  • Example 35: For the polynomial $8x^3 - 12x^2 + 3x - 7$:
  • The leading term is $8x^3$.
  • The leading coefficient is $8$.
  • The degree of the polynomial is $3$.
  • Example 36: For the polynomial $4x^5 - 6x^3 + 8x - 2$:
  • The leading term is $4x^5$.
  • The degree of the polynomial is $5$.

Special Names and Classification of Polynomial Functions

• Special Names Based on Degree:

  • A zero degree function (degree $0$) is called a constant function.
  • A first degree function (degree $1$) is called a linear function.
  • A second degree function (degree $2$) is called a quadratic function.
  • A third degree function (degree $3$) is called a cubic function.
  • A fourth degree function (degree $4$) is called a quartic function.

• Example 37: Identification of Functions and Their Properties:

  • Part a: $f(x) = 3x^4 - 7x^3 - 2x + 4$
  • Classification: Polynomial function.
  • Degree: $4$.
  • Leading term: $3x^4$.
  • Leading coefficient: $3$.
  • Part b: $f(x) = -\frac{3}{5}x^2 - 7x + 2$
  • Classification: Polynomial function.
  • Degree: $2$.
  • Leading term: $-\frac{3}{5}x^2$.
  • Leading coefficient: $-\frac{3}{5}$.
  • Part c: $f(x) = 3\sqrt{x} + 2x - 7$
  • Classification: Not a polynomial function.
  • Reason: Contains a radical term $3\sqrt{x} = 3x^{1/2}$, which has a fractional exponent ($\frac{1}{2}$) rather than a non-negative integer exponent.

Common Properties of All Polynomial Functions

• Domain of Polynomial Functions:

  • The domain of every polynomial function is all real numbers, written in interval notation as $(-\infty, \infty)$.

• Continuity and Smoothness:

  • The graph of a polynomial function is always continuous and smooth.
  • Continuous: There are no holes, breaks, or gaps anywhere in the graph.
  • Smooth: There are no sharp bends, corners, or sharp twists.
  • Non-polynomial graph features (features that polynomial graphs do not possess):
  • Corners (sharp turns or bends).
  • Holes.
  • Breaks or gaps.

End Behavior and the Leading Term Test

• Definition of End Behavior:

  • The end behavior of a function is the direction the graph is headed as $x$ is moving far to the left ($x \rightarrow -\infty$) and as $x$ is moving far to the right ($x \rightarrow \infty$).
  • Determines whether the graph is heading up ($y \rightarrow \infty$) or heading down ($y \rightarrow -\infty$) at each extremity.

• Role of the Leading Term:

  • The leading term is the term in a polynomial that is most responsible for the value of the polynomial for values of $x$ that are far away from 0$.\n * Example 40: For f(x) = 3x^5 - 6x^4 + 8x^3 - 6x^2 + 9x - 2$, if$x$is a very large positive number, the value of the leading term$3x^5$is positive and so large that it overshadows the value of the other terms. As$x gets bigger, the value of the polynomial gets bigger.\n\n* Rules for the Leading Term Test:\n * Even-Degreed Polynomial Functions:\n * The behavior is the same on both ends of the graph.\n * If the leading coefficient is positive (a_n > 0), the graph goes UP on both sides (left up, right up).\n * If the leading coefficient is negative (a_n < 0), the graph goes DOWN on both sides (left down, right down).\n * Odd-Degreed Polynomial Functions:\n * The behavior on the ends is opposite (up on one end and down on the other).\n * If the leading coefficient is positive (a_n > 0), the graph goes DOWN on the left and UP on the right.\n * If the leading coefficient is negative (a_n < 0), the graph goes UP on the left and DOWN on the right.\n\n* Examples Applying the Leading Term Test:\n * Example 41: Describe the end behavior of f(x) = 5x^4 - 7x^3 + 11x^2 - 3x + 2\n * Leading term: 5x^4.\n * Degree: 4 (even).\n * Leading coefficient: 5 (positive).\n * End behavior: Up on the left, up on the right.\n * Example 42: Describe the end behavior of f(x) = -2x^6 + 3x^5 - 12x^3 + 6x^2 - 2x + 19\n * Leading term: -2x^6.\n * Degree: 6 (even).\n * Leading coefficient: -2 (negative).\n * End behavior: Down on the left, down on the right.\n * Example 43: Describe the end behavior of f(x) = -4x^5 + 7x^3 - 6x^3 + 2x^2 - 8x + 1\n * Leading term: -4x^5.\n * Degree: 5 (odd).\n * Leading coefficient: -4 (negative).\n * End behavior: Up on the left, down on the right.\n * Example 44: Describe the end behavior of f(x) = 6x^3 + 7x^2 + 3x - 8\n * Leading term: 6x^3.\n * Degree: 3 (odd).\n * Leading coefficient: 6 (positive).\n * End behavior: Down on the left, up on the right.\n\n# Zeros of Polynomial Functions\n\n* Definition of a Zero:\n * A zero is a number in the domain of a function that makes the function evaluate to zero.\n * Formal definition: A number c$is a zero of a function$f(x)$if$f(c) = 0\n * Example 45: For f(x) = x^2 - 11x + 18$,$x = 2$and$x = 9$are both zeros of$f(x)$because$f(2) = 0$and$f(9) = 0\n * A zero is strictly a domain value (x-value).\n\n* Equivalent Statements:\n * All five of the following statements are equivalent and mean the exact same thing:\n * x = c$is a zero of$f(x)\n * f(c) = 0\n * x = c$is a solution to the equation$f(x) = 0\n * c$is an$x$-intercept of the graph of$f(x)\n * (x - c)$is a factor of$f(x)\n * Example 46: If x = 5$is a zero of the function$f(x) = x^2 - 3x - 10, then:\n * f(5) = 0\n * x = 5$is a solution to$x^2 - 3x - 10 = 0\n * 5$is an$x$-intercept of the graph of$f(x)\n * (x - 5)$is a factor of$f(x)\n\n* Number of Possible Zeros:\n * An n$\text{th} degree polynomial function must have$n zeros.\n * Zeros can be rational (can be written as a fraction), irrational (containing radicals), or imaginary.\n * Zeros may repeat (twin zeros, triple zeros, quadruple zeros, etc.).\n\n* Finding Real Zeros by Factoring:\n * Factoring methods used to find real zeros include:\n * Greatest Monomial Factoring (GMF)\n * Factoring by Grouping\n * Special Binomial Factoring\n * Trinomial Factoring\n * Example 47: Find the real zeros of f(x) = 6x^3 - 33x^2 - 18x\n * Since f(x) is a 3rd degree polynomial, there are 3 zeros.\n * Factor out GMF 3x$:$f(x) = 3x(2x^2 - 11x - 6)\n * Factor trinomial: f(x) = 3x(2x + 1)(x - 6)\n * Set each factor to zero:\n * 3x = 0 \Rightarrow x = 0\n * 2x + 1 = 0 \Rightarrow x = -\frac{1}{2}\n * x - 6 = 0 \Rightarrow x = 6\n * The zeros are: x = 0$,$x = -\frac{1}{2}$,$x = 6\n * Example 48: Find the real zeros of f(x) = 4x^3 + 8x^2 - 144x - 288\n * Factor out GMF 4$:$f(x) = 4(x^3 + 2x^2 - 36x - 72)\n * Factor by grouping inside parentheses: x^2(x + 2) - 36(x + 2) = (x^2 - 36)(x + 2)\n * Factor difference of squares: f(x) = 4(x - 6)(x + 6)(x + 2)\n * Set each factor to zero:\n * x - 6 = 0 \Rightarrow x = 6\n * x + 6 = 0 \Rightarrow x = -6\n * x + 2 = 0 \Rightarrow x = -2\n * The zeros are: x = 6$,$x = -6$,$x = -2\n\n# Multiplicity of Zeros and Graphical Behavior\n\n* Definition of Multiplicity:\n * The multiplicity of a zero is the number of times that zero appears as a solution.\n * If the function is in factored form, the multiplicity tells how many times that zero appears in a factor (the exponent on that factor).\n * Example 49: For f(x) = x^5(x - 2)(x + 7)^3(x - 5):\n * Zero x = 0$has multiplicity$5\n * Zero x = 2$has multiplicity$1\n * Zero x = -7$has multiplicity$3\n * Zero x = 5$has multiplicity$1\n\n* Graphical Behavior According to Multiplicity:\n * Multiplicity of 1:\n * The graph will cross the x$-axis at that zero (go from one side of the$x-axis to the other).\n * Even Multiplicity (2, 4, 6, 8, etc.):\n * The graph will touch the x$-axis and then immediately reverse direction without crossing (the graph is tangent to the$x-axis).\n * Odd Multiplicity Greater than 1 (3, 5, 7, 9, etc.):\n * The graph will approach the x$-axis, flatten out, and then cross the$x-axis.\n\n* Example 50: Analysis of a Factored 6th Degree Polynomial:\n * Function: f(x) = (x - 1)(x - 4)^2(x + 2)^3\n * Behavior at zero x = 1$(multiplicity$1$): Graph crosses the$x-axis.\n * Behavior at zero x = 4$(multiplicity$2$): Graph touches the$x-axis and turns around (tangent).\n * Behavior at zero x = -2$(multiplicity$3$): Graph flattens out and crosses the$x-axis.\n * End Behavior: Degree is 1 + 2 + 3 = 6$(even) and leading coefficient is positive$1, so the graph goes up on both ends.\n\n# Graph Sketching of Polynomial Functions\n\n* Information Required to Project a Polynomial Graph:\n * End behavior\n * Zeros (x-intercepts)\n * Multiplicity of each zero\n\n* Example 51: Complete Analysis of f(x) = -2x^2(x - 4)^3(x + 7)^4\n * Part a: Degree of the function:\n * Calculated by adding the multiplicities (exponents): 2 + 3 + 4 = 9\n * Degree is 9\n * Part b: Leading term:\n * Obtained by multiplying leading coefficients and terms: -2 \cdot x^2 \cdot x^3 \cdot x^4 = -2x^9\n * Leading term is -2x^9\n * Part c: End behavior of the function:\n * Degree 9$(odd) and leading coefficient$-2 (negative).\n * Graph goes up on the left (y \rightarrow \infty$as$x \rightarrow -\infty$) and down on the right ($y \rightarrow -\infty$as$x \rightarrow \infty).\n * Part d: Zeros of f(x):\n * x = 0$,$x = 4$,$x = -7\n * Part e: Multiplicity and behavior for each zero:\n * x = 0$: Multiplicity$2$(even) -> Touches the$x-axis and turns around (tangent).\n * x = 4$: Multiplicity$3$(odd > 1) -> Approaches, flattens out, and crosses the$x-axis.\n * x = -7$: Multiplicity$4$(even) -> Touches the$x-axis and turns around (tangent).\n * Part f: Graph Sketch Pathway:\n * Begins high in Quadrant II (coming down from positive infinity on the left).\n * Meets the x$-axis at$x = -7, touches it, and rebounds back up.\n * Reaches a local peak, turns downward toward x = 0\n * Meets the x$-axis at$x = 0, touches it, and rebounds back up.\n * Reaches a local peak, turns downward toward x = 4\n * Meets the x$-axis at$x = 4$$, flattens out, crosses through the axis into Quadrant IV, and continues downward toward negative infinity on the far right.