📘Intermediate Algebra Unit 5 – Polynomials and Polynomial Functions

Key Concepts and Definitions

• Polynomials are expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication
• Degree of a polynomial refers to the highest power of the variable in the polynomial (quadratic polynomial has degree 2)
• Leading coefficient is the coefficient of the term with the highest degree
• Constant term is the term without a variable
• Monomial is a polynomial with only one term ($3x^2$)
• Binomial is a polynomial with exactly two terms ($2x^3 + 5x$)
• Trinomial is a polynomial with exactly three terms ($4x^2 - 3x + 1$)

Polynomial Structure and Classification

• Standard form of a polynomial arranges terms in descending order of degree ($ax^n + bx^{n-1} + \cdots + kx + c$)
• Polynomials can be classified by degree (linear, quadratic, cubic)
• Polynomials can be classified by number of terms (monomial, binomial, trinomial)
• Zero polynomial is a polynomial where all coefficients are zero ($0x^3 + 0x^2 + 0x + 0$)
• Polynomial functions are functions defined by polynomial expressions ($f(x) = 2x^3 - 5x^2 + 3x - 1$)
  • Domain of a polynomial function is all real numbers
  • Degree of a polynomial function determines its end behavior

Operations with Polynomials

• Adding polynomials involves combining like terms ($(3x^2 + 2x) + (2x^2 - 4x) = 5x^2 - 2x$)
• Subtracting polynomials involves distributing the negative sign and combining like terms ($(4x^3 - 3x^2) - (2x^3 + x^2) = 2x^3 - 4x^2$)
• Multiplying polynomials uses the distributive property and combines like terms ($(2x + 3)(x - 1) = 2x^2 + x - 3$)
  • FOIL method can be used to multiply binomials
• Dividing polynomials can be done using long division or synthetic division
• Remainder Theorem states that the remainder when a polynomial $P(x)$ is divided by $(x - a)$ is equal to $P(a)$
• Factor Theorem states that $(x - a)$ is a factor of a polynomial $P(x)$ if and only if $P(a) = 0$

Graphing Polynomial Functions

• Polynomial functions can be graphed by plotting points or using transformations
• Degree of a polynomial determines the maximum number of turning points (degree 3 has at most 2 turning points)
• Even-degree polynomials have the same end behavior in both directions (both ends up or both ends down)
• Odd-degree polynomials have opposite end behavior (one end up, one end down)
• Multiplicity of a zero determines the behavior of the graph at that x-intercept (even multiplicity bounces off x-axis, odd multiplicity crosses x-axis)
  • Multiplicity is the number of times a factor appears in the factored form of the polynomial
• y-intercept is found by evaluating the function at x = 0

Solving Polynomial Equations

• Setting a polynomial function equal to zero and solving for x finds the x-intercepts or zeros of the function
• Factoring can be used to solve polynomial equations if the polynomial can be factored ($(x - 2)(x + 3) = 0$ gives $x = 2$ or $x = -3$)
• Quadratic Formula can be used to solve quadratic equations ($ax^2 + bx + c = 0$ has solutions $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$)
• Rational Root Theorem generates a list of possible rational zeros of a polynomial (for $P(x) = ax^n + \cdots + c$, possible rational zeros are $\pm \frac{p}{q}$ where $p$ factors of $c$ and $q$ factors of $a$)
• Descartes' Rule of Signs gives the maximum number of positive and negative real zeros of a polynomial based on sign changes in the coefficients
  • Maximum number of positive real zeros equals the number of sign changes in $P(x)$
  • Maximum number of negative real zeros equals the number of sign changes in $P(-x)$

Polynomial Applications

• Polynomial functions can model real-world situations (projectile motion, population growth)
• Polynomial regression can be used to find a polynomial function that best fits a set of data points
• Optimization problems can be solved by finding the maximum or minimum value of a polynomial function
  • Derivative of a polynomial function gives the slope of the tangent line at any point
  • Critical points (where derivative equals zero) can be used to find local maxima and minima
• Polynomial functions are used in computer graphics and animation (Bezier curves)

Common Mistakes and Tips

• Remember to distribute negative signs when subtracting polynomials
• Be careful with the order of operations when evaluating polynomial functions (PEMDAS)
• When factoring, make sure all terms are factored completely (including GCF)
• Check your answers by plugging them back into the original equation
• Graph polynomials to check the reasonableness of your solutions (x-intercepts should match zeros)
  • Use a graphing calculator or computer algebra system to check your work
• When using the Rational Root Theorem, test each possible zero until you find all the actual zeros
  • Synthetic division can be used to test possible zeros more efficiently

Practice Problems and Solutions

1. Find the degree and leading coefficient of $P(x) = 3x^5 - 2x^3 + 7x - 1$

   • Degree: 5, Leading coefficient: 3

2. Multiply $(2x - 3)(x^2 + 4x - 5)$

   • $(2x - 3)(x^2 + 4x - 5)$
   • $= 2x^3 + 8x^2 - 10x - 3x^2 - 12x + 15$
   • $= 2x^3 + 5x^2 - 22x + 15$

3. Find all zeros of $f(x) = x^3 - 5x^2 - 2x + 24$

   • Possible rational zeros (Rational Root Theorem): $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24$

   • $f(1) = 1 - 5 - 2 + 24 = 18 \neq 0$

   • $f(-1) = -1 - 5 + 2 + 24 = 20 \neq 0$

   • $f(2) = 8 - 20 - 4 + 24 = 8 \neq 0$

   • $f(-2) = -8 - 20 + 4 + 24 = 0$, so $x = -2$ is a zero

   • Synthetic division:

     $1$	$-5$	$-2$	$24$
     	$-2$	$6$	$12$
     ------	------	------	------
     $1$	$-7$	$4$	$36$

   • Quadratic factor: $x^2 - 7x + 12 = (x - 3)(x - 4)$

   • Zeros: $x = -2, 3, 4$

4. Graph $y = -x^3 + 5x^2 - 4x + 1$ and state the end behavior

   • Degree 3, odd, so one end up and one end down
   • As $x \to -\infty$, $y \to -\infty$ (down)
   • As $x \to \infty$, $y \to \infty$ (up)
   • y-intercept: $f(0) = 1$
   • Zeros (from factoring): $x = -1, 1, 4$
   • Graph:

      |
   4 -|        /
      |       /
   2 -|      /
      |     /
   0 -|----/---------
      |   /
  -2 -|  /
      | /
  -4 -|/
      |
      +-+-+-+-+-+-+-+-
       -2  0  2  4  6