Algebra II 3.1-3.5 Review Notes

Algebra II 3.1-3.5 Review Notes

Polynomial Functions

• Definition of Polynomial Function: A polynomial function is a function that can be expressed in the form of $f(x) = a<em>n x^n + a</em>{n-1} x^{n-1} + … + a<em>1 x + a</em>0$, where $a<em>n, a</em>{n-1}, …, a<em>0$ are coefficients, $n$ is a non-negative integer, and $an
  eq 0.

Examples of Polynomial Functions:

1. Function: $f(x) = 3x + 5x^3 - 6x^2 + 2$

   • Degree: 3

   • Type: Cubic

   • Leading Coefficient: 5

2. Function: $f(x) = 8x - 6x + 2x^3 - \sqrt{7}x + x^2 - 2$

   • Degree: 3

   • Type: Cubic

   • Leading Coefficient: 2

   • Note: The function has a degree of 3 as the highest power of $x$ is 3.

Evaluating Functions

• Evaluate the Function: The process to determine the value of a function for a given $x$.

  • Example: Evaluate $f(x) = 7x^3 - 10x^2 + 14x - 26$ at $x = -7$:

    • Calculation:
      $f(-7) = 7(-7)^3 - 10(-7)^2 + 14(-7) - 26$
      $= 7(-343) - 10(49) - 98 - 26$
      $= -2401 - 490 - 98 - 26 = -3015$

End Behavior of Functions

• Description of End Behavior: Refers to the behavior of the graph of the polynomial function as $x$ approaches either positive or negative infinity.

  • When analyzing the end behavior, consider the leading term's degree and coefficient.

  • For the function $g(x) = 6x^3 + x^2 - 12x - 3$:

    • The leading term, $6x^3$, has a positive coefficient for an odd degree, indicating that as $x o ext{positive infinity}$, $g(x) o ext{positive infinity}$, and as $x o ext{negative infinity}$, $g(x) o ext{negative infinity}$.

  • Contrast with the function $g(x) = -12x^2 + 4x^3 + 8 + x$:

    • Here, the leading term $4x^3$ also has a positive coefficient. End behavior will be similar.

Graphing Functions

• Graphing a cubic function: Example with $g(x) = x^3 + x + 3$.

  • The graph will display characteristics of cubic functions such as:

    • A single curve generally changing directions once.

  • Ensure to include points for creating accurate representations of the function's behavior.

• Graphing the Parabola: Example with $f(x) = x^2 + 3$:

  • The vertex is at (0,3).

  • The parabola opens upwards, typical for quadratics where the leading coefficient is positive.

Factoring Polynomials

• Finding Factors: Determine if a binomial is a factor of a polynomial and factor the polynomial completely.

  • Example: For $g(x) = x^2 - 22x - 40$:

  • Verify that $x - 5$ is a factor.

  • Factor completely to find $g(x) = (x - 5)(x + 8)$.

Solving Polynomial Equations

• Solving the equations involves finding the values of $x$ that satisfy the equation:

1. Equation: $4x^2 + 12x^2 + 9x = 0$

   • Factoring out common terms leads to solutions for $x$.

2. Equation: $6h^2 = 12h$

   • Rearranging gives $h(h - 2) = 0$, leading to roots $h = 0$ or $h = 2$.

3. Equation: $16p^{13} - 8p^2 + p = 0$

   • Factoring and using the zero product property can help in finding the roots.

4. Equation: $643 - 124 = 0$

   • Solve for variables set to zero.

Writing Polynomial Functions

• Constructing Polynomial Functions: Create a polynomial function of least degree with rational coefficients, leading coefficient of 1, and given zeroes:

  • Given zeroes: 2, 4, 3

  • Resulting function:
    $y = (x - 2)(x + 4)(x - 3)$ is equivalent to $x^3 - x^2 - 14x + 24$.

Finding Zeros of Functions

• Finding Zeros and Graphing: Discover zeros of given polynomials and sketch graphs accordingly.

  • Example function: $f(x) = 3x^2 + 24x^2 + 48x$

  • Find zeros, then graph the function.

• Sketch: Important to represent zeros graphically, indicating where the function crosses the x-axis.

Synthetic Division and Evaluation

1. Synthetic Division: Method of dividing polynomial functions.

   • Example: For $p(x) = 4x^2 - 9x + 2$ using $x = 3$.

2. Synthetic Substitution: Evaluating polynomials using synthetic division.

   • Function: $h(x) = -4x^3 + 2x^2 + 5x - 6; x = -3$ \n - Calculation using synthetic division to evaluate can provide quick results.