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 $f(x) = an x^n + a{n-1} x^{n-1} + … + a1 x + a0$, where $an, a{n-1}, …, a0$ 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)−26f(-7) = 7(-7)^3 - 10(-7)^2 + 14(-7) - 26
        =7(−343)−10(49)−98−26= 7(-343) - 10(49) - 98 - 26
        =−2401−490−98−26=−3015= -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)y = (x - 2)(x + 4)(x - 3) is equivalent to x3−x2−14x+24x^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.