Notes on Simplifying Rational Expressions and Finding Zeros

Rational Expression: An algebraic fraction where both the numerator and the denominator are polynomials.

Excluded Values

Excluded values are specific values for the variable that make the denominator equal to zero, leading to undefined expressions. They must be excluded in calculations.

Examples of Excluded Values:

  1. For the expression \frac{36 + 2}{b + 7}:

    • Set denominator: b + 7 = 0

    • Excluded value: b = -7

  2. For the expression \frac{5x + 2}{x^2 - x - 12}:

    • Factor denominator:
      x^2 - x - 12 = (x - 4)(x + 3)

    • Set each factor to zero:
      x - 4 = 0 \Rightarrow x = 4
      x + 3 = 0 \Rightarrow x = -3

    • Excluded values: x = 4, -3

  3. For the expression \frac{2x + 1}{2x + 1}:

    • Set denominator: 2x + 1 = 0

    • Excluded value: x = -\frac{1}{2}

Simplifying Rational Expressions

To simplify a rational expression, factor both the numerator and the denominator, and then cancel any common factors.

Example Simplification:

Given expression: \frac{32x^3 y^2}{8x^4 y^5}:

  • Factor:
    \frac{32}{8} \cdot \frac{y^2}{y^5} \cdot \frac{x^3}{x^4}

  • Result after simplification:
    = 4 \cdot \frac{1}{y^3} \cdot \frac{1}{x}
    = \frac{4}{xy^3}

Factorization Examples:

  1. For the expression \frac{4x + 16}{x^2 - 5x - 36}:

    • Factor numerator: 4(x + 4)

    • Factor denominator: x^2 - 5x - 36 = (x-9)(x+4)

    • Simplified form:
      \frac{4(x + 4)}{(x - 9)(x + 4)} = \frac{4}{x - 9}

    • Excluded value: x = -4 (from the cancellation)

Difference of Perfect Squares

Recall that a difference of squares like 25 - x^2 can be factored as:
25 - x^2 = (5 - x)(5 + x)

Finding Zeros of a Function

To find the zeros of a function, set the function equal to zero and solve for the variable.

Example:

Given the function f(x) = x^2 + 8x + 7:

  • Set it to zero: 0 = x^2 + 8x + 7

  • Factor:
    0 = (x + 1)(x + 7)

  • Solutions (Zeros):
    x + 1 = 0 \Rightarrow x = -1
    x + 7 = 0 \Rightarrow x = -7
    We identify zeros at x = -1 and x = -7

Finding Intercepts

X-intercepts can be found at the zeros of the function.
To find a Y-intercept, substitute zero for x in the function.