Math Problem Statement

p(x) = (x^2 - 4) / (2x + 1); Find p(-8).

Solution

The given function is:

p(x)=x242x+1p(x) = \frac{x^2 - 4}{2x + 1}

You are asked to find p(8)p(-8), which means substituting x=8x = -8 into the function.

Let's compute it step-by-step:

p(8)=(8)242(8)+1p(-8) = \frac{(-8)^2 - 4}{2(-8) + 1}

  1. First, calculate (8)24(-8)^2 - 4: (8)2=64,644=60(-8)^2 = 64, \quad 64 - 4 = 60

  2. Next, calculate 2(8)+12(-8) + 1: 2(8)=16,16+1=152(-8) = -16, \quad -16 + 1 = -15

  3. Now, substitute these values into the equation: p(8)=6015=4p(-8) = \frac{60}{-15} = -4

Thus, p(8)=4p(-8) = -4.

Would you like more details on any step?

Here are 5 follow-up questions you might consider:

  1. What is the domain of p(x)p(x)?
  2. How do you find the vertical asymptotes of this rational function?
  3. Can you simplify x242x+1\frac{x^2 - 4}{2x + 1} by factoring?
  4. What is the behavior of p(x)p(x) as xx \to \infty?
  5. What would p(x)p(x) be if you change the numerator to x2+4x^2 + 4?

Tip: When evaluating a rational function, always check whether the denominator is zero at the substitution point to avoid undefined values.

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Math Problem Analysis

Mathematical Concepts

Algebra
Rational Functions
Substitution

Formulas

p(x) = (x^2 - 4) / (2x + 1)
Substitute x = -8 into p(x)

Theorems

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Suitable Grade Level

Grades 9-12