g(x)=(x^2-5 x+25)/(x^3+125)

Name: Problem 24, Chapter 1: Calculus of a Single Variable
Uploaded: 2019-07-24T16:38:27-08:00
Duration: 7 min 43 s
Channel: Carlos Pinilla
Description: g(x)=(x^2-5 x+25)/(x^3+125)

step 1 So you have the following function. We have an x squared minus 5 x plus 25 over x cube plus 125.

Question

$$g(x)=\frac{x^{2}-5 x+25}{x^{3}+125}$$

   $$g(x)=\frac{x^{2}-5 x+25}{x^{3}+125}$$

Calculus of a Single Variable

Ron Larson 11th Edition

Chapter 1, Problem 24

Instant Answer

Solved by Expert Carlos Pinilla

07/24/2019

Step 1

The sum of cubes can be factored as $(a+b)(a^2-ab+b^2)$. So, we have $x^3+125 = x^3+5^3 = (x+5)(x^2-5x+25)$. Show more…

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$$g(x)=\frac{x^{2}-5 x+25}{x^{3}+125}$$

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Key Concepts

Polynomial Factorization

This concept involves breaking down polynomials into the product of simpler polynomials. In the context of the expression provided, recognizing that the denominator is a sum of cubes enables one to factor it using the identity a³ + b³ = (a + b)(a² - ab + b²), which simplifies analysis and further manipulation of the function.

Simplification of Rational Functions

Simplification involves canceling out common factors between the numerator and denominator of a rational function. This procedure not only leads to a more compact expression but also highlights features such as removable discontinuities and the true domain of the function. It is important to ensure that cancellation is valid by verifying that the canceled factor does not vanish in the original function's domain.

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