/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 26 Factor the polynomials. $$x^{3... [FREE SOLUTION] | 91影视

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Chapter 0: Problem 26

Factor the polynomials. $$x^{3}+125$$

Short Answer

Expert verified
The factorization of \(x^{3} + 125\) is \((x + 5)(x^{2} - 5x + 25)\).

Step by step solution

01

Identify the polynomial structure

The given polynomial is of the form \(x^{3} + 125\). Notice that 125 is a perfect cube, as \(125 = 5^{3}\). Hence, the polynomial can be seen as a sum of cubes: \(a^{3} + b^{3}\), where \(a = x\) and \(b = 5\).
02

Apply the sum of cubes formula

The sum of cubes formula is: \(a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})\).
03

Substitute a and b into the formula

Here, \(a = x\) and \(b = 5\). Substitute these values into the sum of cubes formula: \[(x^{3} + 5^{3}) = (x + 5)(x^{2} - 5x + 25)\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sum of Cubes Formula
To factor the polynomial, we used the sum of cubes formula. The formula states that the sum of two cubes, \(a^{3} + b^{3}\), can be factored as \( (a + b)(a^{2} - ab + b^{2}) \). The formula is crucial because it breaks down complex cubic expressions into simpler binomial and trinomial forms.
For example, in the exercise, we identified that the polynomial \(x^{3} + 125\) could be seen as \(a^{3} + b^{3}\), where \(a = x\) and \(b = 5\). By substituting these values into the formula, we factored the sum of cubes as \((x + 5)(x^{2} - 5x + 25)\).

This formula is extremely useful for simplifying and solving polynomials that are sums of perfect cubes.
Polynomials
A polynomial is a mathematical expression consisting of variables (also called indeterminates) and coefficients. Polynomials involve only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
In the exercise, the given polynomial was \(x^{3} + 125\), a cubic polynomial because it has a variable raised to the power of three. When working with polynomials, identifying their structure is key to solving them effectively.

Here are some important points about polynomials:
  • A polynomial can have one or more terms. For example, \(5x^{2} + 3x + 1\) is a trinomial with three terms.
  • The degree of a polynomial is the highest power of the variable in the polynomial. In \(x^{3} + 125\), the degree is 3.
  • Polynomials are classified based on their degree: linear (degree 1), quadratic (degree 2), cubic (degree 3), and so on.
Perfect Cubes
Perfect cubes are numbers or expressions that can be written as the cube of an integer or another expression.
In the exercise, we noted that \(125\) is a perfect cube because it can be expressed as \(5^{3}\). This recognition allowed us to treat the polynomial \(x^{3} + 125\) as a sum of cubes.

Here are some more examples of perfect cubes:
  • 1 is a perfect cube because \(1 = 1^{3}\).
  • 8 is a perfect cube because \(8 = 2^{3}\).
  • 27 is a perfect cube because \(27 = 3^{3}\).
Recognizing perfect cubes is valuable when factoring polynomials, making it easier to apply formulas like the sum of cubes or the difference of cubes. Identifying perfect cubes saves time and simplifies the factoring process.

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Most popular questions from this chapter

Let \(f(x)=\sqrt[3]{x}\) and \(g(x)=\frac{1}{x^{2}} .\) Calculate the following functions. Take \(x > 0\). $$[f(x) g(x)]^{3}$$

If \(g(t)=4 t-t^{2},\) find \(\frac{g(t+h)-g(t)}{h}\) and simplify.

Calculate the compound amount from the given data. principal \(=\$ 20,000,\) compounded quarterly, 3 years, annual rate \(=12 \%\)

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. $$\frac{1}{x^{-3}}$$

Use your graphing calculator to find the value of the given function at the indicated values of \(x .\) $$f(x)=\frac{2 x-1}{x^{3}+3 x^{2}+4 x+1} ; \quad x=2, x=6$$
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