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Chapter 5: Problem 11

The following exercises are of mixed variety. Factor each polynomial. $$ x^{3}-1000 $$

Short Answer

Expert verified
The factorization of \x^3 - 1000\ is \(x - 10)(x^2 + 10x + 100)\.

Step by step solution

01

- Identify Polynomial Form

Recognize that the polynomial is of the form \(a^3 - b^3\), where \a = x\ and \b = 10. This helps in applying the appropriate factorization formula for the difference of cubes.
02

- Apply Difference of Cubes Formula

Use the factorization formula for the difference of cubes: \[a^3 - b^3 = (a - b)(a^2 + ab + b^2) \] Substituting the identified values \a = x\ and \b = 10\, we get: \[x^3 - 1000 = (x - 10)(x^2 + 10x + 100) \]
03

- Simplify the Expression

Combine the terms to finalize the factorized form: \[x^3 - 1000 = (x - 10)(x^2 + 10x + 100) \]

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Key Concepts

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

Difference of Cubes
The 'difference of cubes' is a special technique used to factorize polynomials of the form \(a^3 - b^3\). This form is called the 'difference of cubes' because it represents the subtraction (difference) of two cubed terms.

To factorize such expressions, we apply the 'difference of cubes' formula:
\ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \

Using this formula makes the factorization process straightforward:
    \* You identify the values for \ a \ and \ b \. In the example \ x^3 - 1000 \, here \ a = x \ and \ b = 10 \(x = x \ and \ 1000 = 10^3 \) \*You then substitute these values into the formula to get the factorized expression.

Let鈥檚 consider our example again:

When we apply the formula to \ x^3 - 1000 \, we get: \ (x - 10)(x^2 + 10x + 100) \

This results in the factorized form of the given polynomial.
Factorization Formula
Factorization formulas are essential tools in algebra. They help break down complex polynomials into simpler factors. This makes it easier to simplify, solve, or manipulate polynomial expressions without hassle.

The most common factorization formulas include:
    \*Difference of Cubes Formula \*Sum of Cubes Formula \*Difference of Squares Formula <\ul>

    In our example, we used the 'difference of cubes' formula: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). This formula turns the problem of factoring a cubic polynomial into a much simpler task.

    The key steps are:
      <\*Identify the form of the polynomial \(Difference of cubes in our case\)\ \*Substitute the values of \ a \ and \ b \ into the formula \(Here, \ a = x \ and \ b = 10\)\


    Using these steps correctly ensures accurate factorization and prepares you for more complex algebraic manipulations.
Polynomial Expressions
Polynomial expressions consist of variables and coefficients, structured in a way where variables are raised to non-negative integer powers.

Polynomials can appear simple, like \ x^2 + 3x + 2 \ or complex, like \(x^5 + 4x^3 - x + 7 \). Regardless of their structure, we can apply various factorization techniques to simplify them.

The example \ x^3 - 1000 \ shows a polynomial expression that looks complex initially. However, by recognizing it as a 'difference of cubes,' it becomes easier to factor.

Factoring polynomials:
    <\* Reduces larger expressions to their simplest forms \(Here, \ x^3 - 1000\ became \ (x - 10)(x^2 + 10x + 100)\)\ <\* Helps identify roots or solutions to polynomial equations \* Makes it simpler to perform further algebraic operations

By mastering polynomial expressions and their factorization, you'll find many algebra problems much easier to handle.

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Most popular questions from this chapter

Factor each polynomial. $$ 64 m^{2}-512 m^{3}-81 n^{2}+729 n^{3} $$

Solve each equation. $$ 2 x^{2}-12-4 x=x^{2}-3 x $$

Factor each polynomial. $$ 27-1000 x^{9} $$

Solve each equation. $$ 25 x^{3}=64 x $$

The binomial \(x^{6}-y^{6}\) may be considered as either a difference of squares or a difference of cubes. Factor \(x^{6}-y^{6}\) by first factoring as a difference of squares. Then factor further by considering one of the factors as a sum of cubes and the other factor as a difference of cubes.
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