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Chapter 6: Problem 15

Factor completely. $$ t^{3}-1000 $$

Short Answer

Expert verified
The expression factorizes to

Step by step solution

01

- Recognize the difference of cubes

Identify that the expression is a difference of cubes. In this case, it is written as:
02

- Write the difference of cubes formula

Use the difference of cubes formula: where in this instance,
03

- Substitute and simplify

Substitute into the difference of cubes formula: Simplify to get the factorized form:

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Key Concepts

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

Factoring Polynomials
Factoring polynomials involves breaking them down into simpler components that, when multiplied together, give you the original polynomial.
It's like reversing multiplication. To factor a polynomial, we use techniques like finding common factors, using special formulas, and recognizing patterns.
In the case of the exercise, we dealt with the difference of cubes, which is a specific method of factoring.
To factor the polynomial completely, we used a specialized formula that simplifies the expression into more manageable parts.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations (like addition, subtraction, multiplication, and division).
The expression from our exercise, \(t^{3} - 1000\), consists of a variable \(t\) and a number 1000.
When factoring algebraic expressions, we often look for patterns or special cases, like in this instance with the difference of cubes.
Knowing how to manipulate these expressions correctly is key to solving many algebra problems.
Polynomial Identities
Polynomial identities are standard formulas that always hold true for polynomials.
These identities help us simplify and factor polynomials efficiently.
One crucial identity used in the exercise was the difference of cubes formula: \(a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2}) \).
This identity allows us to quickly convert \(t^{3} - 1000\) into its factored form.
By recognizing and applying polynomial identities, we can solve problems more easily.
Cube Roots
Cube roots are the number that, when multiplied by itself three times, gives the original value.
For example, the cube root of 1000 is 10 because \(10 \times 10 \times 10 = 1000\).
In the difference of cubes exercise, we recognize both \(t^{3}\) and 1000 as perfect cubes.
Knowing the cube roots helps us factor them using the difference of cubes formula.
The polynomial \(t^{3} - 1000\) becomes easier to handle once we realize \(t^{3} = (t)^{3}\) and \(1000 = 10^{3}\).

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Most popular questions from this chapter

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ r^{3}-3 r^{2}-9 r+27 $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ q^{2}+1 $$

Determine whether each of the following is a difference of squares. $$ -12+25 t^{2} $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ 16-a^{2}-2 a b-b^{2} $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ t^{3}+8 t^{2}-t-8 $$
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