Factor by any method. See Examples 1–7. $$1000x3+343y31000x^3$+343y^3$$

Ch. R - Review of Basic Concepts

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

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Lial 13th Edition

Ch. R - Review of Basic Concepts

Problem 103

Chapter 1, Problem 103

Factor by any method. See Examples 1–7. $1000 x cubed plus 343 y cubed$

Verified step by step guidance

Recognize that the expression $1000x^3 + 343y^3$ is a sum of cubes because $1000x^3 = (10x)^3$ and $343y^3 = (7y)^3$.

Recall the sum of cubes factoring formula: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.

Identify $a = 10x$ and $b = 7y$ in the given expression.

Apply the formula by substituting $a$ and $b$: write the factorization as $(10x + 7y)((10x)^2 - (10x)(7y) + (7y)^2)$.

Simplify the terms inside the second factor: calculate $(10x)^2$, $(10x)(7y)$, and $(7y)^2$ to complete the factorization.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sum of Cubes Formula

The sum of cubes formula states that a³ + b³ = (a + b)(a² - ab + b²). This formula is used to factor expressions where two terms are each perfect cubes added together. Recognizing 1000x³ and 343y³ as cubes allows applying this formula directly.

Identifying Perfect Cubes

A perfect cube is a number or expression raised to the third power, such as 1000 = 10³ and 343 = 7³. Identifying each term as a perfect cube is essential before applying the sum of cubes formula, ensuring the expression fits the pattern a³ + b³.

Factoring Polynomials

Factoring polynomials involves rewriting an expression as a product of simpler polynomials. Using special formulas like the sum of cubes helps break down complex expressions into factors, which can simplify solving equations or further algebraic manipulation.

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Related Practice

Textbook Question

Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, M = {0, 2, 4, 6, 8}, N = {1, 3, 5, 7, 9, 11, 13}, Q = {0, 2, 4, 6, 8, 10, 12}, and R = {0, 1, 2, 3, 4}.Use these sets to find each of the following. Identify any disjoint sets. Q ∩ (M ∪ N)

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Textbook Question

Add or subtract as indicated. 18 - 2.789

Textbook Question

Perform each division. See Examples 9 and 10.

$\frac{(10 x^{3} + 11 x^{2} - 2 x + 3)}{(5 x + 3)}$