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Chapter 5: Problem 93

Factor each expression. $$(3 x-y)\left(x^{2}-2\right)+\left(x^{2}-2\right)$$

Short Answer

Expert verified
The factorized form of the expression is \( (3x - y + 1)(x^{2} - 2) \).

Step by step solution

01

Identify the common factor

The common factor in both terms is \(x^{2}-2\).
02

Separate common factor

Rewrite the original expression by factoring out the common factor. So, the expression becomes \( (3x - y + 1)(x^{2} - 2) \).\n
03

Check your work

Distribute to check the work. \( (3x - y + 1) \times (x^{2} - 2) = (3x \times x^{2} - (y \times x^{2}) + x^{2} - 2 \times 3x + 2y - 2) = (3x^{3} - yx^{2} + x^{2} - 6x + 2y - 2), which is the original expression, confirming that the factorization is correct.

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Key Concepts

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

Factoring Out Common Factors
Factoring out common factors is an essential step in simplifying algebraic expressions. When working with expressions, the first step is always to look for terms that share a common element. In the given exercise, the common factor is clearly identified as \(x^2 - 2\). This expression appears in both terms, making it the key to successful factoring.

To factor out the common factor:
  • Identify the shared factor across the terms of the expression.
  • Rewrite the expression as a product, with the common factor outside a set of parentheses.
  • This simplifies the original expression into a more manageable form.
By taking \(x^2 - 2\) out of both terms, the expression becomes \((3x - y + 1)(x^2 - 2)\). This step makes the rest of the problem easier to handle and reduces the complexity of further manipulation.
Polynomial Expressions
Polynomial expressions are expressions consisting of variables, coefficients, and constants combined using addition, subtraction, and multiplication. The expression in this exercise, \((3x - y)(x^2 - 2) + (x^2 - 2)\), is a polynomial.

To understand polynomial expressions, remember:
  • Each part of the expression separated by a plus or minus sign is called a term.
  • The degree of a polynomial is determined by the highest power of the variable.
  • Polynomials can be simplified by combining like terms or factoring.
Generally, polynomial expressions are foundational in algebra and are used to express a wide variety of mathematical situations. Recognizing the structure of polynomials, as shown here, helps in further steps of algebraic manipulation.
Algebraic Manipulation
Algebraic manipulation involves the strategic rearrangement and simplification of expressions to make them easier to work with. This skill is crucial for solving problems and understanding the properties of different expressions. In this exercise, the manipulation involved converting the given polynomial into a simpler, factored form.

The key aspects of algebraic manipulation include:
  • Recognizing and extracting common factors, as demonstrated in the initial step.
  • Rewriting the expression to facilitate easier calculations or further simplification.
  • Verifying steps, such as using the distributive property to ensure the factorization is correct, as shown in the final step where the original expression is reformed from its factors.
Practicing these techniques strengthens your ability to solve complex problems and enhances your agility in handling algebraic expressions.

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Most popular questions from this chapter

Factor. Write each trinomial in descending powers of one variable, if necessary. If a polynomial is prime, so indicate. $$u^{2}-3 v^{2}+2 u v$$

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Factor. If the polynomial is prime, so indicate. $$4 x^{2}+10 x-6$$

Factor. $$x^{2}+10 x+25$$

For what values of \(b\) will the trinomial \(5 y^{2}-b y-3\) be factorable?
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