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Chapter 0: Problem 129

What does it mean to factor completely?

Short Answer

Expert verified
Factoring completely means breaking down an algebraic expression into its simplest forms, where no further factoring is possible. It's simplifying an expression to its smallest building blocks, or factors, which are typically prime numbers and variables.

Step by step solution

01

Understanding Factoring

Factoring refers to breaking an algebraic expression down into a product of simpler terms. These 'factors' when multiplied together will result in the original expression.
02

Define Complete Factoring

When we talk about factoring 'completely', it means that we continue the process of factoring until we cannot factor the expression any further. In other words, an expression is said to be factored completely if no common factors other than 1 can be found within the terms of an expression.
03

Example

As an example, consider the expression \(36x^2y\). We can factor this as \(6 \cdot 6 \cdot x \cdot x \cdot y\), but that's not completely factored. Factoring completely would look like this: \(2 \cdot 2 \cdot 3 \cdot 3 \cdot x \cdot x \cdot y\). Now, every factor is a prime number, or a letter, and we have factored completely.

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Most popular questions from this chapter

Evaluate each algebraic expression for the given value or values of the variable(s). $$\frac{2 x+y}{x y-2 x}, \text { for } x=-2 \text { and } y=4$$

$$\text { factor completely.}$$ $$x^{4}-y^{4}-2 x^{3} y+2 x y^{3}$$

Your local electronics store is having an end-of-the-year sale. The price on a plasma television had been reduced by \(30 \% .\) Now the sale price is reduced by another \(30 \%\). If \(x\) is the television's original price, the sale price can be modeled by $$(x-0.3 x)-0.3(x-0.3 x)$$ a. Factor out \((x-0.3 x)\) from each term. Then simplify the resulting expression. b. Use the simplified expression from part (a) to answer these questions. With a \(30 \%\) reduction followed by a \(30 \%\) reduction, is the television selling at \(40 \%\) of its original price? If not, at what percentage of the original price is it selling?

$$\text { factor completely.}$$ $$(x-5)^{-\frac{1}{2}}(x+5)^{-\frac{1}{2}}-(x+5)^{\frac{1}{2}}(x-5)^{-\frac{3}{2}}$$

What is a perfect square trinomial and how is it factored?
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