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Chapter 0: Problem 21

$$ \text { Factor the polynomials in Exercises } \text { . } $$ $$ 3 x-x^{2} $$

Short Answer

Expert verified
The factored form of the polynomial is \(x(3 - x)\).

Step by step solution

01

- Identify the common factor

First, look at the terms of the polynomial and identify the common factor. Here, the polynomial is given as \(3x - x^2\). Both terms share a common factor of \(x\).
02

- Factor out the common factor

Factor out the common factor \(x\) from each term in the polynomial. This gives us:\[3x - x^2 = x(3 - x)\]

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Key Concepts

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

Common Factors
In algebra, identifying common factors is a crucial skill. When you work with polynomials, look for terms that share a common factor. This simplifies the expression and makes subsequent steps easier. In the exercise, we look at the polynomial \(3x - x^2\). Both terms, \(3x\) and \( - x^2\), include \(x\) as a common factor. Notice that \(x\) is the smallest power of \(x\) present in each term. By factoring out the common factor \(x\), we simplify the polynomial to \(x(3 - x)\). When identifying common factors:
  • Look for a variable or number that is a part of each term.
  • Factor it out by placing it outside of a parenthesis.
  • Rewrite the polynomial with the factored expression inside the parenthesis.
This step is like finding a common denominator in fractions, making it easier to manage and simplify algebraic expressions.
Factoring Polynomials
Factoring polynomials involves breaking down a complex polynomial into simpler parts (factors) that multiply together to form the original equation. This process can reveal important properties and roots of the polynomial. In our example, factoring \(3x - x^2\) simplified it by extracting \(x\) as a common factor, leading to the expression \(x(3 - x)\). Think of factoring as splitting the polynomial into pieces that are easier to work with. Here are some key points:
  • Start by identifying and extracting the greatest common factor (GCF).
  • Look for patterns such as the difference of squares or trinomial squares.
  • Rewrite the polynomial in a factored form for simplicity.
Factoring polynomials is an essential tool in algebra, simplifying equations for solving or further manipulation.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations (such as addition, subtraction, multiplication, and division). Understanding how to work with these expressions is vital. In the exercise, we saw the polynomial \(3x - x^2\), which is an algebraic expression involving a variable \(x\) and constants. When dealing with algebraic expressions, keep the following in mind:
  • Follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
  • Keep terms with like variables together for easier manipulation.
  • Simplify expressions by combining like terms.
By recognizing and simplifying algebraic expressions, you can more easily solve equations and understand higher-level math concepts.

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Most popular questions from this chapter

In Exercises \(47-50\), find the zeros of the function. (Use the specified viewing window.) $$ f(x)=x^{2}-x-2 ;[-4,5] \text { by }[-4,10] $$

Annual Compound with Deposits Assume that a couple invests \(\$ 4000\) each year for 4 years in an investment that earns \(8 \%\) compounded annually. What will the value of the investment be 8 years after the first amount is invested?

Semiannual Compound Assume that a \(\$ 1000\) investment earns interest compounded semiannually. Express the value of the investment after 2 years as a polynomial in the annual rate of interest \(r\).

A cellular telephone company estimates that, if it has \(x\) thousand subscribers, its monthly profit is \(P(x)\) thousand dollars, where \(P(x)=12 x-200\). (a) How many subscribers are needed for a monthly profit of 160 thousand dollars? (b) How many new subscribers would be needed to raise the monthly profit from 160 to 166 thousand dollars?

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. \(\frac{-3 x}{15 x^{4}}\)
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