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Chapter 0: Problem 15

Factor each polynomial by factoring out the common monomial factor. $$ 2 x^{2}-2 x $$

Short Answer

Expert verified
2x (x - 1)

Step by step solution

01

- Identify the Common Factor

Look at both terms in the polynomial. The terms are 2x^{2} and -2x. The common factor for both terms is 2x.
02

- Factor Out the Common Factor

Take out the common factor 2x from each term:\[ 2x^{2} - 2x = 2x (x - 1) \]

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Key Concepts

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

Common Monomial Factor
When you factor a polynomial, one of the key techniques you'll use is identifying the common monomial factor.
This is a term that is present in each term of the polynomial.
In the given exercise, the polynomial is \(2x^2 - 2x\).
Both terms, \(2x^2\) and \(-2x\), share a common factor of \(2x\).

**Steps to Find the Common Monomial Factor:**

  • Look at the coefficients (the numerical part of the terms). In our case, both coefficients are \(2\) and \(-2\).
  • Identify the variables present in both terms. Here, both terms include the variable \(x\).
  • Combine the common pieces: the common numeric factor, and the common variable to the lowest power present in all terms. In this example, it's \(2x\).


Finding and factoring out the common monomial factor simplifies your polynomial into a form that is easier to work with.
In our example, after applying this step, we get:

\[2x^2 - 2x = 2x(x - 1)\]
Polynomial Factorization
Polynomial factorization means breaking down a polynomial into simpler 'factors' that, when multiplied together, give you the original polynomial.
This process often makes solving equations and understanding the polynomial much simpler.

In the example \(2x^2 - 2x\), we see two terms:
  • First term: \(2x^2\)
  • Second term: \(-2x\)


Each term can clearly be divided by \(2x\), our common monomial factor.
When we factor out \(2x\), it simplifies to \(2x(x - 1)\).

So, how did we get here? We did this by acknowledging that:
  • \(2x^2 \div 2x = x\)
  • \(-2x \div 2x = -1\)

Thus, we write the polynomial as a product of the common factor and the simplified terms:
\(2x(x - 1)\).
Factorization gives us a streamlined version that's easier to manipulate and solve for variables if needed.
Algebra Basics
In algebra, understanding the basics is crucial to mastering more complex concepts like polynomial factorization.

**Here's what you need to know:**
  • Polynomials are algebraic expressions that consist of terms made up of variables and coefficients.
  • Terms are separated by a plus (+) or minus (-) sign.
  • Exponents indicate how many times a variable is multiplied by itself.
  • A common monomial factor is a term that can evenly divide each term in the polynomial.


**Let's break down the basics using our example \(2x^2 - 2x\):**

  • Polynomial: An expression like \(2x^2 - 2x\) is a polynomial.
  • Terms: The expression \(2x^2 - 2x\) has two terms: \(2x^2\) and \(-2x\).
  • Coefficients: These are the numerical parts of the terms (2 in this case).
  • Exponents: The exponent of \(x\) in the term \(2x^2\) is 2.

By identifying terms, coefficients, and exponents, you can simplify algebraic expressions and solve equations more easily.
Remember, algebra builds on itself鈥攎astering basics like these will help you tackle more complex problems.

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Most popular questions from this chapter

Simplify each expression. Assume that all variables are positive when they appear. $$\sqrt[4]{32 x}+\sqrt[4]{2 x^{5}}$$

Simplify each expression. Assume that all variables are positive when they appear. $$9 \sqrt[3]{24}-\sqrt[3]{81}$$

Rationalize the denominator of each expression. Assume that all variables are positive when they appear. $$\frac{-\sqrt{3}}{\sqrt{5}}$$

Simplify each expression. Express your answer so that only positive exponents occur. Assume that the variables are positive. $$ \frac{\left(4 x^{-1} y^{1 / 3}\right)^{3 / 2}}{(x y)^{3 / 2}} $$

Rationalize the numerator of each expression. Assume that all variables are positive when they appear. $$\frac{4-\sqrt{x-9}}{x-25} x \neq 25$$
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