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Chapter 0: Problem 13

Factor each polynomial by factoring out the common monomial factor. $$ x^{3}+x^{2}+x $$

Short Answer

Expert verified
The polynomial \(x^{3} + x^{2} + x\) factors to \(x(x^{2} + x + 1)\).

Step by step solution

01

Identify the Common Factor

Look for the greatest common factor (GCF) in the polynomial. The terms in the polynomial are \( x^{3}, x^{2}, x \). The GCF of these terms is \( x \).
02

Factor Out the Common Monomial Factor

Once the GCF is identified, factor it out from each term in the polynomial: \[ x^{3} + x^{2} + x = x(x^{2}) + x(x) + x(1) \]
03

Simplify the Expression

Combine the factored terms: \[ x(x^{2} + x + 1) \]This is the factored form of the polynomial.

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Key Concepts

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

greatest common factor
In mathematics, the greatest common factor (GCF) is the highest factor that divides two or more numbers.
When factoring polynomials, finding the GCF of the terms is the first step.
For example, in the polynomial \( x^3 + x^2 + x \), the GCF is \( x \).
This is because each term includes at least one \(\text{x}\). By identifying the GCF, we can simplify the polynomial.
monomials
A monomial is a single term consisting of a coefficient, a variable, or a product of both.
Examples of monomials are \(x^2\), \(3y\), and \(7\).
In our problem, the GCF \(x\) is a monomial.
Understanding monomials is crucial when factoring, as you often identify a common monomial to factor polynomials.
This allows you to simplify complex expressions into simpler parts.
polynomial simplification
Simplifying polynomials involves reducing them to their simplest form.
This can include factoring out the greatest common factor.
For instance, the polynomial \(x^3 + x^2 + x\) can be simplified.
First, we identify the GCF, which is \(x\). Factoring it out, we get \(x(x^2 + x + 1)\).
This is a more simplified and manageable form of the original polynomial.
algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols.
Understanding algebraic concepts like polynomials, monomials, and the greatest common factor is essential.
These concepts form the basis for solving many types of equations.
Polynomials, for example, are expressions consisting of variables and coefficients.
By using algebraic techniques like factoring, you can simplify and solve polynomial equations effectively.
This sets the foundation for more advanced mathematical concepts.

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Most popular questions from this chapter

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

Simplify each expression. $$\left(\frac{8}{9}\right)^{-3 / 2}$$

Expressions that occur in calculus are given. Write each expression as a single quotient in which only positive exponents and radicals appear. $$\frac{\sqrt{x^{2}+1}-x \cdot \frac{2 x}{2 \sqrt{x^{2}+1}}}{x^{2}+1}$$

The period \(T\), in seconds, of a pendulum of length \(l,\) in feet, may be approximated using the formula $$T=2 \pi \sqrt{\frac{l}{32}}$$ Express your answer both as a square root and as a decimal approximation. Find the period \(T\) of a pendulum whose length is 16 feet.

Expressions that occur in calculus are given. Factor each expression. Express your answer so that only positive exponents occur. $$\left(x^{2}+4\right)^{4 / 3}+x \cdot \frac{4}{3}\left(x^{2}+4\right)^{1 / 3} \cdot 2 x$$
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