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Chapter 6: Problem 19

Factor out the greatest common monomial factor from the polynomial. $$ x^{2}+x $$

Short Answer

Expert verified
The polynomial factored out by the greatest common monomial factor is \(x(x+1)\).

Step by step solution

01

Identify the GCF

In the given polynomial \(x^{2} + x\), observe that the monomials \(x^{2}\) and \(x\) have the GCF of \(x\).
02

Factor out the GCF

Factor out \(x\) from both the terms. The expression becomes \(x(x + 1)\). Here, \(x\) is being multiplied by the quantities in the brackets. If you expand the factored expression by multiplying \(x\) with all terms inside the brackets, you should get the original polynomial.

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Key Concepts

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

Greatest Common Factor
The Greatest Common Factor (GCF) is a fundamental concept in algebra, especially when factoring polynomials. It refers to the highest number or expression that divides all terms in a polynomial without leaving a remainder. Finding the GCF is the first step when simplifying polynomial expressions.

To find the GCF:
  • List the factors of each term in the polynomial.
  • Select the largest factor that appears in each list.
In the polynomial \(x^2 + x\), the terms are \(x^2\) and \(x\). Both terms share \(x\) as a factor. Therefore, \(x\) is the GCF of the polynomial. Factoring a polynomial by the GCF simplifies the expression, making it easier to work with. Remember, identifying the GCF correctly is crucial for the accuracy of the factorization.
Monomials
Monomials play a critical role in the structure of polynomials. A monomial is a mathematical expression that consists of a single term. This term can be a constant, a variable, or both multiplied together, such as \(7\), \(x\), or \(5x^3\).

When dealing with polynomials, they are composed of multiple monomials combined through addition or subtraction. For instance, in the polynomial \(x^2 + x\), \(x^2\) and \(x\) are both monomials.
  • Each monomial in a polynomial can have coefficients and literal parts (variables and their powers).
  • Identifying individual monomials is essential for operations such as addition, subtraction, and specifically, factoring.
Grasping the concept of monomials helps in understanding more complex polynomial expressions and their manipulations.
Polynomial Expressions
Polynomial expressions are mathematical expressions that involve multiple terms, or monomials, grouped together. These terms are combined using addition or subtraction.

In the example \(x^2 + x\), there's a polynomial with two terms.
  • The terms of a polynomial consist of coefficients and variables raised to whole number exponents.
  • Polynomials can be simplified by factoring, which often involves finding the GCF.
  • Factoring helps to express the polynomial in a more manageable form, such as \(x(x+1)\) for \(x^2 + x\).
Understanding the structure of polynomials allows one to perform various algebraic operations, such as addition, subtraction, multiplication, and division. Comprehending polynomial expressions is key to solving algebraic equations and functions.

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Most popular questions from this chapter

Explain how to identify and factor a perfect square trinomial.

Find two consecutive positive integers whose product is 240 .

Solve the equation. $$ 3 x^{2}-2 x=9-8 x $$

In Exercises \(45-52\), solve the equation. $$ 5 x^{2}-24=37 x $$

Is the equation \(2 x^{2}+18 x=0\) in general form? Explain.
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