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Chapter 1: Problem 95

Factor the expression completely. $$2 x^{3}+4 x^{2}+x+2$$

Short Answer

Expert verified
The expression factors to \((x + 2)(2x^2 + 1)\).

Step by step solution

01

Group Terms

The expression given is \(2x^3 + 4x^2 + x + 2\). First, we'll group the terms into two pairs: \((2x^3 + 4x^2)\) and \((x + 2)\). This strategic grouping will help us simplify the factorization process.
02

Factor Each Group

For the first group, \(2x^3 + 4x^2\), notice that \(2x^2\) is a common factor. Factoring it out gives us \(2x^2(x + 2)\). For the second group, \(x + 2\), there is no common factor other than 1, so it remains \(x + 2\). Now, the expression is \(2x^2(x + 2) + 1(x + 2)\).
03

Factor Out the Common Binomial Factor

Now, observe that \((x + 2)\) is a common binomial factor in both terms of the expression. Factor \((x + 2)\) out: \((x + 2)(2x^2 + 1)\).
04

Confirm and Simplify

We now have \((x + 2)(2x^2 + 1)\) as the complete factorization of the original expression. Double-check by expanding to ensure it results in the original expression. Expanding gives \(2x^3 + 4x^2 + x + 2\), confirming that the factorization is correct.

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Key Concepts

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

Grouping Terms
Grouping terms is a strategic part of polynomial factorization. It involves arranging the terms in a way that simplifies the process of finding common factors. In our exercise, the aim was to factor the expression \(2x^3 + 4x^2 + x + 2\). To start, we divided these terms into two pairs: \((2x^3 + 4x^2)\) and \((x + 2)\). This is the grouping step.
  • Why do we group? To identify potential common factors that are not apparent when the expression is viewed as a whole.
  • Group terms intelligently. Consider which terms have some similarity, like shared factors or structure.
Whether you are factoring a simple polynomial or a more complex expression, grouping gives a clearer path to factorization and helps in identifying manageable pieces. Always remember, the goal is simplification, which eventually leads to successful factorization.
Common Factor
Identifying a common factor in math expressions is crucial as it simplifies parts of the polynomial, making further factorization more straightforward. In our example, once we grouped \(2x^3 + 4x^2\), we identified \(2x^2\) as a common factor.
  • What is a common factor? It鈥檚 a number or algebraic expression that evenly divides each term in a group.
  • Why is it important? Pulling out a common factor reduces each term, highlighting structures we can further factor or simplify.
  • If no common factor exists other than 1, as in \(x + 2\), consider next steps like other factorization strategies.
After factoring \(2x^2\) from \(2x^3 + 4x^2\), the expression simplifies to \(2x^2(x + 2)\). Finding and using common factors helps us cleanly and efficiently simplify complex polynomials.
Binomial Factor
A binomial factor is an expression consisting of two terms, which can serve as a common factor for a polynomial. In our process of factoring \(2x^3 + 4x^2 + x + 2\), we noticed that after grouping and factoring out common factors, the remnants \((x + 2)\) appeared in both terms of our expression, making it a binomial factor to extract.
  • What makes a factor binomial? It's when it acts as a shared factor between multiple terms or expressions.
  • It's not always evident right away, especially if you haven't grouped or factored out other simpler factors first.
  • By identifying \((x + 2)\) as a binomial factor, we managed to factor out our expression into \((x + 2)(2x^2 + 1)\).
Knowing how to spot and extract binomial factors from polynomial expressions is a key skill in algebra. It allows for successful factorization and simplification, reducing larger expressions into their fundamental components.

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Most popular questions from this chapter

State whether the given equation is true for all values of the variables. (Disregard any value that makes a denominator zero.) $$\frac{1+x+x^{2}}{x}=\frac{1}{x}+1+x$$

Factor the expression completely. $$\left(a^{2}-1\right) b^{2}-4\left(a^{2}-1\right)$$

Complete the squares in the general equation \(x^{2}+a x+y^{2}+b y+c=0\) and simplify the result as much as possible. Under what conditions on the coefficients \(a, b,\) and \(c\) does this equation represent a circle? A single point? The empty set? In the case that the equation does represent a circle, find its center and radius.

Plot the points \(P(0,3)\) \(Q(2,2),\) and \(R(5,3)\) on a coordinate plane. Where should the point \(S\) be located so that the figure \(P Q R S\) is a parallelogram? Write a brief description of the steps you took and your reasons for taking them.

Show that the equation represents a circle, and find the center and radius of the circle. $$3 x^{2}+3 y^{2}+6 x-y=0$$
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