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Chapter 6: Problem 90

Give an example of the use of the Distributive Property to factor by grouping.

Short Answer

Expert verified
By using Distributive Property and the method of factor by grouping, the example \(3x^2 + 6x + 2x - 4\) simplifies to \((x + 2)(3x + 2)\).

Step by step solution

01

Create an Example

Let's take an example of a polynomial, say \(3x^2 + 6x + 2x - 4\). This is the expression on which we'll apply the distributive property and factor by grouping.
02

Group the Terms

Begin by grouping the similar terms together. Here, \(3x^2 + 6x\) and \(2x - 4\) form two distinct groups.
03

Using Distributive Property

Now apply the distributive property by taking out common factors in each group. In the first group \(3x( x + 2 )\) and in second group \(2( x - 2 )\).
04

Factor by Grouping

For both groups, we have a common term \((x + 2)\). Taking it common, we have \((x + 2)(3x + 2)\). This is the final factored form by grouping.

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Key Concepts

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

Factoring by Grouping
Factoring by grouping is a useful technique when dealing with polynomials, particularly those with four terms. It involves grouping terms with common factors, making it easier to simplify the expression.
  • First, examine the polynomial and try identifying pairs of terms that share common factors. You often start with adjacent terms, but sometimes rearranging may be necessary for proper grouping.
  • After grouping, apply the distributive property to each group. This means you factor out the greatest common factor (GCF) from each subset of terms.
  • With the GCF factored out from each group, you will likely find a common binomial factor throughout the polynomial.
  • Finally, rewrite the expression as a product of these common factors to achieve the factored form.
This method is highly efficient when no single common factor exists across all terms of the polynomial. It breaks the problem down into smaller, more manageable parts, making it easier to handle even complex expressions.
Polynomials
Polynomials are expressions consisting of variables and coefficients, involving terms in which the variables have whole number exponents. Understanding polynomials is crucial when working with algebraic expressions.
  • Terms of a polynomial include monomials, binomials, and trinomials, based on the number of terms they have.
  • The degree of a polynomial is determined by the highest power of the variable within its terms. For example, the degree of the polynomial \(3x^2 + 6x + 2x - 4\) is 2.
  • Operations with polynomials include addition, subtraction, multiplication, and division, each requiring precise application of arithmetic rules.
  • Factoring polynomials, like factoring by grouping, simplifies the expression and exposes its roots or solutions.
Understanding how to manipulate polynomials through various operations allows for solving algebraic equations and understanding functions in math.
Algebraic Expressions
Algebraic expressions incorporate constants, variables, and algebraic operations. These expressions form the backbone of solving equations and exploring algebraic concepts.
  • Common elements of algebraic expressions include variables (e.g., \(x\), \(y\)), coefficients (numeric multipliers of variables), and constants (fixed numeric values).
  • Unlike equations, algebraic expressions do not contain an equals sign and therefore do not state a relationship between expressions; they represent a value or set of values.
  • Algebraic expressions can be simplified or factored using various algebraic properties like distribution and combining like terms.
  • Understanding and using properties such as the distributive property, associative property, and commutative property is essential in manipulating and simplifying these expressions.
Proficiency with algebraic expressions paves the way for solving more complex problems and understanding higher-level mathematics concepts.

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Most popular questions from this chapter

In Exercises 67-74, factor the polynomial completely. $$ x^{2}-81 $$

In Exercises 59-62, determine whether the statement is true or false. Justify your answer. The only solution of \(x^{2}=4 x\) is \(x=4\).

Evaluate the quantity mentally using the two samples as models. $$ \begin{array}{rlrl} 29^{2}=(30-1)^{2} & =30^{2}-2 \cdot 30 \cdot 1+1^{2} & 48 \cdot 52 & =(50-2)(50+2) \\ & =900-60+1=841 & & =50^{2}-2^{2}=2496 \end{array} $$ $$ 49^{2} $$

Solve the equation. $$ x(3 x+1)=2 $$

Solve the equation and check your solution.xplain how to identify and factor a perfect square trinomial. $$ 2(x+1)=0 $$
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