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Chapter 6: Problem 53

In the following exercises, factor. $$ 3 x^{3}-7 x^{2}+6 x-14 $$

Short Answer

Expert verified
(3x - 7)(x^2 + 2)

Step by step solution

01

Group Terms

Group the terms in pairs. Group the first two terms and the last two terms: 3x^3 - 7x^2 + 6x - 14 = (3x^3 - 7x^2) + (6x - 14)
02

Factor Out Common Factors

Factor out the greatest common factor (GCF) from each pair of grouped terms. For (3x^3 - 7x^2), the GCF is x^2: x^2(3x - 7) For (6x - 14), the GCF is 2: 2(3x - 7) So, the expression becomes: x^2(3x - 7) + 2(3x - 7)
03

Factor by Grouping

Notice that both terms have a common factor of (3x - 7). Factor this out: (3x - 7)(x^2 + 2)

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Key Concepts

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

greatest common factor
When factoring polynomials, the first step is often to find the Greatest Common Factor (GCF). The GCF is the largest factor that divides all of the terms in the polynomial. This process makes the polynomial simpler and prepares it for further factoring. To find the GCF:
  • List the factors of each term.
  • Identify the largest common factor from these lists.

For example, in the polynomial 3x^3 - 7x^2 + 6x - 14, we can group the terms in pairs: (3x^3 - 7x^2) and (6x - 14).

For the first group (3x^3 - 7x^2), the common factor is x^2. For the second group (6x - 14), the common factor is 2. So, factoring out these common factors, we get:

\(x^2(3x - 7) + 2(3x - 7)\)
factoring by grouping
Factoring by grouping is a useful method when dealing with polynomials that have four or more terms. Here, you group terms that have common factors, factor out those common factors, and then look for a common binomial factor.

Let's apply this to our example:
  • Group the terms in pairs: (3x^3 - 7x^2) and (6x - 14).
  • Factor out the GCF in each group: \(x^2(3x - 7) + 2(3x - 7)\).

Notice that now both grouped terms contain the common binomial factor \((3x - 7)\). So, we can factor \((3x - 7)\) out to receive:
(3x - 7)(x^2 + 2)
polynomial expressions
Polynomials are expressions that consist of variables, coefficients, and exponents summed together. Factoring polynomial expressions involves breaking them down into simpler, multiplied forms.
  • Polynomials can be classified based on their number of terms: monomials (one term), binomials (two terms), and trinomials (three terms).
  • In our exercise, the polynomial 3x^3 - 7x^2 + 6x - 14 is a four-term polynomial.

When factoring polynomials, the goal is to simplify the expression to make solving equations easier. For instance, after factoring \(3x^3 - 7x^2 + 6x - 14\) as \((3x - 7)(x^2 + 2)\), solving for the x values becomes straightforward.

In summary, factoring polynomials by identifying the GCF, using the grouping method, and understanding polynomial expressions are essential skills that simplify complex algebraic problems.

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Most popular questions from this chapter

In the following exercises, factor each expression using any method. $$ 13 z^{2}+39 z-26 $$

In the following exercises, factor completely using trial and error. $$ 21 m^{2}-29 m n+10 n^{2} $$

In the following exercises, factor completely using trial and error. $$ 6 y^{4}+12 y^{3}-48 y^{2} $$

Tommy factored \(x^{2}-x-20\) as \((x+5)(x-4)\). Sara factored it as \((x+4)(x-5)\). Ernesto factored it as \((x-5)(x-4)\). Who is correct? Explain why the other two are wrong.

In the following exercises, factor each expression using any method. $$ q^{2}-29 q r-96 r^{2} $$
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