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Chapter 5: Problem 11

Factor each trinomial. See Examples 1 through 4. $$ 4 x^{2} z+28 x z+40 z $$

Short Answer

Expert verified
The factored form is \(4z(x + 5)(x + 2)\).

Step by step solution

01

Identify a common factor

First, identify the greatest common factor (GCF) of the terms in the trinomial. Here, each term contains a factor of 4 and z. So, the GCF is 4z.
02

Factor out the common factor

Factor out the GCF from the trinomial:\[4xz(x) + 7xz + 10z = 4z(x^2 + 7x + 10)\]
03

Factor the remaining trinomial

Now, focus on factoring \(x^2 + 7x + 10\). Look for two numbers that multiply to 10 and add to 7. These numbers are 5 and 2.
04

Write the complete factored expression

Write the trinomial as a product using the numbers found in Step 3. This gives:\[x^2 + 7x + 10 = (x + 5)(x + 2)\]Therefore, the fully factored form of the original expression is:\[4z(x + 5)(x + 2)\]

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Key Concepts

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

Greatest Common Factor
When you're factoring trinomials, one of the first things you should do is identify the greatest common factor (GCF). This is a process where you look for the largest number and variable that can divide all terms in a polynomial expression without leaving any remainder. In the trinomial given as an example, \(4x^2z + 28xz + 40z\), we observe that the GCF is \(4z\).

You can find the GCF by:
  • Listing all factors of the coefficients and variables.
  • Selecting the largest factor common to each term.
After identifying the GCF, you factor it out of the polynomial, simplifying the expression and making other factoring methods easier to apply.
Factoring by Grouping
Factoring by grouping involves rearranging the terms in a polynomial expression to form groups. This is particularly useful once a common factor is already factored out. In our step-by-step solution, after removing the GCF \(4z\), we are left with \(x^2 + 7x + 10\).

The goal here is to split the trinomial into simpler parts by finding two numbers that not only multiply to give the constant term but also add up to the middle term. Here, we found 5 and 2 to fit this purpose.

Finally, group these into binomials:
  • Combine terms that have a common factor.
  • Factor separately, ensuring no terms are left out.
  • Finish by writing the expression as a product of binomials.
This method is a reliable strategy to deal with trinomials during factoring.
Polynomial Expressions
Polynomial expressions come in various forms and degrees, such as monomials, binomials, and trinomials. Understanding these is key to mastering factoring techniques. A polynomial in one variable looks like a sum of terms each having a certain degree. For instance, in our exercise, \(4x^2z + 28xz + 40z\) is a trinomial since it consists of three terms.

Key aspects of understanding polynomial expressions include:
  • Recognizing the degree of each term based on variables.
  • Identifying coefficients attached to those variables.
  • Understanding how operations like addition, multiplication, and factoring affect polynomials.
Grasping the structure of polynomials allows for more effective manipulation and simplification, crucial for various algebraic operations, including factoring.

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Most popular questions from this chapter

Factor each trinomial. See Examples 5 through 10. $$ 28 y^{2}+22 y+4 $$

Factor each polynomial completely. See Examples 1 through 12. $$ x^{2}+4 x-5 $$

Factor. Assume that variables used as exponents represent positive integers. $$ 4 x^{2 n}-12 x^{n}+9 $$

Factor each polynomial completely. See Examples 1 through 12. $$ x^{4}-5 x^{2}-6 $$

Factor. Assume that variables used as exponents represent positive integers. $$ 9 x^{2 n}+24 x^{n}+16 $$
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