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Chapter 6: Problem 14

For Problems \(11-46\), factor each polynomial completely. (Objective 2) $$ 24 x-40 x y $$

Short Answer

Expert verified
The polynomial factors to \(8x(3 - 5y)\).

Step by step solution

01

Identify the Greatest Common Factor (GCF)

Examine the given polynomial \(24x - 40xy\) and identify the greatest common factor in each term. Both terms have \(x\) as a common variable, and the numbers 24 and 40 share 8 as their greatest common factor. The GCF is therefore \(8x\).
02

Factor Out the GCF

Factor out \(8x\) from each term in the polynomial. Divide each term by \(8x\) to find the resulting polynomial: \(24x \div 8x = 3\) and \(-40xy \div 8x = -5y\). Therefore, when factored out, the expression becomes \(8x(3 - 5y)\).
03

Verify the Factored Form

Expand \(8x(3 - 5y)\) to confirm it returns to the original expression. Multiplying gives \(8x \times 3 = 24x\) and \(8x \times (-5y) = -40xy\). The expanded form \(24x - 40xy\) matches 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
When tackling problems in algebra, a crucial step is identifying the Greatest Common Factor (GCF). The GCF is the largest factor that divides all the terms in an expression without leaving a remainder. It's like finding the common ingredient in multiple recipes so you can cook efficiently.
In our polynomial example, \(24x - 40xy\), to find the GCF:
  • Look for common factors in the numerical coefficients, which are 24 and 40. The greatest common number that divides both is 8.
  • Examine the variable part. Both terms contain the variable \(x\), so \(x\) is a common factor in the variables.

Therefore, the GCF for this polynomial is \(8x\). Factoring out the GCF simplifies the problem and sets the stage for further operations.
Polynomial Expression
A polynomial expression is like a mathematical phrase composed of variables (like \(x\) or \(y\)) and coefficients (like 24 or -40). Polynomials can have one or more terms, and in algebra, we often need to manipulate them by factoring or expanding.
Consider our polynomial expression \(24x - 40xy\):
  • Each term is a product of a number and a variable part. In this case, \(24x\) and \(-40xy\) are the terms.
  • Operations such as addition or subtraction separate these terms.

Understanding how to manage and restructure these expressions is key to solving them, especially when factoring or simplifying.
Algebraic Techniques
Algebraic techniques like factoring can make seemingly complicated expressions much simpler. By using established methods, we rearrange and simplify polynomial expressions. This often involves identifying patterns or using known strategies like factoring by grouping or the distributive property.
In the problem \(24x - 40xy\), we applied a specific technique:
  • Identified the GCF of the entire polynomial.
  • Factored out the GCF from each term, turning the expression into \(8x(3 - 5y)\).
  • Verified the solution by re-expanding the factored expression to ensure it simplifies back to the original polynomial.

Using these algebraic techniques helps solve problems methodically and can aid in solving more complex algebraic challenges later.

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Most popular questions from this chapter

$$ \text { For Problems } 41-70 \text {, solve each equation. (Objective } 3 \text { ) } $$ $$ -2 x^{3}+8 x=0 $$

$$ \text { For Problems } 41-70 \text {, solve each equation. (Objective } 3 \text { ) } $$ $$ 24 x^{2}+74 x-35=0 $$

For Problems \(71-88\), set up an equation and solve each problem. (Objective 4) The sum of the areas of two circles is \(180 \pi\) square inches. The length of a radius of the smaller circle is 6 inches less than the length of a radius of the larger circle. Find the length of a radius of each circle.

$$ \text { For Problems } 41-70 \text {, solve each equation. (Objective } 3 \text { ) } $$ $$ x^{3}+10 x^{2}+25 x=0 $$

Find two consecutive odd whole numbers whose product is 63 .
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